sqrt(x + 2) - sqrt(2x)

Percentage Accurate: 99.3% → 99.3%
Time: 3.9s
Alternatives: 7
Speedup: 1.0×

Specification

?
\[0 \leq x \land x \leq 1.79 \cdot 10^{+308}\]
\[\begin{array}{l} \\ \sqrt{x + 2} - \sqrt{2 \cdot x} \end{array} \]
(FPCore (x) :precision binary64 (- (sqrt (+ x 2.0)) (sqrt (* 2.0 x))))
double code(double x) {
	return sqrt((x + 2.0)) - sqrt((2.0 * x));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x + 2.0d0)) - sqrt((2.0d0 * x))
end function

public static double code(double x) {
	return Math.sqrt((x + 2.0)) - Math.sqrt((2.0 * x));
}

def code(x):
	return math.sqrt((x + 2.0)) - math.sqrt((2.0 * x))

function code(x)
	return Float64(sqrt(Float64(x + 2.0)) - sqrt(Float64(2.0 * x)))
end

function tmp = code(x)
	tmp = sqrt((x + 2.0)) - sqrt((2.0 * x));
end

code[x_] := N[(N[Sqrt[N[(x + 2.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sqrt{x + 2} - \sqrt{2 \cdot x}
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 7 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 99.3% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \sqrt{x + 2} - \sqrt{2 \cdot x} \end{array} \]
(FPCore (x) :precision binary64 (- (sqrt (+ x 2.0)) (sqrt (* 2.0 x))))
double code(double x) {
	return sqrt((x + 2.0)) - sqrt((2.0 * x));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x + 2.0d0)) - sqrt((2.0d0 * x))
end function

public static double code(double x) {
	return Math.sqrt((x + 2.0)) - Math.sqrt((2.0 * x));
}

def code(x):
	return math.sqrt((x + 2.0)) - math.sqrt((2.0 * x))

function code(x)
	return Float64(sqrt(Float64(x + 2.0)) - sqrt(Float64(2.0 * x)))
end

function tmp = code(x)
	tmp = sqrt((x + 2.0)) - sqrt((2.0 * x));
end

code[x_] := N[(N[Sqrt[N[(x + 2.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sqrt{x + 2} - \sqrt{2 \cdot x}
\end{array}

Alternative 1: 99.3% accurate, 0.2× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x \cdot 2}\\ t_1 := \sqrt{x + 2}\\ \mathsf{fma}\left(\frac{t\_0}{-1}, \frac{t\_0}{\mathsf{fma}\left(\sqrt{2}, \sqrt{x}, t\_1\right)}, \left(t\_1 - t\_0\right) \cdot \frac{x + 2}{\mathsf{fma}\left(-2, x, x + 2\right)}\right) \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (* x 2.0))) (t_1 (sqrt (+ x 2.0))))
   (fma
    (/ t_0 -1.0)
    (/ t_0 (fma (sqrt 2.0) (sqrt x) t_1))
    (* (- t_1 t_0) (/ (+ x 2.0) (fma -2.0 x (+ x 2.0)))))))
double code(double x) {
	double t_0 = sqrt((x * 2.0));
	double t_1 = sqrt((x + 2.0));
	return fma((t_0 / -1.0), (t_0 / fma(sqrt(2.0), sqrt(x), t_1)), ((t_1 - t_0) * ((x + 2.0) / fma(-2.0, x, (x + 2.0)))));
}

function code(x)
	t_0 = sqrt(Float64(x * 2.0))
	t_1 = sqrt(Float64(x + 2.0))
	return fma(Float64(t_0 / -1.0), Float64(t_0 / fma(sqrt(2.0), sqrt(x), t_1)), Float64(Float64(t_1 - t_0) * Float64(Float64(x + 2.0) / fma(-2.0, x, Float64(x + 2.0)))))
end

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(x + 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 / -1.0), $MachinePrecision] * N[(t$95$0 / N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - t$95$0), $MachinePrecision] * N[(N[(x + 2.0), $MachinePrecision] / N[(-2.0 * x + N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sqrt{x \cdot 2}\\
t_1 := \sqrt{x + 2}\\
\mathsf{fma}\left(\frac{t\_0}{-1}, \frac{t\_0}{\mathsf{fma}\left(\sqrt{2}, \sqrt{x}, t\_1\right)}, \left(t\_1 - t\_0\right) \cdot \frac{x + 2}{\mathsf{fma}\left(-2, x, x + 2\right)}\right)
\end{array}
\end{array}
Derivation

  1. Initial program 99.3%

    \[\sqrt{x + 2} - \sqrt{2 \cdot x} \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift--.f64N/A

      \[\leadsto \color{blue}{\sqrt{x + 2} - \sqrt{2 \cdot x}} \]

