Ian Simplification

Percentage Accurate: 7.2% → 8.6%

Time: 24.0s

Alternatives: 8

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: 7.2% 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: 8.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{0.5 + x \cdot -0.5}\\ t_1 := \cos^{-1} t_0\\ t_2 := \mathsf{fma}\left(\pi, 0.5, t_1\right)\\ \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} t_0}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{0.25 \cdot {\left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)}^{2}}{t_2} - \frac{{t_1}^{2}}{t_2}, \pi \cdot 0.5\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 0.5 (* x -0.5))))
        (t_1 (acos t_0))
        (t_2 (fma PI 0.5 t_1)))
   (/
    (- (* (pow PI 2.0) 0.25) (* (cbrt (pow (pow (asin t_0) 2.0) 3.0)) 4.0))
    (fma
     2.0
     (-
      (/ (* 0.25 (pow (* (cbrt PI) (pow (cbrt PI) 2.0)) 2.0)) t_2)
      (/ (pow t_1 2.0) t_2))
     (* PI 0.5)))))

C

double code(double x) {
	double t_0 = sqrt((0.5 + (x * -0.5)));
	double t_1 = acos(t_0);
	double t_2 = fma(((double) M_PI), 0.5, t_1);
	return ((pow(((double) M_PI), 2.0) * 0.25) - (cbrt(pow(pow(asin(t_0), 2.0), 3.0)) * 4.0)) / fma(2.0, (((0.25 * pow((cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0)), 2.0)) / t_2) - (pow(t_1, 2.0) / t_2)), (((double) M_PI) * 0.5));
}

Julia

function code(x)
	t_0 = sqrt(Float64(0.5 + Float64(x * -0.5)))
	t_1 = acos(t_0)
	t_2 = fma(pi, 0.5, t_1)
	return Float64(Float64(Float64((pi ^ 2.0) * 0.25) - Float64(cbrt(((asin(t_0) ^ 2.0) ^ 3.0)) * 4.0)) / fma(2.0, Float64(Float64(Float64(0.25 * (Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)) ^ 2.0)) / t_2) - Float64((t_1 ^ 2.0) / t_2)), Float64(pi * 0.5)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcCos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(Pi * 0.5 + t$95$1), $MachinePrecision]}, N[(N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.25), $MachinePrecision] - N[(N[Power[N[Power[N[Power[N[ArcSin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(N[(N[(0.25 * N[Power[N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] - N[(N[Power[t$95$1, 2.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 5.7%

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

   1. flip-- 5.7%

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

3. Applied egg-rr 5.7%

   \[\leadsto \color{blue}{\frac{{\pi}^{2} \cdot 0.25 - {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}} \]
4. Step-by-step derivation

   1. add-cbrt-cube 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \color{blue}{\sqrt[3]{\left({\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2} \cdot {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}\right) \cdot {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   2. pow3 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{\color{blue}{{\left({\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}\right)}^{3}}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   3. sub-neg 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-x \cdot 0.5\right)}}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   4. distribute-rgt-neg-in 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot \left(-0.5\right)}}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   5. metadata-eval 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot \color{blue}{-0.5}}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

5. Applied egg-rr 7.5%

   \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \color{blue}{\sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]
6. Step-by-step derivation

   1. asin-acos 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \color{blue}{\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, \pi \cdot 0.5\right)} \]

   2. flip-- 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}}, \pi \cdot 0.5\right)} \]

   3. div-inv 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, \pi \cdot 0.5\right)} \]

   4. metadata-eval 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{\left(\pi \cdot \color{blue}{0.5}\right) \cdot \frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, \pi \cdot 0.5\right)} \]

   5. div-inv 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, \pi \cdot 0.5\right)} \]

   6. metadata-eval 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot \color{blue}{0.5}\right) - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, \pi \cdot 0.5\right)} \]

   7. swap-sqr 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{\color{blue}{\left(\pi \cdot \pi\right) \cdot \left(0.5 \cdot 0.5\right)} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, \pi \cdot 0.5\right)} \]

   8. unpow2 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{\color{blue}{{\pi}^{2}} \cdot \left(0.5 \cdot 0.5\right) - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, \pi \cdot 0.5\right)} \]

   9. metadata-eval 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{{\pi}^{2} \cdot \color{blue}{0.25} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, \pi \cdot 0.5\right)} \]

   10. div-sub 7.5%

       \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \color{blue}{\frac{{\pi}^{2} \cdot 0.25}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)} - \frac{\cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}}, \pi \cdot 0.5\right)} \]

7. Applied egg-rr 7.5%

   \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \color{blue}{\frac{{\pi}^{2} \cdot 0.25}{\mathsf{fma}\left(\pi, 0.5, \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)} - \frac{{\cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}}, \pi \cdot 0.5\right)} \]
8. Step-by-step derivation

