Ian Simplification

Percentage Accurate: 6.8% → 8.3%

Time: 37.2s

Alternatives: 9

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: 6.8% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ \frac{\frac{\log \left(e^{{\pi}^{6} \cdot 0.015625 - {\left(\mathsf{fma}\left(-2, t\_0, \pi\right)\right)}^{6}}\right)}{{\left(\pi - t\_0 \cdot 2\right)}^{4} + \left({\left(\pi \cdot \left(\pi \cdot 0.5 - t\_0\right)\right)}^{2} + {\left(\pi \cdot 0.5\right)}^{4}\right)}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (sqrt (fma -0.5 x 0.5)))))
   (/
    (/
     (log (exp (- (* (pow PI 6.0) 0.015625) (pow (fma -2.0 t_0 PI) 6.0))))
     (+
      (pow (- PI (* t_0 2.0)) 4.0)
      (+ (pow (* PI (- (* PI 0.5) t_0)) 2.0) (pow (* PI 0.5) 4.0))))
    (fma 2.0 (asin (sqrt (- 0.5 (* x 0.5)))) (* PI 0.5)))))

C

double code(double x) {
	double t_0 = acos(sqrt(fma(-0.5, x, 0.5)));
	return (log(exp(((pow(((double) M_PI), 6.0) * 0.015625) - pow(fma(-2.0, t_0, ((double) M_PI)), 6.0)))) / (pow((((double) M_PI) - (t_0 * 2.0)), 4.0) + (pow((((double) M_PI) * ((((double) M_PI) * 0.5) - t_0)), 2.0) + pow((((double) M_PI) * 0.5), 4.0)))) / fma(2.0, asin(sqrt((0.5 - (x * 0.5)))), (((double) M_PI) * 0.5));
}

Julia

function code(x)
	t_0 = acos(sqrt(fma(-0.5, x, 0.5)))
	return Float64(Float64(log(exp(Float64(Float64((pi ^ 6.0) * 0.015625) - (fma(-2.0, t_0, pi) ^ 6.0)))) / Float64((Float64(pi - Float64(t_0 * 2.0)) ^ 4.0) + Float64((Float64(pi * Float64(Float64(pi * 0.5) - t_0)) ^ 2.0) + (Float64(pi * 0.5) ^ 4.0)))) / fma(2.0, asin(sqrt(Float64(0.5 - Float64(x * 0.5)))), Float64(pi * 0.5)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[Sqrt[N[(-0.5 * x + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Log[N[Exp[N[(N[(N[Power[Pi, 6.0], $MachinePrecision] * 0.015625), $MachinePrecision] - N[Power[N[(-2.0 * t$95$0 + Pi), $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(N[Power[N[(Pi - N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision], 4.0], $MachinePrecision] + N[(N[Power[N[(Pi * N[(N[(Pi * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(Pi * 0.5), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[ArcSin[N[Sqrt[N[(0.5 - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.1%

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

   1. flip-- 7.1%

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

   2. pow2 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

   5. pow2 7.1%

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

   6. div-sub 7.1%

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

   7. metadata-eval 7.1%

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

   8. div-inv 7.1%

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

   9. metadata-eval 7.1%

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

   10. +-commutative 7.1%

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

4. Applied egg-rr 7.1%

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

   1. asin-acos 8.6%

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

   2. div-inv 8.6%

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

   3. metadata-eval 8.6%

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

   4. *-commutative 8.6%

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

   5. cancel-sign-sub-inv 8.6%

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

   6. metadata-eval 8.6%

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

   7. *-commutative 8.6%

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

   8. +-commutative 8.6%

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

   9. fma-define 8.6%

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

6. Applied egg-rr 8.6%

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

   1. flip3-- 8.6%

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

8. Applied egg-rr 8.6%

   \[\leadsto \frac{\color{blue}{\frac{{\pi}^{6} \cdot 0.015625 - {\left(2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)\right)}^{6}}{{\left(\pi \cdot 0.5\right)}^{4} + \left({\left(2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)\right)}^{4} + {\left(\left(\pi \cdot 0.5\right) \cdot \left(2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)\right)\right)}^{2}\right)}}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

10. Simplified 8.6%

    \[\leadsto \frac{\color{blue}{\frac{{\pi}^{6} \cdot 0.015625 - {\left(\pi + \left(-2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{6}}{{\left(\pi + \left(-2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{4} + \left({\left(\pi \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{2} + {\left(\pi \cdot 0.5\right)}^{4}\right)}}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]
11. Step-by-step derivation