    2. flip--N/A

      \[\leadsto \color{blue}{\frac{\sqrt{x + 2} \cdot \sqrt{x + 2} - \sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}}{\sqrt{x + 2} + \sqrt{2 \cdot x}}} \]

    3. lift-sqrt.f64N/A

      \[\leadsto \frac{\color{blue}{\sqrt{x + 2}} \cdot \sqrt{x + 2} - \sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}}{\sqrt{x + 2} + \sqrt{2 \cdot x}} \]

    4. lift-sqrt.f64N/A

      \[\leadsto \frac{\sqrt{x + 2} \cdot \color{blue}{\sqrt{x + 2}} - \sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}}{\sqrt{x + 2} + \sqrt{2 \cdot x}} \]

    5. rem-square-sqrtN/A

      \[\leadsto \frac{\color{blue}{\left(x + 2\right)} - \sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}}{\sqrt{x + 2} + \sqrt{2 \cdot x}} \]

    6. lift-sqrt.f64N/A

      \[\leadsto \frac{\left(x + 2\right) - \color{blue}{\sqrt{2 \cdot x}} \cdot \sqrt{2 \cdot x}}{\sqrt{x + 2} + \sqrt{2 \cdot x}} \]

    7. lift-sqrt.f64N/A

      \[\leadsto \frac{\left(x + 2\right) - \sqrt{2 \cdot x} \cdot \color{blue}{\sqrt{2 \cdot x}}}{\sqrt{x + 2} + \sqrt{2 \cdot x}} \]

    8. rem-square-sqrtN/A

      \[\leadsto \frac{\left(x + 2\right) - \color{blue}{2 \cdot x}}{\sqrt{x + 2} + \sqrt{2 \cdot x}} \]

    9. div-subN/A

      \[\leadsto \color{blue}{\frac{x + 2}{\sqrt{x + 2} + \sqrt{2 \cdot x}} - \frac{2 \cdot x}{\sqrt{x + 2} + \sqrt{2 \cdot x}}} \]

    10. sub-negN/A

      \[\leadsto \color{blue}{\frac{x + 2}{\sqrt{x + 2} + \sqrt{2 \cdot x}} + \left(\mathsf{neg}\left(\frac{2 \cdot x}{\sqrt{x + 2} + \sqrt{2 \cdot x}}\right)\right)} \]

  4. Applied rewrites99.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2 + x}{\mathsf{fma}\left(-2, x, 2 + x\right)}, \sqrt{2 + x} - \sqrt{2 \cdot x}, -\frac{2 \cdot x}{\mathsf{fma}\left(\sqrt{x}, \sqrt{2}, \sqrt{2 + x}\right)}\right)} \]
  5. Step-by-step derivation

    1. lift-fma.f64N/A

      \[\leadsto \color{blue}{\frac{2 + x}{\mathsf{fma}\left(-2, x, 2 + x\right)} \cdot \left(\sqrt{2 + x} - \sqrt{2 \cdot x}\right) + \left(-\frac{2 \cdot x}{\mathsf{fma}\left(\sqrt{x}, \sqrt{2}, \sqrt{2 + x}\right)}\right)} \]

    2. +-commutativeN/A

      \[\leadsto \color{blue}{\left(-\frac{2 \cdot x}{\mathsf{fma}\left(\sqrt{x}, \sqrt{2}, \sqrt{2 + x}\right)}\right) + \frac{2 + x}{\mathsf{fma}\left(-2, x, 2 + x\right)} \cdot \left(\sqrt{2 + x} - \sqrt{2 \cdot x}\right)} \]

    3. lift-neg.f64N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{2 \cdot x}{\mathsf{fma}\left(\sqrt{x}, \sqrt{2}, \sqrt{2 + x}\right)}\right)\right)} + \frac{2 + x}{\mathsf{fma}\left(-2, x, 2 + x\right)} \cdot \left(\sqrt{2 + x} - \sqrt{2 \cdot x}\right) \]

    4. lift-/.f64N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{2 \cdot x}{\mathsf{fma}\left(\sqrt{x}, \sqrt{2}, \sqrt{2 + x}\right)}}\right)\right) + \frac{2 + x}{\mathsf{fma}\left(-2, x, 2 + x\right)} \cdot \left(\sqrt{2 + x} - \sqrt{2 \cdot x}\right) \]

    5. distribute-neg-frac2N/A

      \[\leadsto \color{blue}{\frac{2 \cdot x}{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{x}, \sqrt{2}, \sqrt{2 + x}\right)\right)}} + \frac{2 + x}{\mathsf{fma}\left(-2, x, 2 + x\right)} \cdot \left(\sqrt{2 + x} - \sqrt{2 \cdot x}\right) \]