   1. add-cube-cbrt 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{{\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)}}^{2} \cdot 0.25}{\mathsf{fma}\left(\pi, 0.5, \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)} - \frac{{\cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}, \pi \cdot 0.5\right)} \]

   2. pow2 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{{\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\pi, 0.5, \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)} - \frac{{\cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}, \pi \cdot 0.5\right)} \]

9. Applied egg-rr 7.5%

   \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{{\color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right)}}^{2} \cdot 0.25}{\mathsf{fma}\left(\pi, 0.5, \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)} - \frac{{\cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}, \pi \cdot 0.5\right)} \]

10. Final simplification 7.5%

    \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{0.25 \cdot {\left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)} - \frac{{\cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}, \pi \cdot 0.5\right)} \]

Alternative 2: 8.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\pi}^{2} \cdot 0.25\\ t_1 := \sqrt{0.5 + x \cdot -0.5}\\ t_2 := \cos^{-1} t_1\\ t_3 := \mathsf{fma}\left(\pi, 0.5, t_2\right)\\ \frac{t_0 - \sqrt[3]{{\left({\sin^{-1} t_1}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{t_0}{t_3} - \frac{{t_2}^{2}}{t_3}, \pi \cdot 0.5\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (pow PI 2.0) 0.25))
        (t_1 (sqrt (+ 0.5 (* x -0.5))))
        (t_2 (acos t_1))
        (t_3 (fma PI 0.5 t_2)))
   (/
    (- t_0 (* (cbrt (pow (pow (asin t_1) 2.0) 3.0)) 4.0))
    (fma 2.0 (- (/ t_0 t_3) (/ (pow t_2 2.0) t_3)) (* PI 0.5)))))

C

double code(double x) {
	double t_0 = pow(((double) M_PI), 2.0) * 0.25;
	double t_1 = sqrt((0.5 + (x * -0.5)));
	double t_2 = acos(t_1);
	double t_3 = fma(((double) M_PI), 0.5, t_2);
	return (t_0 - (cbrt(pow(pow(asin(t_1), 2.0), 3.0)) * 4.0)) / fma(2.0, ((t_0 / t_3) - (pow(t_2, 2.0) / t_3)), (((double) M_PI) * 0.5));
}

Julia

function code(x)
	t_0 = Float64((pi ^ 2.0) * 0.25)
	t_1 = sqrt(Float64(0.5 + Float64(x * -0.5)))
	t_2 = acos(t_1)
	t_3 = fma(pi, 0.5, t_2)
	return Float64(Float64(t_0 - Float64(cbrt(((asin(t_1) ^ 2.0) ^ 3.0)) * 4.0)) / fma(2.0, Float64(Float64(t_0 / t_3) - Float64((t_2 ^ 2.0) / t_3)), Float64(pi * 0.5)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.25), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[ArcCos[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(Pi * 0.5 + t$95$2), $MachinePrecision]}, N[(N[(t$95$0 - N[(N[Power[N[Power[N[Power[N[ArcSin[t$95$1], $MachinePrecision], 2.0], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(N[(t$95$0 / t$95$3), $MachinePrecision] - N[(N[Power[t$95$2, 2.0], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 5.7%

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

   1. flip-- 5.7%

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

3. Applied egg-rr 5.7%

   \[\leadsto \color{blue}{\frac{{\pi}^{2} \cdot 0.25 - {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}} \]
4. Step-by-step derivation

   1. add-cbrt-cube 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \color{blue}{\sqrt[3]{\left({\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2} \cdot {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}\right) \cdot {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   2. pow3 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{\color{blue}{{\left({\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}\right)}^{3}}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   3. sub-neg 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-x \cdot 0.5\right)}}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   4. distribute-rgt-neg-in 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot \left(-0.5\right)}}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   5. metadata-eval 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot \color{blue}{-0.5}}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

5. Applied egg-rr 7.5%

   \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \color{blue}{\sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]
6. Step-by-step derivation

   1. asin-acos 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \color{blue}{\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, \pi \cdot 0.5\right)} \]

   2. flip-- 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}}, \pi \cdot 0.5\right)} \]

   3. div-inv 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, \pi \cdot 0.5\right)} \]

   4. metadata-eval 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{\left(\pi \cdot \color{blue}{0.5}\right) \cdot \frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, \pi \cdot 0.5\right)} \]

   5. div-inv 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, \pi \cdot 0.5\right)} \]

   6. metadata-eval 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot \color{blue}{0.5}\right) - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, \pi \cdot 0.5\right)} \]

   7. swap-sqr 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{\color{blue}{\left(\pi \cdot \pi\right) \cdot \left(0.5 \cdot 0.5\right)} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, \pi \cdot 0.5\right)} \]

   8. unpow2 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{\color{blue}{{\pi}^{2}} \cdot \left(0.5 \cdot 0.5\right) - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, \pi \cdot 0.5\right)} \]

   9. metadata-eval 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{{\pi}^{2} \cdot \color{blue}{0.25} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, \pi \cdot 0.5\right)} \]