    1. add-log-exp 8.6%

       \[\leadsto \frac{\frac{\color{blue}{\log \left(e^{{\pi}^{6} \cdot 0.015625 - {\left(\pi + \left(-2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{6}}\right)}}{{\left(\pi + \left(-2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{4} + \left({\left(\pi \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{2} + {\left(\pi \cdot 0.5\right)}^{4}\right)}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

    2. +-commutative 8.6%

       \[\leadsto \frac{\frac{\log \left(e^{{\pi}^{6} \cdot 0.015625 - {\color{blue}{\left(\left(-2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) + \pi\right)}}^{6}}\right)}{{\left(\pi + \left(-2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{4} + \left({\left(\pi \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{2} + {\left(\pi \cdot 0.5\right)}^{4}\right)}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

    3. distribute-lft-neg-in 8.6%

       \[\leadsto \frac{\frac{\log \left(e^{{\pi}^{6} \cdot 0.015625 - {\left(\color{blue}{\left(-2\right) \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)} + \pi\right)}^{6}}\right)}{{\left(\pi + \left(-2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{4} + \left({\left(\pi \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{2} + {\left(\pi \cdot 0.5\right)}^{4}\right)}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

    4. metadata-eval 8.6%

       \[\leadsto \frac{\frac{\log \left(e^{{\pi}^{6} \cdot 0.015625 - {\left(\color{blue}{-2} \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) + \pi\right)}^{6}}\right)}{{\left(\pi + \left(-2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{4} + \left({\left(\pi \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{2} + {\left(\pi \cdot 0.5\right)}^{4}\right)}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

    5. fma-define 8.6%

       \[\leadsto \frac{\frac{\log \left(e^{{\pi}^{6} \cdot 0.015625 - {\color{blue}{\left(\mathsf{fma}\left(-2, \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), \pi\right)\right)}}^{6}}\right)}{{\left(\pi + \left(-2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{4} + \left({\left(\pi \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{2} + {\left(\pi \cdot 0.5\right)}^{4}\right)}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

12. Applied egg-rr 8.6%

    \[\leadsto \frac{\frac{\color{blue}{\log \left(e^{{\pi}^{6} \cdot 0.015625 - {\left(\mathsf{fma}\left(-2, \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), \pi\right)\right)}^{6}}\right)}}{{\left(\pi + \left(-2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{4} + \left({\left(\pi \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{2} + {\left(\pi \cdot 0.5\right)}^{4}\right)}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

13. Final simplification 8.6%

    \[\leadsto \frac{\frac{\log \left(e^{{\pi}^{6} \cdot 0.015625 - {\left(\mathsf{fma}\left(-2, \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), \pi\right)\right)}^{6}}\right)}{{\left(\pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) \cdot 2\right)}^{4} + \left({\left(\pi \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{2} + {\left(\pi \cdot 0.5\right)}^{4}\right)}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]
14. Add Preprocessing

Alternative 2: 8.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ t_1 := \sqrt{0.5 - x \cdot 0.5}\\ \frac{\frac{{\pi}^{6} \cdot 0.015625 - {\left(\pi - 2 \cdot \cos^{-1} t\_1\right)}^{6}}{{\left(\pi - t\_0 \cdot 2\right)}^{4} + \left({\left(\pi \cdot \left(\pi \cdot 0.5 - t\_0\right)\right)}^{2} + {\left(\pi \cdot 0.5\right)}^{4}\right)}}{\mathsf{fma}\left(2, \sin^{-1} t\_1, \pi \cdot 0.5\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (sqrt (fma -0.5 x 0.5)))) (t_1 (sqrt (- 0.5 (* x 0.5)))))
   (/
    (/
     (- (* (pow PI 6.0) 0.015625) (pow (- PI (* 2.0 (acos t_1))) 6.0))
     (+
      (pow (- PI (* t_0 2.0)) 4.0)
      (+ (pow (* PI (- (* PI 0.5) t_0)) 2.0) (pow (* PI 0.5) 4.0))))
    (fma 2.0 (asin t_1) (* PI 0.5)))))

