Result for log((exp(x)

Alternative 1: 97.7% accurate, 0.6× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{y} \leq 0.2:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right) \cdot x, x, x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(e^{y} - e^{x}\right) \cdot -0.5\right)\\ \end{array} \end{array} \]

(FPCore (x y)
 :precision binary64
 (if (<= (- (exp x) (exp y)) 0.2)
   (log (* (fma (* (fma 0.16666666666666666 x 0.5) x) x (- x (expm1 y))) 0.5))
   (log (* (- (exp y) (exp x)) -0.5))))

double code(double x, double y) {
	double tmp;
	if ((exp(x) - exp(y)) <= 0.2) {
		tmp = log((fma((fma(0.16666666666666666, x, 0.5) * x), x, (x - expm1(y))) * 0.5));
	} else {
		tmp = log(((exp(y) - exp(x)) * -0.5));
	}
	return tmp;
}

function code(x, y)
	tmp = 0.0
	if (Float64(exp(x) - exp(y)) <= 0.2)
		tmp = log(Float64(fma(Float64(fma(0.16666666666666666, x, 0.5) * x), x, Float64(x - expm1(y))) * 0.5));
	else
		tmp = log(Float64(Float64(exp(y) - exp(x)) * -0.5));
	end
	return tmp
end

code[x_, y_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[y], $MachinePrecision]), $MachinePrecision], 0.2], N[Log[N[(N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] * x + N[(x - N[(Exp[y] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], N[Log[N[(N[(N[Exp[y], $MachinePrecision] - N[Exp[x], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;e^{x} - e^{y} \leq 0.2:\\
\;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right) \cdot x, x, x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\left(e^{y} - e^{x}\right) \cdot -0.5\right)\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if (-.f64 (exp.f64 x) (exp.f64 y)) < 0.20000000000000001
1. Initial program 7.0%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in x around 0
  \[\leadsto \log \left(\frac{\color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right) - e^{y}}}{2}\right) \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \log \left(\frac{\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1\right)} - e^{y}}{2}\right) \]
  2. associate--l+N/A
    \[\leadsto \log \left(\frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + \left(1 - e^{y}\right)}}{2}\right) \]
  3. +-commutativeN/A
    \[\leadsto \log \left(\frac{x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1\right)} + \left(1 - e^{y}\right)}{2}\right) \]
  4. distribute-lft-inN/A
    \[\leadsto \log \left(\frac{\color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + x \cdot 1\right)} + \left(1 - e^{y}\right)}{2}\right) \]
  5. *-rgt-identityN/A
    \[\leadsto \log \left(\frac{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + \color{blue}{x}\right) + \left(1 - e^{y}\right)}{2}\right) \]
  6. associate-+l+N/A
    \[\leadsto \log \left(\frac{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + \left(x + \left(1 - e^{y}\right)\right)}}{2}\right) \]
  7. associate-*r*N/A
    \[\leadsto \log \left(\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} + \left(x + \left(1 - e^{y}\right)\right)}{2}\right) \]
  8. lower-fma.f64N/A
    \[\leadsto \log \left(\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2} + \frac{1}{6} \cdot x, x + \left(1 - e^{y}\right)\right)}}{2}\right) \]
  9. lower-*.f64N/A
    \[\leadsto \log \left(\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{6} \cdot x, x + \left(1 - e^{y}\right)\right)}{2}\right) \]
  10. +-commutativeN/A
    \[\leadsto \log \left(\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, x + \left(1 - e^{y}\right)\right)}{2}\right) \]
  11. lower-fma.f64N/A
    \[\leadsto \log \left(\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right)}, x + \left(1 - e^{y}\right)\right)}{2}\right) \]
  12. sub-negN/A
    \[\leadsto \log \left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x + \color{blue}{\left(1 + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right)}{2}\right) \]
  13. +-commutativeN/A
    \[\leadsto \log \left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x + \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + 1\right)}\right)}{2}\right) \]
  14. associate-+r+N/A
    \[\leadsto \log \left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), \color{blue}{\left(x + \left(\mathsf{neg}\left(e^{y}\right)\right)\right) + 1}\right)}{2}\right) \]
  15. sub-negN/A
    \[\leadsto \log \left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), \color{blue}{\left(x - e^{y}\right)} + 1\right)}{2}\right) \]
  16. associate-+l-N/A
    \[\leadsto \log \left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), \color{blue}{x - \left(e^{y} - 1\right)}\right)}{2}\right) \]
  17. lower--.f64N/A
    \[\leadsto \log \left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), \color{blue}{x - \left(e^{y} - 1\right)}\right)}{2}\right) \]
  18. lower-expm1.f6499.1
    \[\leadsto \log \left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x - \color{blue}{\mathsf{expm1}\left(y\right)}\right)}{2}\right) \]
5. Applied rewrites99.1%
  \[\leadsto \log \left(\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x - \mathsf{expm1}\left(y\right)\right)}}{2}\right) \]
6. Step-by-step derivation
  1. lift-/.f64N/A
    \[\leadsto \log \color{blue}{\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x - \mathsf{expm1}\left(y\right)\right)}{2}\right)} \]
  2. div-invN/A
    \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x - \mathsf{expm1}\left(y\right)\right) \cdot \frac{1}{2}\right)} \]
  3. metadata-evalN/A
    \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x - \mathsf{expm1}\left(y\right)\right) \cdot \color{blue}{\frac{1}{2}}\right) \]
  4. lower-*.f6499.1
    \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right)} \]
7. Applied rewrites99.1%
  \[\leadsto \color{blue}{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right) \cdot x, x, x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right)} \]
if 0.20000000000000001 < (-.f64 (exp.f64 x) (exp.f64 y))
1. Initial program 98.6%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Step-by-step derivation
  1. lift-/.f64N/A
    \[\leadsto \log \color{blue}{\left(\frac{e^{x} - e^{y}}{2}\right)} \]
  2. frac-2negN/A
    \[\leadsto \log \color{blue}{\left(\frac{\mathsf{neg}\left(\left(e^{x} - e^{y}\right)\right)}{\mathsf{neg}\left(2\right)}\right)} \]
  3. div-invN/A
    \[\leadsto \log \color{blue}{\left(\left(\mathsf{neg}\left(\left(e^{x} - e^{y}\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)} \]
  4. lower-*.f64N/A
    \[\leadsto \log \color{blue}{\left(\left(\mathsf{neg}\left(\left(e^{x} - e^{y}\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)} \]
  5. neg-sub0N/A
    \[\leadsto \log \left(\color{blue}{\left(0 - \left(e^{x} - e^{y}\right)\right)} \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right) \]
  6. metadata-evalN/A
    \[\leadsto \log \left(\left(\color{blue}{\log 1} - \left(e^{x} - e^{y}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right) \]
  7. lift--.f64N/A
    \[\leadsto \log \left(\left(\log 1 - \color{blue}{\left(e^{x} - e^{y}\right)}\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right) \]
  8. sub-negN/A
    \[\leadsto \log \left(\left(\log 1 - \color{blue}{\left(e^{x} + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right) \]
  9. +-commutativeN/A
    \[\leadsto \log \left(\left(\log 1 - \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + e^{x}\right)}\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right) \]
  10. associate--r+N/A
    \[\leadsto \log \left(\color{blue}{\left(\left(\log 1 - \left(\mathsf{neg}\left(e^{y}\right)\right)\right) - e^{x}\right)} \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right) \]
  11. metadata-evalN/A
    \[\leadsto \log \left(\left(\left(\color{blue}{0} - \left(\mathsf{neg}\left(e^{y}\right)\right)\right) - e^{x}\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right) \]
  12. neg-sub0N/A
    \[\leadsto \log \left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(e^{y}\right)\right)\right)\right)} - e^{x}\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right) \]
  13. remove-double-negN/A
    \[\leadsto \log \left(\left(\color{blue}{e^{y}} - e^{x}\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right) \]
  14. lower--.f64N/A
    \[\leadsto \log \left(\color{blue}{\left(e^{y} - e^{x}\right)} \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right) \]
  15. metadata-evalN/A
    \[\leadsto \log \left(\left(e^{y} - e^{x}\right) \cdot \frac{1}{\color{blue}{-2}}\right) \]
  16. metadata-eval98.6
    \[\leadsto \log \left(\left(e^{y} - e^{x}\right) \cdot \color{blue}{-0.5}\right) \]
4. Applied rewrites98.6%
  \[\leadsto \log \color{blue}{\left(\left(e^{y} - e^{x}\right) \cdot -0.5\right)} \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 2: 50.8% accurate, 0.8× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{y} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5\right), y, -1\right), y, x\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{expm1}\left(x\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

(FPCore (x y)
 :precision binary64
 (if (<= (- (exp x) (exp y)) 2e-12)
   (log (* (fma (fma (fma -0.16666666666666666 y -0.5) y -1.0) y x) 0.5))
   (log (* (expm1 x) 0.5))))

double code(double x, double y) {
	double tmp;
	if ((exp(x) - exp(y)) <= 2e-12) {
		tmp = log((fma(fma(fma(-0.16666666666666666, y, -0.5), y, -1.0), y, x) * 0.5));
	} else {
		tmp = log((expm1(x) * 0.5));
	}
	return tmp;
}

function code(x, y)
	tmp = 0.0
	if (Float64(exp(x) - exp(y)) <= 2e-12)
		tmp = log(Float64(fma(fma(fma(-0.16666666666666666, y, -0.5), y, -1.0), y, x) * 0.5));
	else
		tmp = log(Float64(expm1(x) * 0.5));
	end
	return tmp
end

