Ian Simplification

Percentage Accurate: 6.9% → 8.3%

Time: 14.3s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 8.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{0.5 - 0.5 \cdot x}\\ \frac{\frac{\pi \cdot \left(-\pi\right)}{-4} - {\left(2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} t\_0\right)\right)}^{2}}{\mathsf{fma}\left(2, \sin^{-1} t\_0, \pi \cdot 0.5\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (- 0.5 (* 0.5 x)))))
   (/
    (- (/ (* PI (- PI)) -4.0) (pow (* 2.0 (- (* PI 0.5) (acos t_0))) 2.0))
    (fma 2.0 (asin t_0) (* PI 0.5)))))

C

double code(double x) {
	double t_0 = sqrt((0.5 - (0.5 * x)));
	return (((((double) M_PI) * -((double) M_PI)) / -4.0) - pow((2.0 * ((((double) M_PI) * 0.5) - acos(t_0))), 2.0)) / fma(2.0, asin(t_0), (((double) M_PI) * 0.5));
}

Julia

function code(x)
	t_0 = sqrt(Float64(0.5 - Float64(0.5 * x)))
	return Float64(Float64(Float64(Float64(pi * Float64(-pi)) / -4.0) - (Float64(2.0 * Float64(Float64(pi * 0.5) - acos(t_0))) ^ 2.0)) / fma(2.0, asin(t_0), Float64(pi * 0.5)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(Pi * (-Pi)), $MachinePrecision] / -4.0), $MachinePrecision] - N[Power[N[(2.0 * N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[ArcSin[t$95$0], $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.3%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 7.3%

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

   2. pow2 7.3%

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

   3. div-inv 7.3%

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

   4. metadata-eval 7.3%

      \[\leadsto \frac{{\left(\pi \cdot \color{blue}{0.5}\right)}^{2} - \left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \cdot \left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)}{\frac{\pi}{2} + 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \]

   5. pow2 7.3%

      \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{2} - \color{blue}{{\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)}^{2}}}{\frac{\pi}{2} + 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \]

   6. div-sub 7.3%

      \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{2} - \frac{x}{2}}}\right)\right)}^{2}}{\frac{\pi}{2} + 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \]

   7. metadata-eval 7.3%

      \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{\color{blue}{0.5} - \frac{x}{2}}\right)\right)}^{2}}{\frac{\pi}{2} + 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \]

   8. div-inv 7.3%

      \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - \color{blue}{x \cdot \frac{1}{2}}}\right)\right)}^{2}}{\frac{\pi}{2} + 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \]

   9. metadata-eval 7.3%

      \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot \color{blue}{0.5}}\right)\right)}^{2}}{\frac{\pi}{2} + 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \]

   10. +-commutative 7.3%

       \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}{\color{blue}{2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) + \frac{\pi}{2}}} \]

4. Applied egg-rr 7.3%

   \[\leadsto \color{blue}{\frac{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}} \]
5. Step-by-step derivation

   1. metadata-eval 7.3%

      \[\leadsto \frac{{\left(\pi \cdot \color{blue}{\frac{1}{2}}\right)}^{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   2. div-inv 7.3%

      \[\leadsto \frac{{\color{blue}{\left(\frac{\pi}{2}\right)}}^{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   3. pow2 7.3%

      \[\leadsto \frac{\color{blue}{\frac{\pi}{2} \cdot \frac{\pi}{2}} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   4. frac-2neg 7.3%

      \[\leadsto \frac{\color{blue}{\frac{-\pi}{-2}} \cdot \frac{\pi}{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   5. metadata-eval 7.3%

      \[\leadsto \frac{\frac{-\pi}{\color{blue}{-2}} \cdot \frac{\pi}{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   6. frac-times 7.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(-\pi\right) \cdot \pi}{-2 \cdot 2}} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   7. metadata-eval 7.3%

      \[\leadsto \frac{\frac{\left(-\pi\right) \cdot \pi}{\color{blue}{-4}} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

6. Applied egg-rr 7.3%

   \[\leadsto \frac{\color{blue}{\frac{\left(-\pi\right) \cdot \pi}{-4}} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]
7. Step-by-step derivation