    6. rem-square-sqrtN/A

      \[\leadsto \frac{\color{blue}{\sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}}}{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{x}, \sqrt{2}, \sqrt{2 + x}\right)\right)} + \frac{2 + x}{\mathsf{fma}\left(-2, x, 2 + x\right)} \cdot \left(\sqrt{2 + x} - \sqrt{2 \cdot x}\right) \]

    7. lift-sqrt.f64N/A

      \[\leadsto \frac{\color{blue}{\sqrt{2 \cdot x}} \cdot \sqrt{2 \cdot x}}{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{x}, \sqrt{2}, \sqrt{2 + x}\right)\right)} + \frac{2 + x}{\mathsf{fma}\left(-2, x, 2 + x\right)} \cdot \left(\sqrt{2 + x} - \sqrt{2 \cdot x}\right) \]

    8. lift-sqrt.f64N/A

      \[\leadsto \frac{\sqrt{2 \cdot x} \cdot \color{blue}{\sqrt{2 \cdot x}}}{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{x}, \sqrt{2}, \sqrt{2 + x}\right)\right)} + \frac{2 + x}{\mathsf{fma}\left(-2, x, 2 + x\right)} \cdot \left(\sqrt{2 + x} - \sqrt{2 \cdot x}\right) \]

    9. neg-mul-1N/A

      \[\leadsto \frac{\sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}}{\color{blue}{-1 \cdot \mathsf{fma}\left(\sqrt{x}, \sqrt{2}, \sqrt{2 + x}\right)}} + \frac{2 + x}{\mathsf{fma}\left(-2, x, 2 + x\right)} \cdot \left(\sqrt{2 + x} - \sqrt{2 \cdot x}\right) \]

    10. times-fracN/A

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot x}}{-1} \cdot \frac{\sqrt{2 \cdot x}}{\mathsf{fma}\left(\sqrt{x}, \sqrt{2}, \sqrt{2 + x}\right)}} + \frac{2 + x}{\mathsf{fma}\left(-2, x, 2 + x\right)} \cdot \left(\sqrt{2 + x} - \sqrt{2 \cdot x}\right) \]

  6. Applied rewrites99.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt{x \cdot 2}}{-1}, \frac{\sqrt{x \cdot 2}}{\mathsf{fma}\left(\sqrt{2}, \sqrt{x}, \sqrt{x + 2}\right)}, \left(\sqrt{x + 2} - \sqrt{x \cdot 2}\right) \cdot \frac{x + 2}{\mathsf{fma}\left(-2, x, x + 2\right)}\right)} \]
  7. Add Preprocessing

Alternative 2: 99.3% accurate, 0.2× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2 + x}\\ t_1 := \mathsf{fma}\left(\sqrt{x}, \sqrt{2}, t\_0\right)\\ \mathsf{fma}\left(t\_0, \frac{t\_0}{t\_1}, \frac{\left(-2\right) \cdot x}{t\_1}\right) \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 2.0 x))) (t_1 (fma (sqrt x) (sqrt 2.0) t_0)))
   (fma t_0 (/ t_0 t_1) (/ (* (- 2.0) x) t_1))))
double code(double x) {
	double t_0 = sqrt((2.0 + x));
	double t_1 = fma(sqrt(x), sqrt(2.0), t_0);
	return fma(t_0, (t_0 / t_1), ((-2.0 * x) / t_1));
}

function code(x)
	t_0 = sqrt(Float64(2.0 + x))
	t_1 = fma(sqrt(x), sqrt(2.0), t_0)
	return fma(t_0, Float64(t_0 / t_1), Float64(Float64(Float64(-2.0) * x) / t_1))
end

code[x_] := Block[{t$95$0 = N[Sqrt[N[(2.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + t$95$0), $MachinePrecision]}, N[(t$95$0 * N[(t$95$0 / t$95$1), $MachinePrecision] + N[(N[((-2.0) * x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sqrt{2 + x}\\
t_1 := \mathsf{fma}\left(\sqrt{x}, \sqrt{2}, t\_0\right)\\
\mathsf{fma}\left(t\_0, \frac{t\_0}{t\_1}, \frac{\left(-2\right) \cdot x}{t\_1}\right)
\end{array}
\end{array}
Derivation

  1. Initial program 99.3%

    \[\sqrt{x + 2} - \sqrt{2 \cdot x} \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift--.f64N/A

      \[\leadsto \color{blue}{\sqrt{x + 2} - \sqrt{2 \cdot x}} \]

    2. flip--N/A

      \[\leadsto \color{blue}{\frac{\sqrt{x + 2} \cdot \sqrt{x + 2} - \sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}}{\sqrt{x + 2} + \sqrt{2 \cdot x}}} \]