code[x_, y_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[y], $MachinePrecision]), $MachinePrecision], 2e-12], N[Log[N[(N[(N[(N[(-0.16666666666666666 * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y + x), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], N[Log[N[(N[(Exp[x] - 1), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;e^{x} - e^{y} \leq 2 \cdot 10^{-12}:\\
\;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5\right), y, -1\right), y, x\right) \cdot 0.5\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\mathsf{expm1}\left(x\right) \cdot 0.5\right)\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if (-.f64 (exp.f64 x) (exp.f64 y)) < 1.99999999999999996e-12
1. Initial program 4.5%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in x around 0
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
4. Step-by-step derivation
  1. distribute-lft-outN/A
    \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(x + \left(1 - e^{y}\right)\right)\right)} \]
  2. *-commutativeN/A
    \[\leadsto \log \color{blue}{\left(\left(x + \left(1 - e^{y}\right)\right) \cdot \frac{1}{2}\right)} \]
  3. lower-*.f64N/A
    \[\leadsto \log \color{blue}{\left(\left(x + \left(1 - e^{y}\right)\right) \cdot \frac{1}{2}\right)} \]
  4. sub-negN/A
    \[\leadsto \log \left(\left(x + \color{blue}{\left(1 + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right) \cdot \frac{1}{2}\right) \]
  5. +-commutativeN/A
    \[\leadsto \log \left(\left(x + \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + 1\right)}\right) \cdot \frac{1}{2}\right) \]
  6. associate-+r+N/A
    \[\leadsto \log \left(\color{blue}{\left(\left(x + \left(\mathsf{neg}\left(e^{y}\right)\right)\right) + 1\right)} \cdot \frac{1}{2}\right) \]
  7. sub-negN/A
    \[\leadsto \log \left(\left(\color{blue}{\left(x - e^{y}\right)} + 1\right) \cdot \frac{1}{2}\right) \]
  8. associate-+l-N/A
    \[\leadsto \log \left(\color{blue}{\left(x - \left(e^{y} - 1\right)\right)} \cdot \frac{1}{2}\right) \]
  9. lower--.f64N/A
    \[\leadsto \log \left(\color{blue}{\left(x - \left(e^{y} - 1\right)\right)} \cdot \frac{1}{2}\right) \]
  10. lower-expm1.f6499.1
    \[\leadsto \log \left(\left(x - \color{blue}{\mathsf{expm1}\left(y\right)}\right) \cdot 0.5\right) \]
5. Applied rewrites99.1%
  \[\leadsto \log \color{blue}{\left(\left(x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right)} \]
6. Taylor expanded in y around 0
  \[\leadsto \log \left(\left(x + y \cdot \left(y \cdot \left(\frac{-1}{6} \cdot y - \frac{1}{2}\right) - 1\right)\right) \cdot \frac{1}{2}\right) \]
7. Step-by-step derivation
8. Applied rewrites99.1%
  \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5\right), y, -1\right), y, x\right) \cdot 0.5\right) \]
if 1.99999999999999996e-12 < (-.f64 (exp.f64 x) (exp.f64 y))
1. Initial program 98.0%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in y around 0
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(e^{x} - 1\right)\right)} \]
4. Step-by-step derivation
  1. *-commutativeN/A
    \[\leadsto \log \color{blue}{\left(\left(e^{x} - 1\right) \cdot \frac{1}{2}\right)} \]
  2. lower-*.f64N/A
    \[\leadsto \log \color{blue}{\left(\left(e^{x} - 1\right) \cdot \frac{1}{2}\right)} \]
  3. lower-expm1.f6411.4
    \[\leadsto \log \left(\color{blue}{\mathsf{expm1}\left(x\right)} \cdot 0.5\right) \]
5. Applied rewrites11.4%
  \[\leadsto \log \color{blue}{\left(\mathsf{expm1}\left(x\right) \cdot 0.5\right)} \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 3: 52.1% accurate, 0.9× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{y} \leq 0.5:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5\right), y, -1\right), y, x\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right) \cdot x\right) \cdot x\right)\\ \end{array} \end{array} \]

(FPCore (x y)
 :precision binary64
 (if (<= (- (exp x) (exp y)) 0.5)
   (log (* (fma (fma (fma -0.16666666666666666 y -0.5) y -1.0) y x) 0.5))
   (log (* (* (fma 0.08333333333333333 x 0.25) x) x))))

double code(double x, double y) {
	double tmp;
	if ((exp(x) - exp(y)) <= 0.5) {
		tmp = log((fma(fma(fma(-0.16666666666666666, y, -0.5), y, -1.0), y, x) * 0.5));
	} else {
		tmp = log(((fma(0.08333333333333333, x, 0.25) * x) * x));
	}
	return tmp;
}

function code(x, y)
	tmp = 0.0
	if (Float64(exp(x) - exp(y)) <= 0.5)
		tmp = log(Float64(fma(fma(fma(-0.16666666666666666, y, -0.5), y, -1.0), y, x) * 0.5));
	else
		tmp = log(Float64(Float64(fma(0.08333333333333333, x, 0.25) * x) * x));
	end
	return tmp
end

code[x_, y_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[y], $MachinePrecision]), $MachinePrecision], 0.5], N[Log[N[(N[(N[(N[(-0.16666666666666666 * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y + x), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], N[Log[N[(N[(N[(0.08333333333333333 * x + 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;e^{x} - e^{y} \leq 0.5:\\
\;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5\right), y, -1\right), y, x\right) \cdot 0.5\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right) \cdot x\right) \cdot x\right)\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if (-.f64 (exp.f64 x) (exp.f64 y)) < 0.5
1. Initial program 7.8%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in x around 0
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
4. Step-by-step derivation
  1. distribute-lft-outN/A
    \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(x + \left(1 - e^{y}\right)\right)\right)} \]
  2. *-commutativeN/A
    \[\leadsto \log \color{blue}{\left(\left(x + \left(1 - e^{y}\right)\right) \cdot \frac{1}{2}\right)} \]
  3. lower-*.f64N/A
    \[\leadsto \log \color{blue}{\left(\left(x + \left(1 - e^{y}\right)\right) \cdot \frac{1}{2}\right)} \]
  4. sub-negN/A
    \[\leadsto \log \left(\left(x + \color{blue}{\left(1 + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right) \cdot \frac{1}{2}\right) \]
  5. +-commutativeN/A
    \[\leadsto \log \left(\left(x + \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + 1\right)}\right) \cdot \frac{1}{2}\right) \]
  6. associate-+r+N/A
    \[\leadsto \log \left(\color{blue}{\left(\left(x + \left(\mathsf{neg}\left(e^{y}\right)\right)\right) + 1\right)} \cdot \frac{1}{2}\right) \]
  7. sub-negN/A
    \[\leadsto \log \left(\left(\color{blue}{\left(x - e^{y}\right)} + 1\right) \cdot \frac{1}{2}\right) \]
  8. associate-+l-N/A
    \[\leadsto \log \left(\color{blue}{\left(x - \left(e^{y} - 1\right)\right)} \cdot \frac{1}{2}\right) \]
  9. lower--.f64N/A
    \[\leadsto \log \left(\color{blue}{\left(x - \left(e^{y} - 1\right)\right)} \cdot \frac{1}{2}\right) \]
  10. lower-expm1.f6498.2
    \[\leadsto \log \left(\left(x - \color{blue}{\mathsf{expm1}\left(y\right)}\right) \cdot 0.5\right) \]
5. Applied rewrites98.2%
  \[\leadsto \log \color{blue}{\left(\left(x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right)} \]
6. Taylor expanded in y around 0
  \[\leadsto \log \left(\left(x + y \cdot \left(y \cdot \left(\frac{-1}{6} \cdot y - \frac{1}{2}\right) - 1\right)\right) \cdot \frac{1}{2}\right) \]
7. Step-by-step derivation
8. Applied rewrites97.2%
  \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5\right), y, -1\right), y, x\right) \cdot 0.5\right) \]
if 0.5 < (-.f64 (exp.f64 x) (exp.f64 y))
1. Initial program 98.5%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in x around 0
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(1 - e^{y}\right) + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \log \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right)\right) + \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
  2. *-commutativeN/A
    \[\leadsto \log \left(\color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right)\right) \cdot x} + \frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
  3. lower-fma.f64N/A
    \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right), x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right)} \]
  4. +-commutativeN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right) + \frac{1}{2}}, x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  5. *-commutativeN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{4} + \frac{1}{12} \cdot x\right) \cdot x} + \frac{1}{2}, x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  6. lower-fma.f64N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{4} + \frac{1}{12} \cdot x, x, \frac{1}{2}\right)}, x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  7. +-commutativeN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{12} \cdot x + \frac{1}{4}}, x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  8. lower-fma.f64N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right)}, x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  9. sub-negN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right)\right) \]
  10. +-commutativeN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + 1\right)}\right)\right) \]
  11. distribute-lft-inN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \color{blue}{\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{y}\right)\right) + \frac{1}{2} \cdot 1}\right)\right) \]
  12. neg-mul-1N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \color{blue}{\left(-1 \cdot e^{y}\right)} + \frac{1}{2} \cdot 1\right)\right) \]
  13. associate-*r*N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \color{blue}{\left(\frac{1}{2} \cdot -1\right) \cdot e^{y}} + \frac{1}{2} \cdot 1\right)\right) \]
  14. metadata-evalN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \color{blue}{\frac{-1}{2}} \cdot e^{y} + \frac{1}{2} \cdot 1\right)\right) \]
  15. metadata-evalN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \frac{-1}{2} \cdot e^{y} + \color{blue}{\frac{1}{2}}\right)\right) \]
  16. lower-fma.f64N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, e^{y}, \frac{1}{2}\right)}\right)\right) \]
  17. lower-exp.f6495.3
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right), x, 0.5\right), x, \mathsf{fma}\left(-0.5, \color{blue}{e^{y}}, 0.5\right)\right)\right) \]
5. Applied rewrites95.3%
  \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right), x, 0.5\right), x, \mathsf{fma}\left(-0.5, e^{y}, 0.5\right)\right)\right)} \]
6. Taylor expanded in x around inf
  \[\leadsto \log \left({x}^{3} \cdot \color{blue}{\left(\frac{1}{12} + \frac{1}{4} \cdot \frac{1}{x}\right)}\right) \]
7. Step-by-step derivation
8. Applied rewrites8.6%
  \[\leadsto \log \left(\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right) \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 4: 51.5% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{y} \leq 0.5:\\ \;\;\;\;\log \left(\left(x - y\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right) \cdot x\right) \cdot x\right)\\ \end{array} \end{array} \]