   1. asin-acos 9.1%

      \[\leadsto \frac{\frac{\left(-\pi\right) \cdot \pi}{-4} - {\left(2 \cdot \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}\right)}^{2}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   2. div-inv 9.1%

      \[\leadsto \frac{\frac{\left(-\pi\right) \cdot \pi}{-4} - {\left(2 \cdot \left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)\right)}^{2}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   3. metadata-eval 9.1%

      \[\leadsto \frac{\frac{\left(-\pi\right) \cdot \pi}{-4} - {\left(2 \cdot \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)\right)}^{2}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   4. *-commutative 9.1%

      \[\leadsto \frac{\frac{\left(-\pi\right) \cdot \pi}{-4} - {\left(2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - \color{blue}{0.5 \cdot x}}\right)\right)\right)}^{2}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

8. Applied egg-rr 9.1%

   \[\leadsto \frac{\frac{\left(-\pi\right) \cdot \pi}{-4} - {\left(2 \cdot \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}\right)}^{2}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

9. Final simplification 9.1%

   \[\leadsto \frac{\frac{\pi \cdot \left(-\pi\right)}{-4} - {\left(2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}^{2}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right), \pi \cdot 0.5\right)} \]
10. Add Preprocessing

Alternative 2: 5.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-300}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - -2 \cdot \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e-300)
   (- (/ PI 2.0) (* 2.0 (asin (sqrt 0.5))))
   (- (/ PI 2.0) (* -2.0 (asin (sqrt (+ 0.5 (* x -0.5))))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e-300) {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(0.5)));
	} else {
		tmp = (((double) M_PI) / 2.0) - (-2.0 * asin(sqrt((0.5 + (x * -0.5)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e-300) {
		tmp = (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(0.5)));
	} else {
		tmp = (Math.PI / 2.0) - (-2.0 * Math.asin(Math.sqrt((0.5 + (x * -0.5)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.35e-300:
		tmp = (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(0.5)))
	else:
		tmp = (math.pi / 2.0) - (-2.0 * math.asin(math.sqrt((0.5 + (x * -0.5)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e-300)
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(0.5))));
	else
		tmp = Float64(Float64(pi / 2.0) - Float64(-2.0 * asin(sqrt(Float64(0.5 + Float64(x * -0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.35e-300)
		tmp = (pi / 2.0) - (2.0 * asin(sqrt(0.5)));
	else
		tmp = (pi / 2.0) - (-2.0 * asin(sqrt((0.5 + (x * -0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e-300], N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / 2.0), $MachinePrecision] - N[(-2.0 * N[ArcSin[N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.34999999999999998e-300

   1. Initial program 8.2%

      \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 5.7%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \color{blue}{\left(\sqrt{0.5}\right)} \]

   if 1.34999999999999998e-300 < x

   1. Initial program 6.4%

      \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 6.4%

         \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\log \left(e^{\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)} \]

      2. div-sub 6.4%

         \[\leadsto \frac{\pi}{2} - 2 \cdot \log \left(e^{\sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{2} - \frac{x}{2}}}\right)}\right) \]

      3. metadata-eval 6.4%

         \[\leadsto \frac{\pi}{2} - 2 \cdot \log \left(e^{\sin^{-1} \left(\sqrt{\color{blue}{0.5} - \frac{x}{2}}\right)}\right) \]

      4. div-inv 6.4%

         \[\leadsto \frac{\pi}{2} - 2 \cdot \log \left(e^{\sin^{-1} \left(\sqrt{0.5 - \color{blue}{x \cdot \frac{1}{2}}}\right)}\right) \]

      5. metadata-eval 6.4%

         \[\leadsto \frac{\pi}{2} - 2 \cdot \log \left(e^{\sin^{-1} \left(\sqrt{0.5 - x \cdot \color{blue}{0.5}}\right)}\right) \]

   4. Applied egg-rr 6.4%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\log \left(e^{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right)} \]
   5. Step-by-step derivation

      1. metadata-eval 6.4%

         \[\leadsto \frac{\pi}{2} - \color{blue}{\sqrt{4}} \cdot \log \left(e^{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right) \]

      2. metadata-eval 6.4%

         \[\leadsto \frac{\pi}{2} - \sqrt{\color{blue}{{2}^{2}}} \cdot \log \left(e^{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right) \]

      3. rem-log-exp 6.4%

         \[\leadsto \frac{\pi}{2} - \sqrt{{2}^{2}} \cdot \color{blue}{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)} \]

      4. add-sqr-sqrt 6.5%

         \[\leadsto \frac{\pi}{2} - \sqrt{{2}^{2}} \cdot \color{blue}{\left(\sqrt{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)} \cdot \sqrt{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right)} \]