    3. lift-sqrt.f64N/A

      \[\leadsto \frac{\color{blue}{\sqrt{x + 2}} \cdot \sqrt{x + 2} - \sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}}{\sqrt{x + 2} + \sqrt{2 \cdot x}} \]

    4. lift-sqrt.f64N/A

      \[\leadsto \frac{\sqrt{x + 2} \cdot \color{blue}{\sqrt{x + 2}} - \sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}}{\sqrt{x + 2} + \sqrt{2 \cdot x}} \]

    5. rem-square-sqrtN/A

      \[\leadsto \frac{\color{blue}{\left(x + 2\right)} - \sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}}{\sqrt{x + 2} + \sqrt{2 \cdot x}} \]

    6. lift-sqrt.f64N/A

      \[\leadsto \frac{\left(x + 2\right) - \color{blue}{\sqrt{2 \cdot x}} \cdot \sqrt{2 \cdot x}}{\sqrt{x + 2} + \sqrt{2 \cdot x}} \]

    7. lift-sqrt.f64N/A

      \[\leadsto \frac{\left(x + 2\right) - \sqrt{2 \cdot x} \cdot \color{blue}{\sqrt{2 \cdot x}}}{\sqrt{x + 2} + \sqrt{2 \cdot x}} \]

    8. rem-square-sqrtN/A

      \[\leadsto \frac{\left(x + 2\right) - \color{blue}{2 \cdot x}}{\sqrt{x + 2} + \sqrt{2 \cdot x}} \]

    9. div-subN/A

      \[\leadsto \color{blue}{\frac{x + 2}{\sqrt{x + 2} + \sqrt{2 \cdot x}} - \frac{2 \cdot x}{\sqrt{x + 2} + \sqrt{2 \cdot x}}} \]

    10. sub-negN/A

      \[\leadsto \color{blue}{\frac{x + 2}{\sqrt{x + 2} + \sqrt{2 \cdot x}} + \left(\mathsf{neg}\left(\frac{2 \cdot x}{\sqrt{x + 2} + \sqrt{2 \cdot x}}\right)\right)} \]

  4. Applied rewrites99.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2 + x}, \frac{\sqrt{2 + x}}{\mathsf{fma}\left(\sqrt{x}, \sqrt{2}, \sqrt{2 + x}\right)}, -\frac{2 \cdot x}{\mathsf{fma}\left(\sqrt{x}, \sqrt{2}, \sqrt{2 + x}\right)}\right)} \]

  5. Final simplification99.3%

    \[\leadsto \mathsf{fma}\left(\sqrt{2 + x}, \frac{\sqrt{2 + x}}{\mathsf{fma}\left(\sqrt{x}, \sqrt{2}, \sqrt{2 + x}\right)}, \frac{\left(-2\right) \cdot x}{\mathsf{fma}\left(\sqrt{x}, \sqrt{2}, \sqrt{2 + x}\right)}\right) \]
  6. Add Preprocessing

Alternative 3: 97.4% accurate, 0.9× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\left(1 - \sqrt{x}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} - \sqrt{2 \cdot x}\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (if (<= x 2.0) (* (- 1.0 (sqrt x)) (sqrt 2.0)) (- (sqrt x) (sqrt (* 2.0 x)))))
double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = (1.0 - sqrt(x)) * sqrt(2.0);
	} else {
		tmp = sqrt(x) - sqrt((2.0 * x));
	}
	return tmp;
}

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = (1.0d0 - sqrt(x)) * sqrt(2.0d0)
    else
        tmp = sqrt(x) - sqrt((2.0d0 * x))
    end if
    code = tmp
end function

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = (1.0 - Math.sqrt(x)) * Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt(x) - Math.sqrt((2.0 * x));
	}
	return tmp;
}

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = (1.0 - math.sqrt(x)) * math.sqrt(2.0)
	else:
		tmp = math.sqrt(x) - math.sqrt((2.0 * x))
	return tmp

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(1.0 - sqrt(x)) * sqrt(2.0));
	else
		tmp = Float64(sqrt(x) - sqrt(Float64(2.0 * x)));
	end
	return tmp
end

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (1.0 - sqrt(x)) * sqrt(2.0);
	else
		tmp = sqrt(x) - sqrt((2.0 * x));
	end
	tmp_2 = tmp;
end

code[x_] := If[LessEqual[x, 2.0], N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] - N[Sqrt[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq 2:\\
\;\;\;\;\left(1 - \sqrt{x}\right) \cdot \sqrt{2}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{x} - \sqrt{2 \cdot x}\\


\end{array}
\end{array}
Derivation
  1. Split input into 2 regimes

  2. if x < 2

    1. Initial program 99.9%

      \[\sqrt{x + 2} - \sqrt{2 \cdot x} \]
    2. Add Preprocessing

    3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\sqrt{2} - \sqrt{x} \cdot \sqrt{2}} \]
    4. Step-by-step derivation