(FPCore (x y)
 :precision binary64
 (if (<= (- (exp x) (exp y)) 0.5)
   (log (* (- x y) 0.5))
   (log (* (* (fma 0.08333333333333333 x 0.25) x) x))))

double code(double x, double y) {
	double tmp;
	if ((exp(x) - exp(y)) <= 0.5) {
		tmp = log(((x - y) * 0.5));
	} else {
		tmp = log(((fma(0.08333333333333333, x, 0.25) * x) * x));
	}
	return tmp;
}

function code(x, y)
	tmp = 0.0
	if (Float64(exp(x) - exp(y)) <= 0.5)
		tmp = log(Float64(Float64(x - y) * 0.5));
	else
		tmp = log(Float64(Float64(fma(0.08333333333333333, x, 0.25) * x) * x));
	end
	return tmp
end

code[x_, y_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[y], $MachinePrecision]), $MachinePrecision], 0.5], N[Log[N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], N[Log[N[(N[(N[(0.08333333333333333 * x + 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;e^{x} - e^{y} \leq 0.5:\\
\;\;\;\;\log \left(\left(x - y\right) \cdot 0.5\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right) \cdot x\right) \cdot x\right)\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if (-.f64 (exp.f64 x) (exp.f64 y)) < 0.5
1. Initial program 7.8%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in x around 0
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
4. Step-by-step derivation
  1. distribute-lft-outN/A
    \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(x + \left(1 - e^{y}\right)\right)\right)} \]
  2. *-commutativeN/A
    \[\leadsto \log \color{blue}{\left(\left(x + \left(1 - e^{y}\right)\right) \cdot \frac{1}{2}\right)} \]
  3. lower-*.f64N/A
    \[\leadsto \log \color{blue}{\left(\left(x + \left(1 - e^{y}\right)\right) \cdot \frac{1}{2}\right)} \]
  4. sub-negN/A
    \[\leadsto \log \left(\left(x + \color{blue}{\left(1 + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right) \cdot \frac{1}{2}\right) \]
  5. +-commutativeN/A
    \[\leadsto \log \left(\left(x + \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + 1\right)}\right) \cdot \frac{1}{2}\right) \]
  6. associate-+r+N/A
    \[\leadsto \log \left(\color{blue}{\left(\left(x + \left(\mathsf{neg}\left(e^{y}\right)\right)\right) + 1\right)} \cdot \frac{1}{2}\right) \]
  7. sub-negN/A
    \[\leadsto \log \left(\left(\color{blue}{\left(x - e^{y}\right)} + 1\right) \cdot \frac{1}{2}\right) \]
  8. associate-+l-N/A
    \[\leadsto \log \left(\color{blue}{\left(x - \left(e^{y} - 1\right)\right)} \cdot \frac{1}{2}\right) \]
  9. lower--.f64N/A
    \[\leadsto \log \left(\color{blue}{\left(x - \left(e^{y} - 1\right)\right)} \cdot \frac{1}{2}\right) \]
  10. lower-expm1.f6498.2
    \[\leadsto \log \left(\left(x - \color{blue}{\mathsf{expm1}\left(y\right)}\right) \cdot 0.5\right) \]
5. Applied rewrites98.2%
  \[\leadsto \log \color{blue}{\left(\left(x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right)} \]
6. Taylor expanded in y around 0
  \[\leadsto \log \left(\left(x + -1 \cdot y\right) \cdot \frac{1}{2}\right) \]
7. Step-by-step derivation
8. Applied rewrites96.9%
  \[\leadsto \log \left(\left(x - y\right) \cdot 0.5\right) \]
if 0.5 < (-.f64 (exp.f64 x) (exp.f64 y))
1. Initial program 98.5%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in x around 0
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(1 - e^{y}\right) + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \log \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right)\right) + \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
  2. *-commutativeN/A
    \[\leadsto \log \left(\color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right)\right) \cdot x} + \frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
  3. lower-fma.f64N/A
    \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right), x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right)} \]
  4. +-commutativeN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right) + \frac{1}{2}}, x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  5. *-commutativeN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{4} + \frac{1}{12} \cdot x\right) \cdot x} + \frac{1}{2}, x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  6. lower-fma.f64N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{4} + \frac{1}{12} \cdot x, x, \frac{1}{2}\right)}, x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  7. +-commutativeN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{12} \cdot x + \frac{1}{4}}, x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  8. lower-fma.f64N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right)}, x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  9. sub-negN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right)\right) \]
  10. +-commutativeN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + 1\right)}\right)\right) \]
  11. distribute-lft-inN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \color{blue}{\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{y}\right)\right) + \frac{1}{2} \cdot 1}\right)\right) \]
  12. neg-mul-1N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \color{blue}{\left(-1 \cdot e^{y}\right)} + \frac{1}{2} \cdot 1\right)\right) \]
  13. associate-*r*N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \color{blue}{\left(\frac{1}{2} \cdot -1\right) \cdot e^{y}} + \frac{1}{2} \cdot 1\right)\right) \]
  14. metadata-evalN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \color{blue}{\frac{-1}{2}} \cdot e^{y} + \frac{1}{2} \cdot 1\right)\right) \]
  15. metadata-evalN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \frac{-1}{2} \cdot e^{y} + \color{blue}{\frac{1}{2}}\right)\right) \]
  16. lower-fma.f64N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, e^{y}, \frac{1}{2}\right)}\right)\right) \]
  17. lower-exp.f6495.3
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right), x, 0.5\right), x, \mathsf{fma}\left(-0.5, \color{blue}{e^{y}}, 0.5\right)\right)\right) \]
5. Applied rewrites95.3%
  \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right), x, 0.5\right), x, \mathsf{fma}\left(-0.5, e^{y}, 0.5\right)\right)\right)} \]
6. Taylor expanded in x around inf
  \[\leadsto \log \left({x}^{3} \cdot \color{blue}{\left(\frac{1}{12} + \frac{1}{4} \cdot \frac{1}{x}\right)}\right) \]
7. Step-by-step derivation
8. Applied rewrites8.6%
  \[\leadsto \log \left(\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right) \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 5: 50.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{y} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\log \left(\left(x - y\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(0.25, x, 0.5\right) \cdot x\right)\\ \end{array} \end{array} \]

(FPCore (x y)
 :precision binary64
 (if (<= (- (exp x) (exp y)) 2e-12)
   (log (* (- x y) 0.5))
   (log (* (fma 0.25 x 0.5) x))))

double code(double x, double y) {
	double tmp;
	if ((exp(x) - exp(y)) <= 2e-12) {
		tmp = log(((x - y) * 0.5));
	} else {
		tmp = log((fma(0.25, x, 0.5) * x));
	}
	return tmp;
}

function code(x, y)
	tmp = 0.0
	if (Float64(exp(x) - exp(y)) <= 2e-12)
		tmp = log(Float64(Float64(x - y) * 0.5));
	else
		tmp = log(Float64(fma(0.25, x, 0.5) * x));
	end
	return tmp
end

code[x_, y_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[y], $MachinePrecision]), $MachinePrecision], 2e-12], N[Log[N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], N[Log[N[(N[(0.25 * x + 0.5), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;e^{x} - e^{y} \leq 2 \cdot 10^{-12}:\\
\;\;\;\;\log \left(\left(x - y\right) \cdot 0.5\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\mathsf{fma}\left(0.25, x, 0.5\right) \cdot x\right)\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if (-.f64 (exp.f64 x) (exp.f64 y)) < 1.99999999999999996e-12
1. Initial program 4.5%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in x around 0
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
4. Step-by-step derivation
  1. distribute-lft-outN/A
    \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(x + \left(1 - e^{y}\right)\right)\right)} \]
  2. *-commutativeN/A
    \[\leadsto \log \color{blue}{\left(\left(x + \left(1 - e^{y}\right)\right) \cdot \frac{1}{2}\right)} \]
  3. lower-*.f64N/A
    \[\leadsto \log \color{blue}{\left(\left(x + \left(1 - e^{y}\right)\right) \cdot \frac{1}{2}\right)} \]
  4. sub-negN/A
    \[\leadsto \log \left(\left(x + \color{blue}{\left(1 + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right) \cdot \frac{1}{2}\right) \]
  5. +-commutativeN/A
    \[\leadsto \log \left(\left(x + \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + 1\right)}\right) \cdot \frac{1}{2}\right) \]
  6. associate-+r+N/A
    \[\leadsto \log \left(\color{blue}{\left(\left(x + \left(\mathsf{neg}\left(e^{y}\right)\right)\right) + 1\right)} \cdot \frac{1}{2}\right) \]
  7. sub-negN/A
    \[\leadsto \log \left(\left(\color{blue}{\left(x - e^{y}\right)} + 1\right) \cdot \frac{1}{2}\right) \]
  8. associate-+l-N/A
    \[\leadsto \log \left(\color{blue}{\left(x - \left(e^{y} - 1\right)\right)} \cdot \frac{1}{2}\right) \]
  9. lower--.f64N/A
    \[\leadsto \log \left(\color{blue}{\left(x - \left(e^{y} - 1\right)\right)} \cdot \frac{1}{2}\right) \]
  10. lower-expm1.f6499.1
    \[\leadsto \log \left(\left(x - \color{blue}{\mathsf{expm1}\left(y\right)}\right) \cdot 0.5\right) \]
5. Applied rewrites99.1%
  \[\leadsto \log \color{blue}{\left(\left(x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right)} \]
6. Taylor expanded in y around 0
  \[\leadsto \log \left(\left(x + -1 \cdot y\right) \cdot \frac{1}{2}\right) \]
7. Step-by-step derivation
8. Applied rewrites99.0%
  \[\leadsto \log \left(\left(x - y\right) \cdot 0.5\right) \]
if 1.99999999999999996e-12 < (-.f64 (exp.f64 x) (exp.f64 y))
1. Initial program 98.0%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in y around 0
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(e^{x} - 1\right)\right)} \]
4. Step-by-step derivation
  1. *-commutativeN/A
    \[\leadsto \log \color{blue}{\left(\left(e^{x} - 1\right) \cdot \frac{1}{2}\right)} \]
  2. lower-*.f64N/A
    \[\leadsto \log \color{blue}{\left(\left(e^{x} - 1\right) \cdot \frac{1}{2}\right)} \]
  3. lower-expm1.f6411.4
    \[\leadsto \log \left(\color{blue}{\mathsf{expm1}\left(x\right)} \cdot 0.5\right) \]
5. Applied rewrites11.4%
  \[\leadsto \log \color{blue}{\left(\mathsf{expm1}\left(x\right) \cdot 0.5\right)} \]
6. Taylor expanded in x around 0
  \[\leadsto \log \left(x \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{4} \cdot x\right)}\right) \]
7. Step-by-step derivation
8. Applied rewrites9.0%
  \[\leadsto \log \left(\mathsf{fma}\left(0.25, x, 0.5\right) \cdot \color{blue}{x}\right) \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 6: 50.7% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{y} \leq 0.51:\\ \;\;\;\;\log \left(\left(x - y\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(0.5 \cdot x\right)\\ \end{array} \end{array} \]