      5. sqrt-unprod 6.4%

         \[\leadsto \frac{\pi}{2} - \sqrt{{2}^{2}} \cdot \color{blue}{\sqrt{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}} \]

      6. pow2 6.4%

         \[\leadsto \frac{\pi}{2} - \sqrt{{2}^{2}} \cdot \sqrt{\color{blue}{{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}}} \]

      7. sqrt-prod 6.4%

         \[\leadsto \frac{\pi}{2} - \color{blue}{\sqrt{{2}^{2} \cdot {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}}} \]

      8. *-commutative 6.4%

         \[\leadsto \frac{\pi}{2} - \sqrt{\color{blue}{{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2} \cdot {2}^{2}}} \]

      9. pow2 6.4%

         \[\leadsto \frac{\pi}{2} - \sqrt{\color{blue}{\left(\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)} \cdot {2}^{2}} \]

      10. metadata-eval 6.4%

          \[\leadsto \frac{\pi}{2} - \sqrt{\left(\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \cdot \color{blue}{4}} \]

      11. metadata-eval 6.4%

          \[\leadsto \frac{\pi}{2} - \sqrt{\left(\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \cdot \color{blue}{\left(-2 \cdot -2\right)}} \]

      12. swap-sqr 6.4%

          \[\leadsto \frac{\pi}{2} - \sqrt{\color{blue}{\left(\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)}} \]

      13. sqrt-unprod 0.0%

          \[\leadsto \frac{\pi}{2} - \color{blue}{\sqrt{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2} \cdot \sqrt{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2}} \]

      14. add-sqr-sqrt 5.6%

          \[\leadsto \frac{\pi}{2} - \color{blue}{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2} \]

      15. metadata-eval 5.6%

          \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \color{blue}{\left(-2\right)} \]

      16. distribute-rgt-neg-in 5.6%

          \[\leadsto \frac{\pi}{2} - \color{blue}{\left(-\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot 2\right)} \]

   6. Applied egg-rr 5.6%

      \[\leadsto \frac{\pi}{2} - \color{blue}{\left(0 - 2 \cdot \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)} \]
   7. Step-by-step derivation

      1. neg-sub0 5.6%

         \[\leadsto \frac{\pi}{2} - \color{blue}{\left(-2 \cdot \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)} \]

      2. distribute-lft-neg-in 5.6%

         \[\leadsto \frac{\pi}{2} - \color{blue}{\left(-2\right) \cdot \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)} \]

      3. metadata-eval 5.6%

         \[\leadsto \frac{\pi}{2} - \color{blue}{-2} \cdot \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right) \]

   8. Simplified 5.6%

      \[\leadsto \frac{\pi}{2} - \color{blue}{-2 \cdot \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 8.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} + 2 \cdot \left(\cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) - \pi \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (/ PI 2.0) (* 2.0 (- (acos (sqrt (- 0.5 (* 0.5 x)))) (* PI 0.5)))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) + (2.0 * (acos(sqrt((0.5 - (0.5 * x)))) - (((double) M_PI) * 0.5)));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) + (2.0 * (Math.acos(Math.sqrt((0.5 - (0.5 * x)))) - (Math.PI * 0.5)));
}

Python

def code(x):
	return (math.pi / 2.0) + (2.0 * (math.acos(math.sqrt((0.5 - (0.5 * x)))) - (math.pi * 0.5)))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) + Float64(2.0 * Float64(acos(sqrt(Float64(0.5 - Float64(0.5 * x)))) - Float64(pi * 0.5))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) + (2.0 * (acos(sqrt((0.5 - (0.5 * x)))) - (pi * 0.5)));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] + N[(2.0 * N[(N[ArcCos[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] - N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. asin-acos 9.1%

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

   2. div-inv 9.1%

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

   3. metadata-eval 9.1%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \]

   4. div-sub 9.1%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{\frac{1}{2} - \frac{x}{2}}}\right)\right) \]

   5. metadata-eval 9.1%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5} - \frac{x}{2}}\right)\right) \]

   6. div-inv 9.1%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - \color{blue}{x \cdot \frac{1}{2}}}\right)\right) \]

   7. metadata-eval 9.1%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - x \cdot \color{blue}{0.5}}\right)\right) \]