C

double code(double x) {
	double t_0 = acos(sqrt(fma(-0.5, x, 0.5)));
	double t_1 = sqrt((0.5 - (x * 0.5)));
	return (((pow(((double) M_PI), 6.0) * 0.015625) - pow((((double) M_PI) - (2.0 * acos(t_1))), 6.0)) / (pow((((double) M_PI) - (t_0 * 2.0)), 4.0) + (pow((((double) M_PI) * ((((double) M_PI) * 0.5) - t_0)), 2.0) + pow((((double) M_PI) * 0.5), 4.0)))) / fma(2.0, asin(t_1), (((double) M_PI) * 0.5));
}

Julia

function code(x)
	t_0 = acos(sqrt(fma(-0.5, x, 0.5)))
	t_1 = sqrt(Float64(0.5 - Float64(x * 0.5)))
	return Float64(Float64(Float64(Float64((pi ^ 6.0) * 0.015625) - (Float64(pi - Float64(2.0 * acos(t_1))) ^ 6.0)) / Float64((Float64(pi - Float64(t_0 * 2.0)) ^ 4.0) + Float64((Float64(pi * Float64(Float64(pi * 0.5) - t_0)) ^ 2.0) + (Float64(pi * 0.5) ^ 4.0)))) / fma(2.0, asin(t_1), Float64(pi * 0.5)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[Sqrt[N[(-0.5 * x + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(0.5 - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(N[Power[Pi, 6.0], $MachinePrecision] * 0.015625), $MachinePrecision] - N[Power[N[(Pi - N[(2.0 * N[ArcCos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(Pi - N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision], 4.0], $MachinePrecision] + N[(N[Power[N[(Pi * N[(N[(Pi * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(Pi * 0.5), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[ArcSin[t$95$1], $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 7.1%

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

   1. flip-- 7.1%

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

   2. pow2 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

   5. pow2 7.1%

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

   6. div-sub 7.1%

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

   7. metadata-eval 7.1%

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

   8. div-inv 7.1%

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

   9. metadata-eval 7.1%

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

   10. +-commutative 7.1%

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

4. Applied egg-rr 7.1%

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

   1. asin-acos 8.6%

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

   2. div-inv 8.6%

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

   3. metadata-eval 8.6%

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

   4. *-commutative 8.6%

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

   5. cancel-sign-sub-inv 8.6%

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

   6. metadata-eval 8.6%

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

   7. *-commutative 8.6%

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

   8. +-commutative 8.6%

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

   9. fma-define 8.6%

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

6. Applied egg-rr 8.6%

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

   1. flip3-- 8.6%

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

8. Applied egg-rr 8.6%

   \[\leadsto \frac{\color{blue}{\frac{{\pi}^{6} \cdot 0.015625 - {\left(2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)\right)}^{6}}{{\left(\pi \cdot 0.5\right)}^{4} + \left({\left(2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)\right)}^{4} + {\left(\left(\pi \cdot 0.5\right) \cdot \left(2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)\right)\right)}^{2}\right)}}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

10. Simplified 8.6%

    \[\leadsto \frac{\color{blue}{\frac{{\pi}^{6} \cdot 0.015625 - {\left(\pi + \left(-2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{6}}{{\left(\pi + \left(-2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{4} + \left({\left(\pi \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{2} + {\left(\pi \cdot 0.5\right)}^{4}\right)}}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

11. Taylor expanded in x around inf 8.6%

    \[\leadsto \frac{\frac{{\pi}^{6} \cdot 0.015625 - \color{blue}{{\left(\pi - 2 \cdot \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}^{6}}}{{\left(\pi + \left(-2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{4} + \left({\left(\pi \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{2} + {\left(\pi \cdot 0.5\right)}^{4}\right)}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]

12. Final simplification 8.6%

    \[\leadsto \frac{\frac{{\pi}^{6} \cdot 0.015625 - {\left(\pi - 2 \cdot \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{6}}{{\left(\pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) \cdot 2\right)}^{4} + \left({\left(\pi \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)\right)}^{2} + {\left(\pi \cdot 0.5\right)}^{4}\right)}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]
13. Add Preprocessing

Alternative 3: 8.3% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/
  (log
   (exp
    (-
     (* (pow PI 2.0) 0.25)
     (* 4.0 (pow (- (* PI 0.5) (acos (sqrt (fma x -0.5 0.5)))) 2.0)))))
  (fma 2.0 (asin (sqrt (- 0.5 (* x 0.5)))) (* PI 0.5))))

C

double code(double x) {
	return log(exp(((pow(((double) M_PI), 2.0) * 0.25) - (4.0 * pow(((((double) M_PI) * 0.5) - acos(sqrt(fma(x, -0.5, 0.5)))), 2.0))))) / fma(2.0, asin(sqrt((0.5 - (x * 0.5)))), (((double) M_PI) * 0.5));
}