(FPCore (x y)
 :precision binary64
 (if (<= (- (exp x) (exp y)) 0.51) (log (* (- x y) 0.5)) (log (* 0.5 x))))

double code(double x, double y) {
	double tmp;
	if ((exp(x) - exp(y)) <= 0.51) {
		tmp = log(((x - y) * 0.5));
	} else {
		tmp = log((0.5 * x));
	}
	return tmp;
}

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((exp(x) - exp(y)) <= 0.51d0) then
        tmp = log(((x - y) * 0.5d0))
    else
        tmp = log((0.5d0 * x))
    end if
    code = tmp
end function

public static double code(double x, double y) {
	double tmp;
	if ((Math.exp(x) - Math.exp(y)) <= 0.51) {
		tmp = Math.log(((x - y) * 0.5));
	} else {
		tmp = Math.log((0.5 * x));
	}
	return tmp;
}

def code(x, y):
	tmp = 0
	if (math.exp(x) - math.exp(y)) <= 0.51:
		tmp = math.log(((x - y) * 0.5))
	else:
		tmp = math.log((0.5 * x))
	return tmp

function code(x, y)
	tmp = 0.0
	if (Float64(exp(x) - exp(y)) <= 0.51)
		tmp = log(Float64(Float64(x - y) * 0.5));
	else
		tmp = log(Float64(0.5 * x));
	end
	return tmp
end

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((exp(x) - exp(y)) <= 0.51)
		tmp = log(((x - y) * 0.5));
	else
		tmp = log((0.5 * x));
	end
	tmp_2 = tmp;
end

code[x_, y_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[y], $MachinePrecision]), $MachinePrecision], 0.51], N[Log[N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], N[Log[N[(0.5 * x), $MachinePrecision]], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;e^{x} - e^{y} \leq 0.51:\\
\;\;\;\;\log \left(\left(x - y\right) \cdot 0.5\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(0.5 \cdot x\right)\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if (-.f64 (exp.f64 x) (exp.f64 y)) < 0.51000000000000001
1. Initial program 8.5%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in x around 0
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
4. Step-by-step derivation
  1. distribute-lft-outN/A
    \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(x + \left(1 - e^{y}\right)\right)\right)} \]
  2. *-commutativeN/A
    \[\leadsto \log \color{blue}{\left(\left(x + \left(1 - e^{y}\right)\right) \cdot \frac{1}{2}\right)} \]
  3. lower-*.f64N/A
    \[\leadsto \log \color{blue}{\left(\left(x + \left(1 - e^{y}\right)\right) \cdot \frac{1}{2}\right)} \]
  4. sub-negN/A
    \[\leadsto \log \left(\left(x + \color{blue}{\left(1 + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right) \cdot \frac{1}{2}\right) \]
  5. +-commutativeN/A
    \[\leadsto \log \left(\left(x + \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + 1\right)}\right) \cdot \frac{1}{2}\right) \]
  6. associate-+r+N/A
    \[\leadsto \log \left(\color{blue}{\left(\left(x + \left(\mathsf{neg}\left(e^{y}\right)\right)\right) + 1\right)} \cdot \frac{1}{2}\right) \]
  7. sub-negN/A
    \[\leadsto \log \left(\left(\color{blue}{\left(x - e^{y}\right)} + 1\right) \cdot \frac{1}{2}\right) \]
  8. associate-+l-N/A
    \[\leadsto \log \left(\color{blue}{\left(x - \left(e^{y} - 1\right)\right)} \cdot \frac{1}{2}\right) \]
  9. lower--.f64N/A
    \[\leadsto \log \left(\color{blue}{\left(x - \left(e^{y} - 1\right)\right)} \cdot \frac{1}{2}\right) \]
  10. lower-expm1.f6497.5
    \[\leadsto \log \left(\left(x - \color{blue}{\mathsf{expm1}\left(y\right)}\right) \cdot 0.5\right) \]
5. Applied rewrites97.5%
  \[\leadsto \log \color{blue}{\left(\left(x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right)} \]
6. Taylor expanded in y around 0
  \[\leadsto \log \left(\left(x + -1 \cdot y\right) \cdot \frac{1}{2}\right) \]
7. Step-by-step derivation
8. Applied rewrites96.1%
  \[\leadsto \log \left(\left(x - y\right) \cdot 0.5\right) \]
if 0.51000000000000001 < (-.f64 (exp.f64 x) (exp.f64 y))
1. Initial program 98.5%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in y around 0
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(e^{x} - 1\right)\right)} \]
4. Step-by-step derivation
  1. *-commutativeN/A
    \[\leadsto \log \color{blue}{\left(\left(e^{x} - 1\right) \cdot \frac{1}{2}\right)} \]
  2. lower-*.f64N/A
    \[\leadsto \log \color{blue}{\left(\left(e^{x} - 1\right) \cdot \frac{1}{2}\right)} \]
  3. lower-expm1.f649.5
    \[\leadsto \log \left(\color{blue}{\mathsf{expm1}\left(x\right)} \cdot 0.5\right) \]
5. Applied rewrites9.5%
  \[\leadsto \log \color{blue}{\left(\mathsf{expm1}\left(x\right) \cdot 0.5\right)} \]
6. Taylor expanded in x around 0
  \[\leadsto \log \left(\frac{1}{2} \cdot \color{blue}{x}\right) \]
7. Step-by-step derivation
8. Applied rewrites7.2%
  \[\leadsto \log \left(0.5 \cdot \color{blue}{x}\right) \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 7: 95.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y} \leq 0.9:\\ \;\;\;\;\log \left(\mathsf{fma}\left(-0.5, e^{y}, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right), y, x\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

(FPCore (x y)
 :precision binary64
 (if (<= (exp y) 0.9)
   (log (fma -0.5 (exp y) 0.5))
   (log (* (fma (fma -0.5 y -1.0) y x) 0.5))))

double code(double x, double y) {
	double tmp;
	if (exp(y) <= 0.9) {
		tmp = log(fma(-0.5, exp(y), 0.5));
	} else {
		tmp = log((fma(fma(-0.5, y, -1.0), y, x) * 0.5));
	}
	return tmp;
}

function code(x, y)
	tmp = 0.0
	if (exp(y) <= 0.9)
		tmp = log(fma(-0.5, exp(y), 0.5));
	else
		tmp = log(Float64(fma(fma(-0.5, y, -1.0), y, x) * 0.5));
	end
	return tmp
end

code[x_, y_] := If[LessEqual[N[Exp[y], $MachinePrecision], 0.9], N[Log[N[(-0.5 * N[Exp[y], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], N[Log[N[(N[(N[(-0.5 * y + -1.0), $MachinePrecision] * y + x), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;e^{y} \leq 0.9:\\
\;\;\;\;\log \left(\mathsf{fma}\left(-0.5, e^{y}, 0.5\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right), y, x\right) \cdot 0.5\right)\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if (exp.f64 y) < 0.900000000000000022
1. Initial program 97.8%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in x around 0
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
4. Step-by-step derivation
  1. sub-negN/A
    \[\leadsto \log \left(\frac{1}{2} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right) \]
  2. +-commutativeN/A
    \[\leadsto \log \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + 1\right)}\right) \]
  3. distribute-lft-inN/A
    \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{y}\right)\right) + \frac{1}{2} \cdot 1\right)} \]
  4. neg-mul-1N/A
    \[\leadsto \log \left(\frac{1}{2} \cdot \color{blue}{\left(-1 \cdot e^{y}\right)} + \frac{1}{2} \cdot 1\right) \]
  5. associate-*r*N/A
    \[\leadsto \log \left(\color{blue}{\left(\frac{1}{2} \cdot -1\right) \cdot e^{y}} + \frac{1}{2} \cdot 1\right) \]
  6. metadata-evalN/A
    \[\leadsto \log \left(\color{blue}{\frac{-1}{2}} \cdot e^{y} + \frac{1}{2} \cdot 1\right) \]
  7. metadata-evalN/A
    \[\leadsto \log \left(\frac{-1}{2} \cdot e^{y} + \color{blue}{\frac{1}{2}}\right) \]
  8. lower-fma.f64N/A
    \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{2}, e^{y}, \frac{1}{2}\right)\right)} \]
  9. lower-exp.f6494.5
    \[\leadsto \log \left(\mathsf{fma}\left(-0.5, \color{blue}{e^{y}}, 0.5\right)\right) \]
5. Applied rewrites94.5%
  \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(-0.5, e^{y}, 0.5\right)\right)} \]
if 0.900000000000000022 < (exp.f64 y)
1. Initial program 9.4%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in x around 0
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
4. Step-by-step derivation
  1. distribute-lft-outN/A
    \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(x + \left(1 - e^{y}\right)\right)\right)} \]
  2. *-commutativeN/A
    \[\leadsto \log \color{blue}{\left(\left(x + \left(1 - e^{y}\right)\right) \cdot \frac{1}{2}\right)} \]
  3. lower-*.f64N/A
    \[\leadsto \log \color{blue}{\left(\left(x + \left(1 - e^{y}\right)\right) \cdot \frac{1}{2}\right)} \]
  4. sub-negN/A
    \[\leadsto \log \left(\left(x + \color{blue}{\left(1 + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right) \cdot \frac{1}{2}\right) \]
  5. +-commutativeN/A
    \[\leadsto \log \left(\left(x + \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + 1\right)}\right) \cdot \frac{1}{2}\right) \]
  6. associate-+r+N/A
    \[\leadsto \log \left(\color{blue}{\left(\left(x + \left(\mathsf{neg}\left(e^{y}\right)\right)\right) + 1\right)} \cdot \frac{1}{2}\right) \]
  7. sub-negN/A
    \[\leadsto \log \left(\left(\color{blue}{\left(x - e^{y}\right)} + 1\right) \cdot \frac{1}{2}\right) \]
  8. associate-+l-N/A
    \[\leadsto \log \left(\color{blue}{\left(x - \left(e^{y} - 1\right)\right)} \cdot \frac{1}{2}\right) \]
  9. lower--.f64N/A
    \[\leadsto \log \left(\color{blue}{\left(x - \left(e^{y} - 1\right)\right)} \cdot \frac{1}{2}\right) \]
  10. lower-expm1.f6496.3
    \[\leadsto \log \left(\left(x - \color{blue}{\mathsf{expm1}\left(y\right)}\right) \cdot 0.5\right) \]
5. Applied rewrites96.3%
  \[\leadsto \log \color{blue}{\left(\left(x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right)} \]
6. Taylor expanded in y around 0
  \[\leadsto \log \left(\left(x + y \cdot \left(\frac{-1}{2} \cdot y - 1\right)\right) \cdot \frac{1}{2}\right) \]
7. Step-by-step derivation
8. Applied rewrites96.3%
  \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right), y, x\right) \cdot 0.5\right) \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 8: 52.2% accurate, 1.3× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y} \leq 0.1:\\ \;\;\;\;\log \left(\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right) \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x - \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y, 0.16666666666666666\right), y, 0.5\right), y, 1\right) \cdot y\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