4. Applied egg-rr 9.1%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)} \]

5. Final simplification 9.1%

   \[\leadsto \frac{\pi}{2} + 2 \cdot \left(\cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) - \pi \cdot 0.5\right) \]
6. Add Preprocessing

Alternative 4: 6.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{1}{\sqrt{\frac{2}{1 - x}}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (/ 1.0 (sqrt (/ 2.0 (- 1.0 x))))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin((1.0 / sqrt((2.0 / (1.0 - x))))));
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.3%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num 7.3%

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

   2. sqrt-div 7.5%

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

   3. metadata-eval 7.5%

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

4. Applied egg-rr 7.5%

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

Alternative 5: 5.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(\sqrt{0.5}\right)\\ \mathbf{if}\;x \leq 1.35 \cdot 10^{-300}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - t\_0 \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (sqrt 0.5))))
   (if (<= x 1.35e-300)
     (- (/ PI 2.0) (* 2.0 t_0))
     (- (/ PI 2.0) (* t_0 -2.0)))))

C

double code(double x) {
	double t_0 = asin(sqrt(0.5));
	double tmp;
	if (x <= 1.35e-300) {
		tmp = (((double) M_PI) / 2.0) - (2.0 * t_0);
	} else {
		tmp = (((double) M_PI) / 2.0) - (t_0 * -2.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.asin(Math.sqrt(0.5));
	double tmp;
	if (x <= 1.35e-300) {
		tmp = (Math.PI / 2.0) - (2.0 * t_0);
	} else {
		tmp = (Math.PI / 2.0) - (t_0 * -2.0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.asin(math.sqrt(0.5))
	tmp = 0
	if x <= 1.35e-300:
		tmp = (math.pi / 2.0) - (2.0 * t_0)
	else:
		tmp = (math.pi / 2.0) - (t_0 * -2.0)
	return tmp

Julia

function code(x)
	t_0 = asin(sqrt(0.5))
	tmp = 0.0
	if (x <= 1.35e-300)
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * t_0));
	else
		tmp = Float64(Float64(pi / 2.0) - Float64(t_0 * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = asin(sqrt(0.5));
	tmp = 0.0;
	if (x <= 1.35e-300)
		tmp = (pi / 2.0) - (2.0 * t_0);
	else
		tmp = (pi / 2.0) - (t_0 * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.35e-300], N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / 2.0), $MachinePrecision] - N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.34999999999999998e-300

   1. Initial program 8.2%

      \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 5.7%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \color{blue}{\left(\sqrt{0.5}\right)} \]

   if 1.34999999999999998e-300 < x

   1. Initial program 6.4%

      \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 6.4%

         \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\log \left(e^{\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)} \]

      2. div-sub 6.4%

         \[\leadsto \frac{\pi}{2} - 2 \cdot \log \left(e^{\sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{2} - \frac{x}{2}}}\right)}\right) \]

      3. metadata-eval 6.4%

         \[\leadsto \frac{\pi}{2} - 2 \cdot \log \left(e^{\sin^{-1} \left(\sqrt{\color{blue}{0.5} - \frac{x}{2}}\right)}\right) \]

      4. div-inv 6.4%

         \[\leadsto \frac{\pi}{2} - 2 \cdot \log \left(e^{\sin^{-1} \left(\sqrt{0.5 - \color{blue}{x \cdot \frac{1}{2}}}\right)}\right) \]

      5. metadata-eval 6.4%

         \[\leadsto \frac{\pi}{2} - 2 \cdot \log \left(e^{\sin^{-1} \left(\sqrt{0.5 - x \cdot \color{blue}{0.5}}\right)}\right) \]

   4. Applied egg-rr 6.4%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\log \left(e^{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right)} \]
   5. Step-by-step derivation

      1. metadata-eval 6.4%

         \[\leadsto \frac{\pi}{2} - \color{blue}{\sqrt{4}} \cdot \log \left(e^{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right) \]

      2. metadata-eval 6.4%

         \[\leadsto \frac{\pi}{2} - \sqrt{\color{blue}{{2}^{2}}} \cdot \log \left(e^{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right) \]

      3. rem-log-exp 6.4%

         \[\leadsto \frac{\pi}{2} - \sqrt{{2}^{2}} \cdot \color{blue}{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)} \]

      4. add-sqr-sqrt 6.5%

         \[\leadsto \frac{\pi}{2} - \sqrt{{2}^{2}} \cdot \color{blue}{\left(\sqrt{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)} \cdot \sqrt{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right)} \]