Julia

function code(x)
	return Float64(log(exp(Float64(Float64((pi ^ 2.0) * 0.25) - Float64(4.0 * (Float64(Float64(pi * 0.5) - acos(sqrt(fma(x, -0.5, 0.5)))) ^ 2.0))))) / fma(2.0, asin(sqrt(Float64(0.5 - Float64(x * 0.5)))), Float64(pi * 0.5)))
end

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. flip-- 7.1%

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

   2. pow2 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

   5. pow2 7.1%

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

   6. div-sub 7.1%

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

   7. metadata-eval 7.1%

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

   8. div-inv 7.1%

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

   9. metadata-eval 7.1%

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

   10. +-commutative 7.1%

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

4. Applied egg-rr 7.1%

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

   1. asin-acos 8.6%

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

   2. div-inv 8.6%

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

   3. metadata-eval 8.6%

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

   4. *-commutative 8.6%

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

   5. cancel-sign-sub-inv 8.6%

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

   6. metadata-eval 8.6%

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

   7. *-commutative 8.6%

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

   8. +-commutative 8.6%

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

   9. fma-define 8.6%

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

6. Applied egg-rr 8.6%

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

   1. add-log-exp 8.6%

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

   2. unpow-prod-down 8.6%

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

   3. metadata-eval 8.6%

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

   4. unpow-prod-down 8.6%

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

   5. metadata-eval 8.6%

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

8. Applied egg-rr 8.6%

   \[\leadsto \frac{\color{blue}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{2}}\right)}}{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)} \]
9. Add Preprocessing

Alternative 4: 8.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{0.5 - x \cdot 0.5}\\ \frac{{\pi}^{2} \cdot 0.25 - 4 \cdot {\left(\pi \cdot 0.5 - \cos^{-1} t\_0\right)}^{2}}{\pi \cdot 0.5 + 2 \cdot \sin^{-1} t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = Math.sqrt((0.5 - (x * 0.5)));
	return ((Math.pow(Math.PI, 2.0) * 0.25) - (4.0 * Math.pow(((Math.PI * 0.5) - Math.acos(t_0)), 2.0))) / ((Math.PI * 0.5) + (2.0 * Math.asin(t_0)));
}

Python

def code(x):
	t_0 = math.sqrt((0.5 - (x * 0.5)))
	return ((math.pow(math.pi, 2.0) * 0.25) - (4.0 * math.pow(((math.pi * 0.5) - math.acos(t_0)), 2.0))) / ((math.pi * 0.5) + (2.0 * math.asin(t_0)))

Julia

function code(x)
	t_0 = sqrt(Float64(0.5 - Float64(x * 0.5)))
	return Float64(Float64(Float64((pi ^ 2.0) * 0.25) - Float64(4.0 * (Float64(Float64(pi * 0.5) - acos(t_0)) ^ 2.0))) / Float64(Float64(pi * 0.5) + Float64(2.0 * asin(t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((0.5 - (x * 0.5)));
	tmp = (((pi ^ 2.0) * 0.25) - (4.0 * (((pi * 0.5) - acos(t_0)) ^ 2.0))) / ((pi * 0.5) + (2.0 * asin(t_0)));
end

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. flip-- 7.1%

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

   2. pow2 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

   5. pow2 7.1%

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

   6. div-sub 7.1%

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

   7. metadata-eval 7.1%

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

   8. div-inv 7.1%

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

   9. metadata-eval 7.1%

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

   10. +-commutative 7.1%

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

4. Applied egg-rr 7.1%

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

   1. asin-acos 8.6%

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

   2. div-inv 8.6%

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

   3. metadata-eval 8.6%

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

   4. *-commutative 8.6%

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

   5. cancel-sign-sub-inv 8.6%

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

   6. metadata-eval 8.6%

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

   7. *-commutative 8.6%

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

   8. +-commutative 8.6%

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

   9. fma-define 8.6%

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

6. Applied egg-rr 8.6%

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

7. Taylor expanded in x around inf 8.6%

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

8. Final simplification 8.6%

   \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - 4 \cdot {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}{\pi \cdot 0.5 + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)} \]
9. Add Preprocessing

Alternative 5: 8.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{\pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (cbrt (+ (* PI 0.5) (* -2.0 (- (* PI 0.5) (acos (sqrt (- 0.5 (* x 0.5))))))))
  3.0))