      1. cancel-sign-sub-invN/A

        \[\leadsto \color{blue}{\sqrt{2} + \left(\mathsf{neg}\left(\sqrt{x}\right)\right) \cdot \sqrt{2}} \]

      2. neg-mul-1N/A

        \[\leadsto \sqrt{2} + \color{blue}{\left(-1 \cdot \sqrt{x}\right)} \cdot \sqrt{2} \]

      3. distribute-rgt1-inN/A

        \[\leadsto \color{blue}{\left(-1 \cdot \sqrt{x} + 1\right) \cdot \sqrt{2}} \]

      4. lower-*.f64N/A

        \[\leadsto \color{blue}{\left(-1 \cdot \sqrt{x} + 1\right) \cdot \sqrt{2}} \]

      5. +-commutativeN/A

        \[\leadsto \color{blue}{\left(1 + -1 \cdot \sqrt{x}\right)} \cdot \sqrt{2} \]

      6. neg-mul-1N/A

        \[\leadsto \left(1 + \color{blue}{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)}\right) \cdot \sqrt{2} \]

      7. unsub-negN/A

        \[\leadsto \color{blue}{\left(1 - \sqrt{x}\right)} \cdot \sqrt{2} \]

      8. lower--.f64N/A

        \[\leadsto \color{blue}{\left(1 - \sqrt{x}\right)} \cdot \sqrt{2} \]

      9. lower-sqrt.f64N/A

        \[\leadsto \left(1 - \color{blue}{\sqrt{x}}\right) \cdot \sqrt{2} \]

      10. lower-sqrt.f6497.7

        \[\leadsto \left(1 - \sqrt{x}\right) \cdot \color{blue}{\sqrt{2}} \]

    5. Applied rewrites97.7%

      \[\leadsto \color{blue}{\left(1 - \sqrt{x}\right) \cdot \sqrt{2}} \]

    if 2 < x

    1. Initial program 98.7%

      \[\sqrt{x + 2} - \sqrt{2 \cdot x} \]
    2. Add Preprocessing

    3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\sqrt{x}} - \sqrt{2 \cdot x} \]
    4. Step-by-step derivation

      1. lower-sqrt.f6497.2

        \[\leadsto \color{blue}{\sqrt{x}} - \sqrt{2 \cdot x} \]

    5. Applied rewrites97.2%

      \[\leadsto \color{blue}{\sqrt{x}} - \sqrt{2 \cdot x} \]
  3. Recombined 2 regimes into one program.
  4. Add Preprocessing

Alternative 4: 97.1% accurate, 0.9× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\left(1 - \sqrt{x}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \sqrt{2}\right) \cdot \sqrt{x}\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (if (<= x 2.0)
   (* (- 1.0 (sqrt x)) (sqrt 2.0))
   (* (- 1.0 (sqrt 2.0)) (sqrt x))))
double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = (1.0 - sqrt(x)) * sqrt(2.0);
	} else {
		tmp = (1.0 - sqrt(2.0)) * sqrt(x);
	}
	return tmp;
}

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = (1.0d0 - sqrt(x)) * sqrt(2.0d0)
    else
        tmp = (1.0d0 - sqrt(2.0d0)) * sqrt(x)
    end if
    code = tmp
end function

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = (1.0 - Math.sqrt(x)) * Math.sqrt(2.0);
	} else {
		tmp = (1.0 - Math.sqrt(2.0)) * Math.sqrt(x);
	}
	return tmp;
}

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = (1.0 - math.sqrt(x)) * math.sqrt(2.0)
	else:
		tmp = (1.0 - math.sqrt(2.0)) * math.sqrt(x)
	return tmp

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(1.0 - sqrt(x)) * sqrt(2.0));
	else
		tmp = Float64(Float64(1.0 - sqrt(2.0)) * sqrt(x));
	end
	return tmp
end

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (1.0 - sqrt(x)) * sqrt(2.0);
	else
		tmp = (1.0 - sqrt(2.0)) * sqrt(x);
	end
	tmp_2 = tmp;
end

code[x_] := If[LessEqual[x, 2.0], N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq 2:\\
\;\;\;\;\left(1 - \sqrt{x}\right) \cdot \sqrt{2}\\

\mathbf{else}:\\
\;\;\;\;\left(1 - \sqrt{2}\right) \cdot \sqrt{x}\\


\end{array}
\end{array}
Derivation
  1. Split input into 2 regimes

  2. if x < 2

    1. Initial program 99.9%

      \[\sqrt{x + 2} - \sqrt{2 \cdot x} \]
    2. Add Preprocessing

    3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\sqrt{2} - \sqrt{x} \cdot \sqrt{2}} \]
    4. Step-by-step derivation