(FPCore (x y)
 :precision binary64
 (if (<= (exp y) 0.1)
   (log (* (* (fma 0.08333333333333333 x 0.25) x) x))
   (log
    (*
     (-
      x
      (*
       (fma (fma (fma 0.041666666666666664 y 0.16666666666666666) y 0.5) y 1.0)
       y))
     0.5))))

double code(double x, double y) {
	double tmp;
	if (exp(y) <= 0.1) {
		tmp = log(((fma(0.08333333333333333, x, 0.25) * x) * x));
	} else {
		tmp = log(((x - (fma(fma(fma(0.041666666666666664, y, 0.16666666666666666), y, 0.5), y, 1.0) * y)) * 0.5));
	}
	return tmp;
}

function code(x, y)
	tmp = 0.0
	if (exp(y) <= 0.1)
		tmp = log(Float64(Float64(fma(0.08333333333333333, x, 0.25) * x) * x));
	else
		tmp = log(Float64(Float64(x - Float64(fma(fma(fma(0.041666666666666664, y, 0.16666666666666666), y, 0.5), y, 1.0) * y)) * 0.5));
	end
	return tmp
end

code[x_, y_] := If[LessEqual[N[Exp[y], $MachinePrecision], 0.1], N[Log[N[(N[(N[(0.08333333333333333 * x + 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], N[Log[N[(N[(x - N[(N[(N[(N[(0.041666666666666664 * y + 0.16666666666666666), $MachinePrecision] * y + 0.5), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;e^{y} \leq 0.1:\\
\;\;\;\;\log \left(\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right) \cdot x\right) \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\left(x - \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y, 0.16666666666666666\right), y, 0.5\right), y, 1\right) \cdot y\right) \cdot 0.5\right)\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if (exp.f64 y) < 0.10000000000000001
1. Initial program 97.8%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in x around 0
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(1 - e^{y}\right) + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \log \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right)\right) + \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
  2. *-commutativeN/A
    \[\leadsto \log \left(\color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right)\right) \cdot x} + \frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
  3. lower-fma.f64N/A
    \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right), x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right)} \]
  4. +-commutativeN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right) + \frac{1}{2}}, x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  5. *-commutativeN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{4} + \frac{1}{12} \cdot x\right) \cdot x} + \frac{1}{2}, x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  6. lower-fma.f64N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{4} + \frac{1}{12} \cdot x, x, \frac{1}{2}\right)}, x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  7. +-commutativeN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{12} \cdot x + \frac{1}{4}}, x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  8. lower-fma.f64N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right)}, x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  9. sub-negN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right)\right) \]
  10. +-commutativeN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + 1\right)}\right)\right) \]
  11. distribute-lft-inN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \color{blue}{\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{y}\right)\right) + \frac{1}{2} \cdot 1}\right)\right) \]
  12. neg-mul-1N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \color{blue}{\left(-1 \cdot e^{y}\right)} + \frac{1}{2} \cdot 1\right)\right) \]
  13. associate-*r*N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \color{blue}{\left(\frac{1}{2} \cdot -1\right) \cdot e^{y}} + \frac{1}{2} \cdot 1\right)\right) \]
  14. metadata-evalN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \color{blue}{\frac{-1}{2}} \cdot e^{y} + \frac{1}{2} \cdot 1\right)\right) \]
  15. metadata-evalN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \frac{-1}{2} \cdot e^{y} + \color{blue}{\frac{1}{2}}\right)\right) \]
  16. lower-fma.f64N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, e^{y}, \frac{1}{2}\right)}\right)\right) \]
  17. lower-exp.f6496.8
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right), x, 0.5\right), x, \mathsf{fma}\left(-0.5, \color{blue}{e^{y}}, 0.5\right)\right)\right) \]
5. Applied rewrites96.8%
  \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right), x, 0.5\right), x, \mathsf{fma}\left(-0.5, e^{y}, 0.5\right)\right)\right)} \]
6. Taylor expanded in x around inf
  \[\leadsto \log \left({x}^{3} \cdot \color{blue}{\left(\frac{1}{12} + \frac{1}{4} \cdot \frac{1}{x}\right)}\right) \]
7. Step-by-step derivation
8. Applied rewrites8.3%
  \[\leadsto \log \left(\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right) \]
if 0.10000000000000001 < (exp.f64 y)
1. Initial program 10.8%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in x around 0
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
4. Step-by-step derivation
  1. distribute-lft-outN/A
    \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(x + \left(1 - e^{y}\right)\right)\right)} \]
  2. *-commutativeN/A
    \[\leadsto \log \color{blue}{\left(\left(x + \left(1 - e^{y}\right)\right) \cdot \frac{1}{2}\right)} \]
  3. lower-*.f64N/A
    \[\leadsto \log \color{blue}{\left(\left(x + \left(1 - e^{y}\right)\right) \cdot \frac{1}{2}\right)} \]
  4. sub-negN/A
    \[\leadsto \log \left(\left(x + \color{blue}{\left(1 + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right) \cdot \frac{1}{2}\right) \]
  5. +-commutativeN/A
    \[\leadsto \log \left(\left(x + \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + 1\right)}\right) \cdot \frac{1}{2}\right) \]
  6. associate-+r+N/A
    \[\leadsto \log \left(\color{blue}{\left(\left(x + \left(\mathsf{neg}\left(e^{y}\right)\right)\right) + 1\right)} \cdot \frac{1}{2}\right) \]
  7. sub-negN/A
    \[\leadsto \log \left(\left(\color{blue}{\left(x - e^{y}\right)} + 1\right) \cdot \frac{1}{2}\right) \]
  8. associate-+l-N/A
    \[\leadsto \log \left(\color{blue}{\left(x - \left(e^{y} - 1\right)\right)} \cdot \frac{1}{2}\right) \]
  9. lower--.f64N/A
    \[\leadsto \log \left(\color{blue}{\left(x - \left(e^{y} - 1\right)\right)} \cdot \frac{1}{2}\right) \]
  10. lower-expm1.f6496.4
    \[\leadsto \log \left(\left(x - \color{blue}{\mathsf{expm1}\left(y\right)}\right) \cdot 0.5\right) \]
5. Applied rewrites96.4%
  \[\leadsto \log \color{blue}{\left(\left(x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right)} \]
6. Taylor expanded in y around 0
  \[\leadsto \log \left(\left(x - y \cdot \left(1 + y \cdot \left(\frac{1}{2} + y \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot y\right)\right)\right)\right) \cdot \frac{1}{2}\right) \]
7. Step-by-step derivation
8. Applied rewrites95.4%
  \[\leadsto \log \left(\left(x - \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y, 0.16666666666666666\right), y, 0.5\right), y, 1\right) \cdot y\right) \cdot 0.5\right) \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 9: 96.9% accurate, 1.4× speedup?

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right) \cdot x, x, x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right) \end{array} \]

(FPCore (x y)
 :precision binary64
 (log (* (fma (* (fma 0.16666666666666666 x 0.5) x) x (- x (expm1 y))) 0.5)))

double code(double x, double y) {
	return log((fma((fma(0.16666666666666666, x, 0.5) * x), x, (x - expm1(y))) * 0.5));
}

function code(x, y)
	return log(Float64(fma(Float64(fma(0.16666666666666666, x, 0.5) * x), x, Float64(x - expm1(y))) * 0.5))
end

code[x_, y_] := N[Log[N[(N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] * x + N[(x - N[(Exp[y] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\log \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right) \cdot x, x, x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right)
\end{array}

Derivation

Initial program 56.0%
\[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
Add Preprocessing
Taylor expanded in x around 0
\[\leadsto \log \left(\frac{\color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right) - e^{y}}}{2}\right) \]
Step-by-step derivation
1. +-commutativeN/A
  \[\leadsto \log \left(\frac{\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1\right)} - e^{y}}{2}\right) \]
2. associate--l+N/A
  \[\leadsto \log \left(\frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + \left(1 - e^{y}\right)}}{2}\right) \]
3. +-commutativeN/A
  \[\leadsto \log \left(\frac{x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1\right)} + \left(1 - e^{y}\right)}{2}\right) \]
4. distribute-lft-inN/A
  \[\leadsto \log \left(\frac{\color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + x \cdot 1\right)} + \left(1 - e^{y}\right)}{2}\right) \]
5. *-rgt-identityN/A
  \[\leadsto \log \left(\frac{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + \color{blue}{x}\right) + \left(1 - e^{y}\right)}{2}\right) \]
6. associate-+l+N/A
  \[\leadsto \log \left(\frac{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + \left(x + \left(1 - e^{y}\right)\right)}}{2}\right) \]
7. associate-*r*N/A
  \[\leadsto \log \left(\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} + \left(x + \left(1 - e^{y}\right)\right)}{2}\right) \]
8. lower-fma.f64N/A
  \[\leadsto \log \left(\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2} + \frac{1}{6} \cdot x, x + \left(1 - e^{y}\right)\right)}}{2}\right) \]
9. lower-*.f64N/A
  \[\leadsto \log \left(\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{6} \cdot x, x + \left(1 - e^{y}\right)\right)}{2}\right) \]
10. +-commutativeN/A
  \[\leadsto \log \left(\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, x + \left(1 - e^{y}\right)\right)}{2}\right) \]
11. lower-fma.f64N/A
  \[\leadsto \log \left(\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right)}, x + \left(1 - e^{y}\right)\right)}{2}\right) \]
12. sub-negN/A
  \[\leadsto \log \left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x + \color{blue}{\left(1 + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right)}{2}\right) \]
13. +-commutativeN/A
  \[\leadsto \log \left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x + \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + 1\right)}\right)}{2}\right) \]
14. associate-+r+N/A
  \[\leadsto \log \left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), \color{blue}{\left(x + \left(\mathsf{neg}\left(e^{y}\right)\right)\right) + 1}\right)}{2}\right) \]
15. sub-negN/A
  \[\leadsto \log \left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), \color{blue}{\left(x - e^{y}\right)} + 1\right)}{2}\right) \]
16. associate-+l-N/A
  \[\leadsto \log \left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), \color{blue}{x - \left(e^{y} - 1\right)}\right)}{2}\right) \]
17. lower--.f64N/A
  \[\leadsto \log \left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), \color{blue}{x - \left(e^{y} - 1\right)}\right)}{2}\right) \]
18. lower-expm1.f6497.1
  \[\leadsto \log \left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x - \color{blue}{\mathsf{expm1}\left(y\right)}\right)}{2}\right) \]
Applied rewrites97.1%
\[\leadsto \log \left(\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x - \mathsf{expm1}\left(y\right)\right)}}{2}\right) \]
Step-by-step derivation
1. lift-/.f64N/A
  \[\leadsto \log \color{blue}{\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x - \mathsf{expm1}\left(y\right)\right)}{2}\right)} \]
2. div-invN/A
  \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x - \mathsf{expm1}\left(y\right)\right) \cdot \frac{1}{2}\right)} \]
3. metadata-evalN/A
  \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x - \mathsf{expm1}\left(y\right)\right) \cdot \color{blue}{\frac{1}{2}}\right) \]
4. lower-*.f6497.1
  \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right)} \]
Applied rewrites97.1%
\[\leadsto \color{blue}{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right) \cdot x, x, x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right)} \]
Add Preprocessing