      5. sqrt-unprod 6.4%

         \[\leadsto \frac{\pi}{2} - \sqrt{{2}^{2}} \cdot \color{blue}{\sqrt{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}} \]

      6. pow2 6.4%

         \[\leadsto \frac{\pi}{2} - \sqrt{{2}^{2}} \cdot \sqrt{\color{blue}{{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}}} \]

      7. sqrt-prod 6.4%

         \[\leadsto \frac{\pi}{2} - \color{blue}{\sqrt{{2}^{2} \cdot {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}}} \]

      8. *-commutative 6.4%

         \[\leadsto \frac{\pi}{2} - \sqrt{\color{blue}{{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2} \cdot {2}^{2}}} \]

      9. pow2 6.4%

         \[\leadsto \frac{\pi}{2} - \sqrt{\color{blue}{\left(\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)} \cdot {2}^{2}} \]

      10. metadata-eval 6.4%

          \[\leadsto \frac{\pi}{2} - \sqrt{\left(\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \cdot \color{blue}{4}} \]

      11. metadata-eval 6.4%

          \[\leadsto \frac{\pi}{2} - \sqrt{\left(\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \cdot \color{blue}{\left(-2 \cdot -2\right)}} \]

      12. swap-sqr 6.4%

          \[\leadsto \frac{\pi}{2} - \sqrt{\color{blue}{\left(\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)}} \]

      13. sqrt-unprod 0.0%

          \[\leadsto \frac{\pi}{2} - \color{blue}{\sqrt{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2} \cdot \sqrt{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2}} \]

      14. add-sqr-sqrt 5.6%

          \[\leadsto \frac{\pi}{2} - \color{blue}{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2} \]

      15. metadata-eval 5.6%

          \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \color{blue}{\left(-2\right)} \]

      16. distribute-rgt-neg-in 5.6%

          \[\leadsto \frac{\pi}{2} - \color{blue}{\left(-\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot 2\right)} \]

   6. Applied egg-rr 5.6%

      \[\leadsto \frac{\pi}{2} - \color{blue}{\left(0 - 2 \cdot \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)} \]
   7. Step-by-step derivation

      1. neg-sub0 5.6%

         \[\leadsto \frac{\pi}{2} - \color{blue}{\left(-2 \cdot \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)} \]

      2. distribute-lft-neg-in 5.6%

         \[\leadsto \frac{\pi}{2} - \color{blue}{\left(-2\right) \cdot \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)} \]

      3. metadata-eval 5.6%

         \[\leadsto \frac{\pi}{2} - \color{blue}{-2} \cdot \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right) \]

   8. Simplified 5.6%

      \[\leadsto \frac{\pi}{2} - \color{blue}{-2 \cdot \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)} \]

   9. Taylor expanded in x around 0 5.6%

      \[\leadsto \frac{\pi}{2} - -2 \cdot \sin^{-1} \color{blue}{\left(\sqrt{0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-300}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - \sin^{-1} \left(\sqrt{0.5}\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 6.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.3%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 7: 3.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - \sin^{-1} \left(\sqrt{0.5}\right) \cdot -2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ PI 2.0) (* (asin (sqrt 0.5)) -2.0)))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (asin(sqrt(0.5)) * -2.0);
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (Math.asin(Math.sqrt(0.5)) * -2.0);
}

Python

def code(x):
	return (math.pi / 2.0) - (math.asin(math.sqrt(0.5)) * -2.0)

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(asin(sqrt(0.5)) * -2.0))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (asin(sqrt(0.5)) * -2.0);
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 7.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\log \left(e^{\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)} \]

   2. div-sub 7.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \log \left(e^{\sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{2} - \frac{x}{2}}}\right)}\right) \]

   3. metadata-eval 7.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \log \left(e^{\sin^{-1} \left(\sqrt{\color{blue}{0.5} - \frac{x}{2}}\right)}\right) \]

   4. div-inv 7.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \log \left(e^{\sin^{-1} \left(\sqrt{0.5 - \color{blue}{x \cdot \frac{1}{2}}}\right)}\right) \]

   5. metadata-eval 7.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \log \left(e^{\sin^{-1} \left(\sqrt{0.5 - x \cdot \color{blue}{0.5}}\right)}\right) \]