      1. cancel-sign-sub-invN/A

        \[\leadsto \color{blue}{\sqrt{2} + \left(\mathsf{neg}\left(\sqrt{x}\right)\right) \cdot \sqrt{2}} \]

      2. neg-mul-1N/A

        \[\leadsto \sqrt{2} + \color{blue}{\left(-1 \cdot \sqrt{x}\right)} \cdot \sqrt{2} \]

      3. distribute-rgt1-inN/A

        \[\leadsto \color{blue}{\left(-1 \cdot \sqrt{x} + 1\right) \cdot \sqrt{2}} \]

      4. lower-*.f64N/A

        \[\leadsto \color{blue}{\left(-1 \cdot \sqrt{x} + 1\right) \cdot \sqrt{2}} \]

      5. +-commutativeN/A

        \[\leadsto \color{blue}{\left(1 + -1 \cdot \sqrt{x}\right)} \cdot \sqrt{2} \]

      6. neg-mul-1N/A

        \[\leadsto \left(1 + \color{blue}{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)}\right) \cdot \sqrt{2} \]

      7. unsub-negN/A

        \[\leadsto \color{blue}{\left(1 - \sqrt{x}\right)} \cdot \sqrt{2} \]

      8. lower--.f64N/A

        \[\leadsto \color{blue}{\left(1 - \sqrt{x}\right)} \cdot \sqrt{2} \]

      9. lower-sqrt.f64N/A

        \[\leadsto \left(1 - \color{blue}{\sqrt{x}}\right) \cdot \sqrt{2} \]

      10. lower-sqrt.f6497.7

        \[\leadsto \left(1 - \sqrt{x}\right) \cdot \color{blue}{\sqrt{2}} \]

    5. Applied rewrites97.7%

      \[\leadsto \color{blue}{\left(1 - \sqrt{x}\right) \cdot \sqrt{2}} \]

    if 2 < x

    1. Initial program 98.7%

      \[\sqrt{x + 2} - \sqrt{2 \cdot x} \]
    2. Add Preprocessing

    3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\sqrt{\frac{1}{x}} - \sqrt{\frac{1}{x}} \cdot \sqrt{2}\right)} \]
    4. Step-by-step derivation

      1. sub-negN/A

        \[\leadsto x \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \left(\mathsf{neg}\left(\sqrt{\frac{1}{x}} \cdot \sqrt{2}\right)\right)\right)} \]

      2. distribute-lft-inN/A

        \[\leadsto \color{blue}{x \cdot \sqrt{\frac{1}{x}} + x \cdot \left(\mathsf{neg}\left(\sqrt{\frac{1}{x}} \cdot \sqrt{2}\right)\right)} \]

      3. *-commutativeN/A

        \[\leadsto x \cdot \sqrt{\frac{1}{x}} + x \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1}{x}}}\right)\right) \]

      4. distribute-lft-neg-inN/A

        \[\leadsto x \cdot \sqrt{\frac{1}{x}} + x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{x}}\right)} \]

      5. associate-*r*N/A

        \[\leadsto x \cdot \sqrt{\frac{1}{x}} + \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right) \cdot \sqrt{\frac{1}{x}}} \]

      6. distribute-rgt-outN/A

        \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(x + x \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right)} \]

      7. lower-*.f64N/A

        \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(x + x \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right)} \]

      8. lower-sqrt.f64N/A

        \[\leadsto \color{blue}{\sqrt{\frac{1}{x}}} \cdot \left(x + x \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right) \]

      9. lower-/.f64N/A

        \[\leadsto \sqrt{\color{blue}{\frac{1}{x}}} \cdot \left(x + x \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right) \]

      10. distribute-rgt-neg-outN/A

        \[\leadsto \sqrt{\frac{1}{x}} \cdot \left(x + \color{blue}{\left(\mathsf{neg}\left(x \cdot \sqrt{2}\right)\right)}\right) \]

      11. unsub-negN/A

        \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{\left(x - x \cdot \sqrt{2}\right)} \]

      12. lower--.f64N/A

        \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{\left(x - x \cdot \sqrt{2}\right)} \]

      13. lower-*.f64N/A

        \[\leadsto \sqrt{\frac{1}{x}} \cdot \left(x - \color{blue}{x \cdot \sqrt{2}}\right) \]

      14. lower-sqrt.f6496.5

        \[\leadsto \sqrt{\frac{1}{x}} \cdot \left(x - x \cdot \color{blue}{\sqrt{2}}\right) \]

    5. Applied rewrites96.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(x - x \cdot \sqrt{2}\right)} \]

    6. Taylor expanded in x around 0

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(1 - \sqrt{2}\right)} \]
    7. Step-by-step derivation