Alternative 10: 52.0% accurate, 1.4× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y} \leq 0.1:\\ \;\;\;\;\log \left(\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right) \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right), y, x\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

(FPCore (x y)
 :precision binary64
 (if (<= (exp y) 0.1)
   (log (* (* (fma 0.08333333333333333 x 0.25) x) x))
   (log (* (fma (fma -0.5 y -1.0) y x) 0.5))))

double code(double x, double y) {
	double tmp;
	if (exp(y) <= 0.1) {
		tmp = log(((fma(0.08333333333333333, x, 0.25) * x) * x));
	} else {
		tmp = log((fma(fma(-0.5, y, -1.0), y, x) * 0.5));
	}
	return tmp;
}

function code(x, y)
	tmp = 0.0
	if (exp(y) <= 0.1)
		tmp = log(Float64(Float64(fma(0.08333333333333333, x, 0.25) * x) * x));
	else
		tmp = log(Float64(fma(fma(-0.5, y, -1.0), y, x) * 0.5));
	end
	return tmp
end

code[x_, y_] := If[LessEqual[N[Exp[y], $MachinePrecision], 0.1], N[Log[N[(N[(N[(0.08333333333333333 * x + 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], N[Log[N[(N[(N[(-0.5 * y + -1.0), $MachinePrecision] * y + x), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;e^{y} \leq 0.1:\\
\;\;\;\;\log \left(\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right) \cdot x\right) \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right), y, x\right) \cdot 0.5\right)\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if (exp.f64 y) < 0.10000000000000001
1. Initial program 97.8%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in x around 0
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(1 - e^{y}\right) + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \log \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right)\right) + \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
  2. *-commutativeN/A
    \[\leadsto \log \left(\color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right)\right) \cdot x} + \frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
  3. lower-fma.f64N/A
    \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right), x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right)} \]
  4. +-commutativeN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{4} + \frac{1}{12} \cdot x\right) + \frac{1}{2}}, x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  5. *-commutativeN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{4} + \frac{1}{12} \cdot x\right) \cdot x} + \frac{1}{2}, x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  6. lower-fma.f64N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{4} + \frac{1}{12} \cdot x, x, \frac{1}{2}\right)}, x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  7. +-commutativeN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{12} \cdot x + \frac{1}{4}}, x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  8. lower-fma.f64N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right)}, x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)\right) \]
  9. sub-negN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right)\right) \]
  10. +-commutativeN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + 1\right)}\right)\right) \]
  11. distribute-lft-inN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \color{blue}{\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{y}\right)\right) + \frac{1}{2} \cdot 1}\right)\right) \]
  12. neg-mul-1N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \frac{1}{2} \cdot \color{blue}{\left(-1 \cdot e^{y}\right)} + \frac{1}{2} \cdot 1\right)\right) \]
  13. associate-*r*N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \color{blue}{\left(\frac{1}{2} \cdot -1\right) \cdot e^{y}} + \frac{1}{2} \cdot 1\right)\right) \]
  14. metadata-evalN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \color{blue}{\frac{-1}{2}} \cdot e^{y} + \frac{1}{2} \cdot 1\right)\right) \]
  15. metadata-evalN/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \frac{-1}{2} \cdot e^{y} + \color{blue}{\frac{1}{2}}\right)\right) \]
  16. lower-fma.f64N/A
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{4}\right), x, \frac{1}{2}\right), x, \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, e^{y}, \frac{1}{2}\right)}\right)\right) \]
  17. lower-exp.f6496.8
    \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right), x, 0.5\right), x, \mathsf{fma}\left(-0.5, \color{blue}{e^{y}}, 0.5\right)\right)\right) \]
5. Applied rewrites96.8%
  \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right), x, 0.5\right), x, \mathsf{fma}\left(-0.5, e^{y}, 0.5\right)\right)\right)} \]
6. Taylor expanded in x around inf
  \[\leadsto \log \left({x}^{3} \cdot \color{blue}{\left(\frac{1}{12} + \frac{1}{4} \cdot \frac{1}{x}\right)}\right) \]
7. Step-by-step derivation
8. Applied rewrites8.3%
  \[\leadsto \log \left(\left(\mathsf{fma}\left(0.08333333333333333, x, 0.25\right) \cdot x\right) \cdot \color{blue}{x}\right) \]
if 0.10000000000000001 < (exp.f64 y)
1. Initial program 10.8%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in x around 0
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
4. Step-by-step derivation
  1. distribute-lft-outN/A
    \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(x + \left(1 - e^{y}\right)\right)\right)} \]
  2. *-commutativeN/A
    \[\leadsto \log \color{blue}{\left(\left(x + \left(1 - e^{y}\right)\right) \cdot \frac{1}{2}\right)} \]
  3. lower-*.f64N/A
    \[\leadsto \log \color{blue}{\left(\left(x + \left(1 - e^{y}\right)\right) \cdot \frac{1}{2}\right)} \]
  4. sub-negN/A
    \[\leadsto \log \left(\left(x + \color{blue}{\left(1 + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right) \cdot \frac{1}{2}\right) \]
  5. +-commutativeN/A
    \[\leadsto \log \left(\left(x + \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + 1\right)}\right) \cdot \frac{1}{2}\right) \]
  6. associate-+r+N/A
    \[\leadsto \log \left(\color{blue}{\left(\left(x + \left(\mathsf{neg}\left(e^{y}\right)\right)\right) + 1\right)} \cdot \frac{1}{2}\right) \]
  7. sub-negN/A
    \[\leadsto \log \left(\left(\color{blue}{\left(x - e^{y}\right)} + 1\right) \cdot \frac{1}{2}\right) \]
  8. associate-+l-N/A
    \[\leadsto \log \left(\color{blue}{\left(x - \left(e^{y} - 1\right)\right)} \cdot \frac{1}{2}\right) \]
  9. lower--.f64N/A
    \[\leadsto \log \left(\color{blue}{\left(x - \left(e^{y} - 1\right)\right)} \cdot \frac{1}{2}\right) \]
  10. lower-expm1.f6496.4
    \[\leadsto \log \left(\left(x - \color{blue}{\mathsf{expm1}\left(y\right)}\right) \cdot 0.5\right) \]
5. Applied rewrites96.4%
  \[\leadsto \log \color{blue}{\left(\left(x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right)} \]
6. Taylor expanded in y around 0
  \[\leadsto \log \left(\left(x + y \cdot \left(\frac{-1}{2} \cdot y - 1\right)\right) \cdot \frac{1}{2}\right) \]
7. Step-by-step derivation
8. Applied rewrites95.2%
  \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right), y, x\right) \cdot 0.5\right) \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 11: 96.0% accurate, 1.5× speedup?

\[\begin{array}{l} \\ \log \left(\left(x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right) \end{array} \]

(FPCore (x y) :precision binary64 (log (* (- x (expm1 y)) 0.5)))

double code(double x, double y) {
	return log(((x - expm1(y)) * 0.5));
}

public static double code(double x, double y) {
	return Math.log(((x - Math.expm1(y)) * 0.5));
}

def code(x, y):
	return math.log(((x - math.expm1(y)) * 0.5))

function code(x, y)
	return log(Float64(Float64(x - expm1(y)) * 0.5))
end

code[x_, y_] := N[Log[N[(N[(x - N[(Exp[y] - 1), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\log \left(\left(x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right)
\end{array}

Derivation

Initial program 56.0%
\[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
Add Preprocessing
Taylor expanded in x around 0
\[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
Step-by-step derivation
1. distribute-lft-outN/A
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(x + \left(1 - e^{y}\right)\right)\right)} \]
2. *-commutativeN/A
  \[\leadsto \log \color{blue}{\left(\left(x + \left(1 - e^{y}\right)\right) \cdot \frac{1}{2}\right)} \]
3. lower-*.f64N/A
  \[\leadsto \log \color{blue}{\left(\left(x + \left(1 - e^{y}\right)\right) \cdot \frac{1}{2}\right)} \]
4. sub-negN/A
  \[\leadsto \log \left(\left(x + \color{blue}{\left(1 + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right) \cdot \frac{1}{2}\right) \]
5. +-commutativeN/A
  \[\leadsto \log \left(\left(x + \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + 1\right)}\right) \cdot \frac{1}{2}\right) \]
6. associate-+r+N/A
  \[\leadsto \log \left(\color{blue}{\left(\left(x + \left(\mathsf{neg}\left(e^{y}\right)\right)\right) + 1\right)} \cdot \frac{1}{2}\right) \]
7. sub-negN/A
  \[\leadsto \log \left(\left(\color{blue}{\left(x - e^{y}\right)} + 1\right) \cdot \frac{1}{2}\right) \]
8. associate-+l-N/A
  \[\leadsto \log \left(\color{blue}{\left(x - \left(e^{y} - 1\right)\right)} \cdot \frac{1}{2}\right) \]
9. lower--.f64N/A
  \[\leadsto \log \left(\color{blue}{\left(x - \left(e^{y} - 1\right)\right)} \cdot \frac{1}{2}\right) \]
10. lower-expm1.f6496.2
  \[\leadsto \log \left(\left(x - \color{blue}{\mathsf{expm1}\left(y\right)}\right) \cdot 0.5\right) \]
Applied rewrites96.2%
\[\leadsto \log \color{blue}{\left(\left(x - \mathsf{expm1}\left(y\right)\right) \cdot 0.5\right)} \]
Add Preprocessing