C

double code(double x) {
	return pow(cbrt(((((double) M_PI) * 0.5) + (-2.0 * ((((double) M_PI) * 0.5) - acos(sqrt((0.5 - (x * 0.5)))))))), 3.0);
}

Java

public static double code(double x) {
	return Math.pow(Math.cbrt(((Math.PI * 0.5) + (-2.0 * ((Math.PI * 0.5) - Math.acos(Math.sqrt((0.5 - (x * 0.5)))))))), 3.0);
}

Julia

function code(x)
	return cbrt(Float64(Float64(pi * 0.5) + Float64(-2.0 * Float64(Float64(pi * 0.5) - acos(sqrt(Float64(0.5 - Float64(x * 0.5)))))))) ^ 3.0
end

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. add-cube-cbrt 7.1%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right) \cdot \sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}} \]

   2. pow3 7.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)}^{3}} \]

4. Applied egg-rr 7.1%

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

   1. asin-acos 8.6%

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

   2. div-inv 8.6%

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

   3. metadata-eval 8.6%

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

   4. *-commutative 8.6%

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

   5. cancel-sign-sub-inv 8.6%

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

   6. metadata-eval 8.6%

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

   7. *-commutative 8.6%

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

   8. +-commutative 8.6%

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

   9. fma-define 8.6%

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

6. Applied egg-rr 8.6%

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

7. Taylor expanded in x around inf 8.6%

   \[\leadsto {\left(\sqrt[3]{\color{blue}{-2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right) + 0.5 \cdot \pi}}\right)}^{3} \]

8. Final simplification 8.6%

   \[\leadsto {\left(\sqrt[3]{\pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}\right)}^{3} \]
9. Add Preprocessing

Alternative 6: 8.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / 2.0)));
}

Python

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

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 / 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) + (2.0 * (acos(sqrt((0.5 + (-0.5 * x)))) - (pi / 2.0)));
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 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

   1. asin-acos 8.6%

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

   2. add-cube-cbrt 6.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\frac{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}}}{2} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \]

   3. associate-/l* 6.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \frac{\sqrt[3]{\pi}}{2}} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \]

   4. fma-neg 6.5%

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

   5. pow2 6.5%

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

   6. div-sub 6.5%

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

   7. metadata-eval 6.5%

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

   8. div-inv 6.5%

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

   9. metadata-eval 6.5%

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

4. Applied egg-rr 6.5%

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

   1. fma-neg 6.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \frac{\sqrt[3]{\pi}}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)} \]

   2. associate-*r/ 6.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\color{blue}{\frac{{\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}}{2}} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \]

   3. unpow2 6.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\frac{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)} \cdot \sqrt[3]{\pi}}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \]

   4. rem-3cbrt-lft 8.6%

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

   5. sub-neg 8.6%

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

   6. distribute-rgt-neg-in 8.6%

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

   7. metadata-eval 8.6%

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

6. Simplified 8.6%

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

7. Final simplification 8.6%

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

Alternative 7: 5.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{1}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-309)
   (- (/ PI 2.0) (* 2.0 (asin (sqrt 0.5))))
   (- (/ PI 2.0) (* 2.0 (asin (/ 1.0 (sqrt 2.0)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(0.5)));
	} else {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin((1.0 / sqrt(2.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(0.5)));
	} else {
		tmp = (Math.PI / 2.0) - (2.0 * Math.asin((1.0 / Math.sqrt(2.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1e-309:
		tmp = (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(0.5)))
	else:
		tmp = (math.pi / 2.0) - (2.0 * math.asin((1.0 / math.sqrt(2.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(0.5))));
	else
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(Float64(1.0 / sqrt(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1e-309)
		tmp = (pi / 2.0) - (2.0 * asin(sqrt(0.5)));
	else
		tmp = (pi / 2.0) - (2.0 * asin((1.0 / sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], 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[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 7.5%

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

   3. Taylor expanded in x around 0 5.6%

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

   if -1.000000000000002e-309 < x

   1. Initial program 6.7%

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

      1. clear-num 6.6%

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

      2. sqrt-div 9.3%

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

      3. metadata-eval 9.3%

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

   4. Applied egg-rr 9.3%

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

   5. Taylor expanded in x around 0 5.7%

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

Alternative 8: 6.8% 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 7.1%

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

Alternative 9: 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 7.1%

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

3. Taylor expanded in x around 0 4.0%

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

Developer target: 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 2024091 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :alt
  (asin x)

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