4. Applied egg-rr 7.3%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\log \left(e^{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right)} \]
5. Step-by-step derivation

   1. metadata-eval 7.3%

      \[\leadsto \frac{\pi}{2} - \color{blue}{\sqrt{4}} \cdot \log \left(e^{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right) \]

   2. metadata-eval 7.3%

      \[\leadsto \frac{\pi}{2} - \sqrt{\color{blue}{{2}^{2}}} \cdot \log \left(e^{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right) \]

   3. rem-log-exp 7.3%

      \[\leadsto \frac{\pi}{2} - \sqrt{{2}^{2}} \cdot \color{blue}{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)} \]

   4. add-sqr-sqrt 7.3%

      \[\leadsto \frac{\pi}{2} - \sqrt{{2}^{2}} \cdot \color{blue}{\left(\sqrt{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)} \cdot \sqrt{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}\right)} \]

   5. sqrt-unprod 7.3%

      \[\leadsto \frac{\pi}{2} - \sqrt{{2}^{2}} \cdot \color{blue}{\sqrt{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}} \]

   6. pow2 7.3%

      \[\leadsto \frac{\pi}{2} - \sqrt{{2}^{2}} \cdot \sqrt{\color{blue}{{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}}} \]

   7. sqrt-prod 7.3%

      \[\leadsto \frac{\pi}{2} - \color{blue}{\sqrt{{2}^{2} \cdot {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}}} \]

   8. *-commutative 7.3%

      \[\leadsto \frac{\pi}{2} - \sqrt{\color{blue}{{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2} \cdot {2}^{2}}} \]

   9. pow2 7.3%

      \[\leadsto \frac{\pi}{2} - \sqrt{\color{blue}{\left(\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)} \cdot {2}^{2}} \]

   10. metadata-eval 7.3%

       \[\leadsto \frac{\pi}{2} - \sqrt{\left(\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \cdot \color{blue}{4}} \]

   11. metadata-eval 7.3%

       \[\leadsto \frac{\pi}{2} - \sqrt{\left(\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \cdot \color{blue}{\left(-2 \cdot -2\right)}} \]

   12. swap-sqr 7.3%

       \[\leadsto \frac{\pi}{2} - \sqrt{\color{blue}{\left(\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)}} \]

   13. sqrt-unprod 0.0%

       \[\leadsto \frac{\pi}{2} - \color{blue}{\sqrt{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2} \cdot \sqrt{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2}} \]

   14. add-sqr-sqrt 4.0%

       \[\leadsto \frac{\pi}{2} - \color{blue}{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2} \]

   15. metadata-eval 4.0%

       \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \color{blue}{\left(-2\right)} \]

   16. distribute-rgt-neg-in 4.0%

       \[\leadsto \frac{\pi}{2} - \color{blue}{\left(-\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot 2\right)} \]

6. Applied egg-rr 4.0%

   \[\leadsto \frac{\pi}{2} - \color{blue}{\left(0 - 2 \cdot \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)} \]
7. Step-by-step derivation

   1. neg-sub0 4.0%

      \[\leadsto \frac{\pi}{2} - \color{blue}{\left(-2 \cdot \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)} \]

   2. distribute-lft-neg-in 4.0%

      \[\leadsto \frac{\pi}{2} - \color{blue}{\left(-2\right) \cdot \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)} \]

   3. metadata-eval 4.0%

      \[\leadsto \frac{\pi}{2} - \color{blue}{-2} \cdot \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right) \]

8. Simplified 4.0%

   \[\leadsto \frac{\pi}{2} - \color{blue}{-2 \cdot \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)} \]

9. Taylor expanded in x around 0 4.0%

   \[\leadsto \frac{\pi}{2} - -2 \cdot \sin^{-1} \color{blue}{\left(\sqrt{0.5}\right)} \]

10. Final simplification 4.0%

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(\sqrt{0.5}\right) \cdot -2 \]
11. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (asin x))

C

double code(double x) {
	return asin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = asin(x)
end function

Java

public static double code(double x) {
	return Math.asin(x);
}

Python

def code(x):
	return math.asin(x)

Julia

function code(x)
	return asin(x)
end

MATLAB

function tmp = code(x)
	tmp = asin(x);
end

Wolfram

code[x_] := N[ArcSin[x], $MachinePrecision]

Reproduce

herbie shell --seed 2024141 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :alt
  (! :herbie-platform default (asin x))

  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))