Alternative 12: 31.0% accurate, 2.8× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;\log \left(0.5 \cdot x\right)\\ \end{array} \end{array} \]

(FPCore (x y)
 :precision binary64
 (if (<= x -5e-310) (/ x (- y)) (log (* 0.5 x))))

double code(double x, double y) {
	double tmp;
	if (x <= -5e-310) {
		tmp = x / -y;
	} else {
		tmp = log((0.5 * x));
	}
	return tmp;
}

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = x / -y
    else
        tmp = log((0.5d0 * x))
    end if
    code = tmp
end function

public static double code(double x, double y) {
	double tmp;
	if (x <= -5e-310) {
		tmp = x / -y;
	} else {
		tmp = Math.log((0.5 * x));
	}
	return tmp;
}

def code(x, y):
	tmp = 0
	if x <= -5e-310:
		tmp = x / -y
	else:
		tmp = math.log((0.5 * x))
	return tmp

function code(x, y)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(x / Float64(-y));
	else
		tmp = log(Float64(0.5 * x));
	end
	return tmp
end

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = x / -y;
	else
		tmp = log((0.5 * x));
	end
	tmp_2 = tmp;
end

code[x_, y_] := If[LessEqual[x, -5e-310], N[(x / (-y)), $MachinePrecision], N[Log[N[(0.5 * x), $MachinePrecision]], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\
\;\;\;\;\frac{x}{-y}\\

\mathbf{else}:\\
\;\;\;\;\log \left(0.5 \cdot x\right)\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if x < -4.999999999999985e-310
1. Initial program 72.9%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in x around 0
  \[\leadsto \color{blue}{\log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) + \frac{x}{1 - e^{y}}} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{\frac{x}{1 - e^{y}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
  2. *-rgt-identityN/A
    \[\leadsto \frac{\color{blue}{x \cdot 1}}{1 - e^{y}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
  3. associate-*r/N/A
    \[\leadsto \color{blue}{x \cdot \frac{1}{1 - e^{y}}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
  4. lower-+.f64N/A
    \[\leadsto \color{blue}{x \cdot \frac{1}{1 - e^{y}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
  5. associate-*r/N/A
    \[\leadsto \color{blue}{\frac{x \cdot 1}{1 - e^{y}}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
  6. *-rgt-identityN/A
    \[\leadsto \frac{\color{blue}{x}}{1 - e^{y}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
  7. lower-/.f64N/A
    \[\leadsto \color{blue}{\frac{x}{1 - e^{y}}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
  8. sub-negN/A
    \[\leadsto \frac{x}{\color{blue}{1 + \left(\mathsf{neg}\left(e^{y}\right)\right)}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
  9. +-commutativeN/A
    \[\leadsto \frac{x}{\color{blue}{\left(\mathsf{neg}\left(e^{y}\right)\right) + 1}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
  10. neg-sub0N/A
    \[\leadsto \frac{x}{\color{blue}{\left(0 - e^{y}\right)} + 1} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
  11. associate-+l-N/A
    \[\leadsto \frac{x}{\color{blue}{0 - \left(e^{y} - 1\right)}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
  12. sub0-negN/A
    \[\leadsto \frac{x}{\color{blue}{\mathsf{neg}\left(\left(e^{y} - 1\right)\right)}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
  13. lower-neg.f64N/A
    \[\leadsto \frac{x}{\color{blue}{-\left(e^{y} - 1\right)}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
  14. lower-expm1.f64N/A
    \[\leadsto \frac{x}{-\color{blue}{\mathsf{expm1}\left(y\right)}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
  15. lower-log.f64N/A
    \[\leadsto \frac{x}{-\mathsf{expm1}\left(y\right)} + \color{blue}{\log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
  16. sub-negN/A
    \[\leadsto \frac{x}{-\mathsf{expm1}\left(y\right)} + \log \left(\frac{1}{2} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right) \]
  17. +-commutativeN/A
    \[\leadsto \frac{x}{-\mathsf{expm1}\left(y\right)} + \log \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + 1\right)}\right) \]
  18. distribute-lft-inN/A
    \[\leadsto \frac{x}{-\mathsf{expm1}\left(y\right)} + \log \color{blue}{\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{y}\right)\right) + \frac{1}{2} \cdot 1\right)} \]
  19. neg-mul-1N/A
    \[\leadsto \frac{x}{-\mathsf{expm1}\left(y\right)} + \log \left(\frac{1}{2} \cdot \color{blue}{\left(-1 \cdot e^{y}\right)} + \frac{1}{2} \cdot 1\right) \]
  20. associate-*r*N/A
    \[\leadsto \frac{x}{-\mathsf{expm1}\left(y\right)} + \log \left(\color{blue}{\left(\frac{1}{2} \cdot -1\right) \cdot e^{y}} + \frac{1}{2} \cdot 1\right) \]
  21. metadata-evalN/A
    \[\leadsto \frac{x}{-\mathsf{expm1}\left(y\right)} + \log \left(\color{blue}{\frac{-1}{2}} \cdot e^{y} + \frac{1}{2} \cdot 1\right) \]
  22. metadata-evalN/A
    \[\leadsto \frac{x}{-\mathsf{expm1}\left(y\right)} + \log \left(\frac{-1}{2} \cdot e^{y} + \color{blue}{\frac{1}{2}}\right) \]
5. Applied rewrites73.9%
  \[\leadsto \color{blue}{\frac{x}{-\mathsf{expm1}\left(y\right)} + \log \left(\mathsf{fma}\left(-0.5, e^{y}, 0.5\right)\right)} \]
6. Taylor expanded in y around 0
  \[\leadsto -1 \cdot \color{blue}{\frac{x}{y}} \]
7. Step-by-step derivation
8. Applied rewrites4.6%
  \[\leadsto \frac{x}{\color{blue}{-y}} \]
if -4.999999999999985e-310 < x
1. Initial program 46.4%
  \[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
2. Add Preprocessing
3. Taylor expanded in y around 0
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot \left(e^{x} - 1\right)\right)} \]
4. Step-by-step derivation
  1. *-commutativeN/A
    \[\leadsto \log \color{blue}{\left(\left(e^{x} - 1\right) \cdot \frac{1}{2}\right)} \]
  2. lower-*.f64N/A
    \[\leadsto \log \color{blue}{\left(\left(e^{x} - 1\right) \cdot \frac{1}{2}\right)} \]
  3. lower-expm1.f6453.7
    \[\leadsto \log \left(\color{blue}{\mathsf{expm1}\left(x\right)} \cdot 0.5\right) \]
5. Applied rewrites53.7%
  \[\leadsto \log \color{blue}{\left(\mathsf{expm1}\left(x\right) \cdot 0.5\right)} \]
6. Taylor expanded in x around 0
  \[\leadsto \log \left(\frac{1}{2} \cdot \color{blue}{x}\right) \]
7. Step-by-step derivation
8. Applied rewrites51.0%
  \[\leadsto \log \left(0.5 \cdot \color{blue}{x}\right) \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 13: 3.7% accurate, 22.5× speedup?

\[\begin{array}{l} \\ \frac{x}{-y} \end{array} \]

(FPCore (x y) :precision binary64 (/ x (- y)))

double code(double x, double y) {
	return x / -y;
}

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / -y
end function

public static double code(double x, double y) {
	return x / -y;
}

def code(x, y):
	return x / -y

function code(x, y)
	return Float64(x / Float64(-y))
end

function tmp = code(x, y)
	tmp = x / -y;
end

code[x_, y_] := N[(x / (-y)), $MachinePrecision]