      1. Applied rewrites96.6%

        \[\leadsto \left(1 - \sqrt{2}\right) \cdot \color{blue}{\sqrt{x}} \]
    8. Recombined 2 regimes into one program.
    9. Add Preprocessing

    Alternative 5: 99.3% accurate, 1.0× speedup?

    \[\begin{array}{l} \\ \sqrt{x + 2} - \sqrt{2 \cdot x} \end{array} \]
    (FPCore (x) :precision binary64 (- (sqrt (+ x 2.0)) (sqrt (* 2.0 x))))
    double code(double x) {
	return sqrt((x + 2.0)) - sqrt((2.0 * x));
}

    real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x + 2.0d0)) - sqrt((2.0d0 * x))
end function

    public static double code(double x) {
	return Math.sqrt((x + 2.0)) - Math.sqrt((2.0 * x));
}

    def code(x):
	return math.sqrt((x + 2.0)) - math.sqrt((2.0 * x))

    function code(x)
	return Float64(sqrt(Float64(x + 2.0)) - sqrt(Float64(2.0 * x)))
end

    function tmp = code(x)
	tmp = sqrt((x + 2.0)) - sqrt((2.0 * x));
end

    code[x_] := N[(N[Sqrt[N[(x + 2.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

    \begin{array}{l}

\\
\sqrt{x + 2} - \sqrt{2 \cdot x}
\end{array}
    Derivation

    1. Initial program 99.3%

      \[\sqrt{x + 2} - \sqrt{2 \cdot x} \]
    2. Add Preprocessing
    3. Add Preprocessing

    Alternative 6: 49.1% accurate, 1.1× speedup?

    \[\begin{array}{l} \\ \left(1 - \sqrt{2}\right) \cdot \sqrt{x} \end{array} \]
    (FPCore (x) :precision binary64 (* (- 1.0 (sqrt 2.0)) (sqrt x)))
    double code(double x) {
	return (1.0 - sqrt(2.0)) * sqrt(x);
}

    real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - sqrt(2.0d0)) * sqrt(x)
end function

    public static double code(double x) {
	return (1.0 - Math.sqrt(2.0)) * Math.sqrt(x);
}

    def code(x):
	return (1.0 - math.sqrt(2.0)) * math.sqrt(x)

    function code(x)
	return Float64(Float64(1.0 - sqrt(2.0)) * sqrt(x))
end

    function tmp = code(x)
	tmp = (1.0 - sqrt(2.0)) * sqrt(x);
end

    code[x_] := N[(N[(1.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

    \begin{array}{l}

\\
\left(1 - \sqrt{2}\right) \cdot \sqrt{x}
\end{array}
    Derivation

    1. Initial program 99.3%

      \[\sqrt{x + 2} - \sqrt{2 \cdot x} \]
    2. Add Preprocessing

    3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\sqrt{\frac{1}{x}} - \sqrt{\frac{1}{x}} \cdot \sqrt{2}\right)} \]
    4. Step-by-step derivation

      1. sub-negN/A

        \[\leadsto x \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \left(\mathsf{neg}\left(\sqrt{\frac{1}{x}} \cdot \sqrt{2}\right)\right)\right)} \]

      2. distribute-lft-inN/A

        \[\leadsto \color{blue}{x \cdot \sqrt{\frac{1}{x}} + x \cdot \left(\mathsf{neg}\left(\sqrt{\frac{1}{x}} \cdot \sqrt{2}\right)\right)} \]

      3. *-commutativeN/A

        \[\leadsto x \cdot \sqrt{\frac{1}{x}} + x \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1}{x}}}\right)\right) \]

      4. distribute-lft-neg-inN/A

        \[\leadsto x \cdot \sqrt{\frac{1}{x}} + x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{x}}\right)} \]

      5. associate-*r*N/A

        \[\leadsto x \cdot \sqrt{\frac{1}{x}} + \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right) \cdot \sqrt{\frac{1}{x}}} \]

      6. distribute-rgt-outN/A

        \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(x + x \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right)} \]

      7. lower-*.f64N/A

        \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(x + x \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right)} \]

      8. lower-sqrt.f64N/A

        \[\leadsto \color{blue}{\sqrt{\frac{1}{x}}} \cdot \left(x + x \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right) \]

      9. lower-/.f64N/A

        \[\leadsto \sqrt{\color{blue}{\frac{1}{x}}} \cdot \left(x + x \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right) \]

      10. distribute-rgt-neg-outN/A

        \[\leadsto \sqrt{\frac{1}{x}} \cdot \left(x + \color{blue}{\left(\mathsf{neg}\left(x \cdot \sqrt{2}\right)\right)}\right) \]

      11. unsub-negN/A

        \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{\left(x - x \cdot \sqrt{2}\right)} \]