   10. div-sub 7.5%

       \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \color{blue}{\frac{{\pi}^{2} \cdot 0.25}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)} - \frac{\cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}{\frac{\pi}{2} + \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}}, \pi \cdot 0.5\right)} \]

7. Applied egg-rr 7.5%

   \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \color{blue}{\frac{{\pi}^{2} \cdot 0.25}{\mathsf{fma}\left(\pi, 0.5, \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)} - \frac{{\cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}}, \pi \cdot 0.5\right)} \]

8. Final simplification 7.5%

   \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \frac{{\pi}^{2} \cdot 0.25}{\mathsf{fma}\left(\pi, 0.5, \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)} - \frac{{\cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}, \pi \cdot 0.5\right)} \]

Alternative 3: 8.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right), \pi \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (-
   (* (pow PI 2.0) 0.25)
   (* (cbrt (pow (pow (asin (sqrt (+ 0.5 (* x -0.5)))) 2.0) 3.0)) 4.0))
  (fma 2.0 (asin (sqrt (- 0.5 (* 0.5 x)))) (* PI 0.5))))

C

double code(double x) {
	return ((pow(((double) M_PI), 2.0) * 0.25) - (cbrt(pow(pow(asin(sqrt((0.5 + (x * -0.5)))), 2.0), 3.0)) * 4.0)) / fma(2.0, asin(sqrt((0.5 - (0.5 * x)))), (((double) M_PI) * 0.5));
}

Julia

function code(x)
	return Float64(Float64(Float64((pi ^ 2.0) * 0.25) - Float64(cbrt(((asin(sqrt(Float64(0.5 + Float64(x * -0.5)))) ^ 2.0) ^ 3.0)) * 4.0)) / fma(2.0, asin(sqrt(Float64(0.5 - Float64(0.5 * x)))), Float64(pi * 0.5)))
end

Wolfram

code[x_] := N[(N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.25), $MachinePrecision] - N[(N[Power[N[Power[N[Power[N[ArcSin[N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[ArcSin[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.7%

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

   1. flip-- 5.7%

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

3. Applied egg-rr 5.7%

   \[\leadsto \color{blue}{\frac{{\pi}^{2} \cdot 0.25 - {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}} \]
4. Step-by-step derivation

   1. add-cbrt-cube 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \color{blue}{\sqrt[3]{\left({\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2} \cdot {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}\right) \cdot {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   2. pow3 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{\color{blue}{{\left({\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}\right)}^{3}}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   3. sub-neg 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-x \cdot 0.5\right)}}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   4. distribute-rgt-neg-in 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot \left(-0.5\right)}}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

   5. metadata-eval 7.5%

      \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot \color{blue}{-0.5}}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

5. Applied egg-rr 7.5%

   \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \color{blue}{\sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

6. Final simplification 7.5%

   \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - \sqrt[3]{{\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}^{3}} \cdot 4}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right), \pi \cdot 0.5\right)} \]

Alternative 4: 8.7% accurate, 0.8× 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 5.7%

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

   1. asin-acos 7.5%

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

   2. div-inv 7.5%

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

   3. metadata-eval 7.5%

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

   4. div-sub 7.5%

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

   5. metadata-eval 7.5%

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

   6. div-inv 7.5%

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

   7. metadata-eval 7.5%

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

3. Applied egg-rr 7.5%

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

4. Final simplification 7.5%

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

Alternative 5: 7.2% 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 5.7%

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

   1. clear-num 5.7%

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

   2. sqrt-div 6.1%

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

   3. metadata-eval 6.1%

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

3. Applied egg-rr 6.1%

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

4. Final simplification 6.1%

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

Alternative 6: 7.2% 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 5.7%

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

2. Final simplification 5.7%

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

Alternative 7: 3.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.7%

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

   1. asin-acos 7.5%

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

   2. div-inv 7.5%

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

   3. metadata-eval 7.5%

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

   4. div-sub 7.5%

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

   5. metadata-eval 7.5%

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

   6. div-inv 7.5%

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

   7. metadata-eval 7.5%

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

3. Applied egg-rr 7.5%

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

   1. cancel-sign-sub-inv 7.5%

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

   2. metadata-eval 7.5%

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

   3. metadata-eval 7.5%

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

   4. div-inv 7.5%

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

   5. asin-acos 5.7%

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

   6. *-commutative 5.7%

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

   7. div-inv 5.7%

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

   8. metadata-eval 5.7%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 3.8%

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

   11. swap-sqr 3.8%

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

   12. unpow2 3.8%

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

   13. metadata-eval 3.8%

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

5. Applied egg-rr 3.8%

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

6. Taylor expanded in x around 0 3.8%

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

7. Final simplification 3.8%

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

Alternative 8: 4.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.7%

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

2. Taylor expanded in x around 0 3.9%

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

3. Final simplification 3.9%

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

Developer target: 100.0% accurate, 3.1× 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 2023333 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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