\begin{array}{l}

\\
\frac{x}{-y}
\end{array}

Derivation

Initial program 56.0%
\[\log \left(\frac{e^{x} - e^{y}}{2}\right) \]
Add Preprocessing
Taylor expanded in x around 0
\[\leadsto \color{blue}{\log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) + \frac{x}{1 - e^{y}}} \]
Step-by-step derivation
1. +-commutativeN/A
  \[\leadsto \color{blue}{\frac{x}{1 - e^{y}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
2. *-rgt-identityN/A
  \[\leadsto \frac{\color{blue}{x \cdot 1}}{1 - e^{y}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
3. associate-*r/N/A
  \[\leadsto \color{blue}{x \cdot \frac{1}{1 - e^{y}}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
4. lower-+.f64N/A
  \[\leadsto \color{blue}{x \cdot \frac{1}{1 - e^{y}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
5. associate-*r/N/A
  \[\leadsto \color{blue}{\frac{x \cdot 1}{1 - e^{y}}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
6. *-rgt-identityN/A
  \[\leadsto \frac{\color{blue}{x}}{1 - e^{y}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
7. lower-/.f64N/A
  \[\leadsto \color{blue}{\frac{x}{1 - e^{y}}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
8. sub-negN/A
  \[\leadsto \frac{x}{\color{blue}{1 + \left(\mathsf{neg}\left(e^{y}\right)\right)}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
9. +-commutativeN/A
  \[\leadsto \frac{x}{\color{blue}{\left(\mathsf{neg}\left(e^{y}\right)\right) + 1}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
10. neg-sub0N/A
  \[\leadsto \frac{x}{\color{blue}{\left(0 - e^{y}\right)} + 1} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
11. associate-+l-N/A
  \[\leadsto \frac{x}{\color{blue}{0 - \left(e^{y} - 1\right)}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
12. sub0-negN/A
  \[\leadsto \frac{x}{\color{blue}{\mathsf{neg}\left(\left(e^{y} - 1\right)\right)}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
13. lower-neg.f64N/A
  \[\leadsto \frac{x}{\color{blue}{-\left(e^{y} - 1\right)}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
14. lower-expm1.f64N/A
  \[\leadsto \frac{x}{-\color{blue}{\mathsf{expm1}\left(y\right)}} + \log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right) \]
15. lower-log.f64N/A
  \[\leadsto \frac{x}{-\mathsf{expm1}\left(y\right)} + \color{blue}{\log \left(\frac{1}{2} \cdot \left(1 - e^{y}\right)\right)} \]
16. sub-negN/A
  \[\leadsto \frac{x}{-\mathsf{expm1}\left(y\right)} + \log \left(\frac{1}{2} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(e^{y}\right)\right)\right)}\right) \]
17. +-commutativeN/A
  \[\leadsto \frac{x}{-\mathsf{expm1}\left(y\right)} + \log \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(e^{y}\right)\right) + 1\right)}\right) \]
18. distribute-lft-inN/A
  \[\leadsto \frac{x}{-\mathsf{expm1}\left(y\right)} + \log \color{blue}{\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{y}\right)\right) + \frac{1}{2} \cdot 1\right)} \]
19. neg-mul-1N/A
  \[\leadsto \frac{x}{-\mathsf{expm1}\left(y\right)} + \log \left(\frac{1}{2} \cdot \color{blue}{\left(-1 \cdot e^{y}\right)} + \frac{1}{2} \cdot 1\right) \]
20. associate-*r*N/A
  \[\leadsto \frac{x}{-\mathsf{expm1}\left(y\right)} + \log \left(\color{blue}{\left(\frac{1}{2} \cdot -1\right) \cdot e^{y}} + \frac{1}{2} \cdot 1\right) \]
21. metadata-evalN/A
  \[\leadsto \frac{x}{-\mathsf{expm1}\left(y\right)} + \log \left(\color{blue}{\frac{-1}{2}} \cdot e^{y} + \frac{1}{2} \cdot 1\right) \]
22. metadata-evalN/A
  \[\leadsto \frac{x}{-\mathsf{expm1}\left(y\right)} + \log \left(\frac{-1}{2} \cdot e^{y} + \color{blue}{\frac{1}{2}}\right) \]
Applied rewrites54.4%
\[\leadsto \color{blue}{\frac{x}{-\mathsf{expm1}\left(y\right)} + \log \left(\mathsf{fma}\left(-0.5, e^{y}, 0.5\right)\right)} \]
Taylor expanded in y around 0
\[\leadsto -1 \cdot \color{blue}{\frac{x}{y}} \]
Step-by-step derivation
Applied rewrites3.7%
\[\leadsto \frac{x}{\color{blue}{-y}} \]
Add Preprocessing

log((exp(x) - exp(y))/2)

could not determine a ground truth (more)

Specification

Local Percentage Accuracy vs ?

Accuracy vs Speed?

Initial Program: 54.2% accurate, 1.0× speedup?

Alternative 1: 97.7% accurate, 0.6× speedup?

`if (-.f64 (exp.f64 x) (exp.f64 y)) < 0.20000000000000001`

`if 0.20000000000000001 < (-.f64 (exp.f64 x) (exp.f64 y))`

Alternative 2: 50.8% accurate, 0.8× speedup?

`if (-.f64 (exp.f64 x) (exp.f64 y)) < 1.99999999999999996e-12`

`if 1.99999999999999996e-12 < (-.f64 (exp.f64 x) (exp.f64 y))`

Alternative 3: 52.1% accurate, 0.9× speedup?

`if (-.f64 (exp.f64 x) (exp.f64 y)) < 0.5`

`if 0.5 < (-.f64 (exp.f64 x) (exp.f64 y))`

Alternative 4: 51.5% accurate, 1.0× speedup?

`if (-.f64 (exp.f64 x) (exp.f64 y)) < 0.5`

`if 0.5 < (-.f64 (exp.f64 x) (exp.f64 y))`

Alternative 5: 50.1% accurate, 1.0× speedup?

`if (-.f64 (exp.f64 x) (exp.f64 y)) < 1.99999999999999996e-12`

`if 1.99999999999999996e-12 < (-.f64 (exp.f64 x) (exp.f64 y))`

Alternative 6: 50.7% accurate, 1.0× speedup?

`if (-.f64 (exp.f64 x) (exp.f64 y)) < 0.51000000000000001`

`if 0.51000000000000001 < (-.f64 (exp.f64 x) (exp.f64 y))`

Alternative 7: 95.1% accurate, 1.0× speedup?

`if (exp.f64 y) < 0.900000000000000022`

`if 0.900000000000000022 < (exp.f64 y)`

Alternative 8: 52.2% accurate, 1.3× speedup?

`if (exp.f64 y) < 0.10000000000000001`

`if 0.10000000000000001 < (exp.f64 y)`

Alternative 9: 96.9% accurate, 1.4× speedup?

Alternative 10: 52.0% accurate, 1.4× speedup?

`if (exp.f64 y) < 0.10000000000000001`

`if 0.10000000000000001 < (exp.f64 y)`

Alternative 11: 96.0% accurate, 1.5× speedup?

Alternative 12: 31.0% accurate, 2.8× speedup?

`if x < -4.999999999999985e-310`

`if -4.999999999999985e-310 < x`

Alternative 13: 3.7% accurate, 22.5× speedup?

Reproduce

could not determine a ground truth (more)

Specification

Local Percentage Accuracy vs ?

Accuracy vs Speed?

Initial Program: 54.2% accurate, 1.0× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

Alternative 1: 97.7% accurate, 0.6× speedupMathFPCoreCJuliaWolframTeX?

if (-.f64 (exp.f64 x) (exp.f64 y)) < 0.20000000000000001

if 0.20000000000000001 < (-.f64 (exp.f64 x) (exp.f64 y))

Alternative 2: 50.8% accurate, 0.8× speedupMathFPCoreCJuliaWolframTeX?

if (-.f64 (exp.f64 x) (exp.f64 y)) < 1.99999999999999996e-12

if 1.99999999999999996e-12 < (-.f64 (exp.f64 x) (exp.f64 y))

Alternative 3: 52.1% accurate, 0.9× speedupMathFPCoreCJuliaWolframTeX?

if (-.f64 (exp.f64 x) (exp.f64 y)) < 0.5

if 0.5 < (-.f64 (exp.f64 x) (exp.f64 y))

Alternative 4: 51.5% accurate, 1.0× speedupMathFPCoreCJuliaWolframTeX?

if (-.f64 (exp.f64 x) (exp.f64 y)) < 0.5

if 0.5 < (-.f64 (exp.f64 x) (exp.f64 y))

Alternative 5: 50.1% accurate, 1.0× speedupMathFPCoreCJuliaWolframTeX?

if (-.f64 (exp.f64 x) (exp.f64 y)) < 1.99999999999999996e-12

if 1.99999999999999996e-12 < (-.f64 (exp.f64 x) (exp.f64 y))

Alternative 6: 50.7% accurate, 1.0× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

if (-.f64 (exp.f64 x) (exp.f64 y)) < 0.51000000000000001

if 0.51000000000000001 < (-.f64 (exp.f64 x) (exp.f64 y))

Alternative 7: 95.1% accurate, 1.0× speedupMathFPCoreCJuliaWolframTeX?

if (exp.f64 y) < 0.900000000000000022

if 0.900000000000000022 < (exp.f64 y)

Alternative 8: 52.2% accurate, 1.3× speedupMathFPCoreCJuliaWolframTeX?

if (exp.f64 y) < 0.10000000000000001

if 0.10000000000000001 < (exp.f64 y)

Alternative 9: 96.9% accurate, 1.4× speedupMathFPCoreCJuliaWolframTeX?

Alternative 10: 52.0% accurate, 1.4× speedupMathFPCoreCJuliaWolframTeX?

if (exp.f64 y) < 0.10000000000000001

if 0.10000000000000001 < (exp.f64 y)

Alternative 11: 96.0% accurate, 1.5× speedupMathFPCoreCJavaPythonJuliaWolframTeX?

Alternative 12: 31.0% accurate, 2.8× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

if x < -4.999999999999985e-310

if -4.999999999999985e-310 < x

Alternative 13: 3.7% accurate, 22.5× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

Reproduce

Initial Program: 54.2% accurate, 1.0× speedup?

Alternative 1: 97.7% accurate, 0.6× speedup?

`if (-.f64 (exp.f64 x) (exp.f64 y)) < 0.20000000000000001`

`if 0.20000000000000001 < (-.f64 (exp.f64 x) (exp.f64 y))`

Alternative 2: 50.8% accurate, 0.8× speedup?

`if (-.f64 (exp.f64 x) (exp.f64 y)) < 1.99999999999999996e-12`

`if 1.99999999999999996e-12 < (-.f64 (exp.f64 x) (exp.f64 y))`

Alternative 3: 52.1% accurate, 0.9× speedup?

`if (-.f64 (exp.f64 x) (exp.f64 y)) < 0.5`

`if 0.5 < (-.f64 (exp.f64 x) (exp.f64 y))`

Alternative 4: 51.5% accurate, 1.0× speedup?

`if (-.f64 (exp.f64 x) (exp.f64 y)) < 0.5`

`if 0.5 < (-.f64 (exp.f64 x) (exp.f64 y))`

Alternative 5: 50.1% accurate, 1.0× speedup?

`if (-.f64 (exp.f64 x) (exp.f64 y)) < 1.99999999999999996e-12`

`if 1.99999999999999996e-12 < (-.f64 (exp.f64 x) (exp.f64 y))`

Alternative 6: 50.7% accurate, 1.0× speedup?

`if (-.f64 (exp.f64 x) (exp.f64 y)) < 0.51000000000000001`

`if 0.51000000000000001 < (-.f64 (exp.f64 x) (exp.f64 y))`

Alternative 7: 95.1% accurate, 1.0× speedup?

`if (exp.f64 y) < 0.900000000000000022`

`if 0.900000000000000022 < (exp.f64 y)`

Alternative 8: 52.2% accurate, 1.3× speedup?

`if (exp.f64 y) < 0.10000000000000001`

`if 0.10000000000000001 < (exp.f64 y)`

Alternative 9: 96.9% accurate, 1.4× speedup?

Alternative 10: 52.0% accurate, 1.4× speedup?

`if (exp.f64 y) < 0.10000000000000001`

`if 0.10000000000000001 < (exp.f64 y)`

Alternative 11: 96.0% accurate, 1.5× speedup?

Alternative 12: 31.0% accurate, 2.8× speedup?

`if x < -4.999999999999985e-310`

`if -4.999999999999985e-310 < x`

Alternative 13: 3.7% accurate, 22.5× speedup?