      12. lower--.f64N/A

        \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{\left(x - x \cdot \sqrt{2}\right)} \]

      13. lower-*.f64N/A

        \[\leadsto \sqrt{\frac{1}{x}} \cdot \left(x - \color{blue}{x \cdot \sqrt{2}}\right) \]

      14. lower-sqrt.f6449.5

        \[\leadsto \sqrt{\frac{1}{x}} \cdot \left(x - x \cdot \color{blue}{\sqrt{2}}\right) \]

    5. Applied rewrites49.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(x - x \cdot \sqrt{2}\right)} \]

    6. Taylor expanded in x around 0

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(1 - \sqrt{2}\right)} \]
    7. Step-by-step derivation

      1. Applied rewrites49.6%

        \[\leadsto \left(1 - \sqrt{2}\right) \cdot \color{blue}{\sqrt{x}} \]
      2. Add Preprocessing

      Alternative 7: 9.7% accurate, 1.1× speedup?

      \[\begin{array}{l} \\ \left(-\sqrt{x}\right) \cdot \sqrt{2} \end{array} \]
      (FPCore (x) :precision binary64 (* (- (sqrt x)) (sqrt 2.0)))
      double code(double x) {
	return -sqrt(x) * sqrt(2.0);
}

      real(8) function code(x)
    real(8), intent (in) :: x
    code = -sqrt(x) * sqrt(2.0d0)
end function

      public static double code(double x) {
	return -Math.sqrt(x) * Math.sqrt(2.0);
}

      def code(x):
	return -math.sqrt(x) * math.sqrt(2.0)

      function code(x)
	return Float64(Float64(-sqrt(x)) * sqrt(2.0))
end

      function tmp = code(x)
	tmp = -sqrt(x) * sqrt(2.0);
end

      code[x_] := N[((-N[Sqrt[x], $MachinePrecision]) * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

      \begin{array}{l}

\\
\left(-\sqrt{x}\right) \cdot \sqrt{2}
\end{array}
      Derivation

      1. Initial program 99.3%

        \[\sqrt{x + 2} - \sqrt{2 \cdot x} \]
      2. Add Preprocessing

      3. Taylor expanded in x around 0

        \[\leadsto \color{blue}{\sqrt{2} - \sqrt{x} \cdot \sqrt{2}} \]
      4. Step-by-step derivation

        1. cancel-sign-sub-invN/A

          \[\leadsto \color{blue}{\sqrt{2} + \left(\mathsf{neg}\left(\sqrt{x}\right)\right) \cdot \sqrt{2}} \]

        2. neg-mul-1N/A

          \[\leadsto \sqrt{2} + \color{blue}{\left(-1 \cdot \sqrt{x}\right)} \cdot \sqrt{2} \]

        3. distribute-rgt1-inN/A

          \[\leadsto \color{blue}{\left(-1 \cdot \sqrt{x} + 1\right) \cdot \sqrt{2}} \]

        4. lower-*.f64N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \sqrt{x} + 1\right) \cdot \sqrt{2}} \]

        5. +-commutativeN/A

          \[\leadsto \color{blue}{\left(1 + -1 \cdot \sqrt{x}\right)} \cdot \sqrt{2} \]

        6. neg-mul-1N/A

          \[\leadsto \left(1 + \color{blue}{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)}\right) \cdot \sqrt{2} \]

        7. unsub-negN/A

          \[\leadsto \color{blue}{\left(1 - \sqrt{x}\right)} \cdot \sqrt{2} \]

        8. lower--.f64N/A

          \[\leadsto \color{blue}{\left(1 - \sqrt{x}\right)} \cdot \sqrt{2} \]

        9. lower-sqrt.f64N/A

          \[\leadsto \left(1 - \color{blue}{\sqrt{x}}\right) \cdot \sqrt{2} \]

        10. lower-sqrt.f6457.3

          \[\leadsto \left(1 - \sqrt{x}\right) \cdot \color{blue}{\sqrt{2}} \]

      5. Applied rewrites57.3%

        \[\leadsto \color{blue}{\left(1 - \sqrt{x}\right) \cdot \sqrt{2}} \]

      6. Taylor expanded in x around inf

        \[\leadsto \left(-1 \cdot \sqrt{x}\right) \cdot \sqrt{\color{blue}{2}} \]
      7. Step-by-step derivation

        1. Applied rewrites9.7%

          \[\leadsto \left(-\sqrt{x}\right) \cdot \sqrt{\color{blue}{2}} \]
        2. Add Preprocessing

        Reproduce

        ?
        herbie shell --seed 1 
(FPCore (x)
  :name "sqrt(x + 2) - sqrt(2x)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 1.79e+308))
  (- (sqrt (+ x 2.0)) (sqrt (* 2.0 x))))