bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.4%

Time: 20.3s

Alternatives: 8

Speedup: 1.0×

Specification

\[0 \leq x \land x \leq 0.5\]

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

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

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Initial Program: 6.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

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

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Alternative 1: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {t\_0}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + {\left(\sqrt{{\left(\sqrt[3]{t\_0}\right)}^{3}}\right)}^{2} \cdot \mathsf{fma}\left(\pi, 0.5, t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/
    (- (* (* (pow PI 1.5) (pow PI 1.5)) 0.125) (pow t_0 3.0))
    (+
     (pow (* PI 0.5) 2.0)
     (* (pow (sqrt (pow (cbrt t_0) 3.0)) 2.0) (fma PI 0.5 t_0))))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return (((pow(((double) M_PI), 1.5) * pow(((double) M_PI), 1.5)) * 0.125) - pow(t_0, 3.0)) / (pow((((double) M_PI) * 0.5), 2.0) + (pow(sqrt(pow(cbrt(t_0), 3.0)), 2.0) * fma(((double) M_PI), 0.5, t_0)));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(Float64(Float64(Float64((pi ^ 1.5) * (pi ^ 1.5)) * 0.125) - (t_0 ^ 3.0)) / Float64((Float64(pi * 0.5) ^ 2.0) + Float64((sqrt((cbrt(t_0) ^ 3.0)) ^ 2.0) * fma(pi, 0.5, t_0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(N[Power[Pi, 1.5], $MachinePrecision] * N[Power[Pi, 1.5], $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(Pi * 0.5), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[N[Sqrt[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(Pi * 0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.4%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. acos-asin 6.4%

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

   2. flip3-- 6.4%

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

   3. div-inv 6.4%

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

   4. metadata-eval 6.4%

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

   5. unpow-prod-down 6.4%

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

   6. metadata-eval 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot \color{blue}{0.125} - {\sin^{-1} \left(1 - x\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   7. pow2 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\color{blue}{{\left(\frac{\pi}{2}\right)}^{2}} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   8. div-inv 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)}}^{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   9. metadata-eval 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot \color{blue}{0.5}\right)}^{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   10. pow2 6.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \left(\color{blue}{{\sin^{-1} \left(1 - x\right)}^{2}} + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   11. div-inv 6.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \left({\sin^{-1} \left(1 - x\right)}^{2} + \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   12. metadata-eval 6.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \left({\sin^{-1} \left(1 - x\right)}^{2} + \left(\pi \cdot \color{blue}{0.5}\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

4. Applied egg-rr 6.4%

   \[\leadsto \color{blue}{\frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \left({\sin^{-1} \left(1 - x\right)}^{2} + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)}} \]
5. Step-by-step derivation

   1. unpow2 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \left(\color{blue}{\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)} + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   2. distribute-rgt-out 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \color{blue}{\sin^{-1} \left(1 - x\right) \cdot \left(\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\right)}} \]

   3. +-commutative 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\left(\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\right)}} \]

   4. fma-udef 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]

6. Simplified 6.4%

   \[\leadsto \color{blue}{\frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]
7. Step-by-step derivation

   1. sqr-pow 9.9%

      \[\leadsto \frac{\color{blue}{\left({\pi}^{\left(\frac{3}{2}\right)} \cdot {\pi}^{\left(\frac{3}{2}\right)}\right)} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   2. metadata-eval 9.9%

      \[\leadsto \frac{\left({\pi}^{\color{blue}{1.5}} \cdot {\pi}^{\left(\frac{3}{2}\right)}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   3. metadata-eval 9.9%

      \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{\color{blue}{1.5}}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

8. Applied egg-rr 9.9%

   \[\leadsto \frac{\color{blue}{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right)} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
9. Step-by-step derivation

   1. add-sqr-sqrt 9.9%

      \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \color{blue}{\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   2. pow2 9.9%

      \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

10. Applied egg-rr 9.9%

    \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
11. Step-by-step derivation

    1. add-cube-cbrt 9.9%

       \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + {\left(\sqrt{\color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}}}\right)}^{2} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

    2. pow3 9.9%

       \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + {\left(\sqrt{\color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}}}\right)}^{2} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

12. Applied egg-rr 9.9%

    \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + {\left(\sqrt{\color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}}}\right)}^{2} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

13. Final simplification 9.9%

    \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + {\left(\sqrt{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}}\right)}^{2} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
14. Add Preprocessing

Alternative 2: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {t\_0}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \mathsf{fma}\left(\pi, 0.5, t\_0\right) \cdot {\left(\sqrt{t\_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/
    (- (* (* (pow PI 1.5) (pow PI 1.5)) 0.125) (pow t_0 3.0))
    (+ (pow (* PI 0.5) 2.0) (* (fma PI 0.5 t_0) (pow (sqrt t_0) 2.0))))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return (((pow(((double) M_PI), 1.5) * pow(((double) M_PI), 1.5)) * 0.125) - pow(t_0, 3.0)) / (pow((((double) M_PI) * 0.5), 2.0) + (fma(((double) M_PI), 0.5, t_0) * pow(sqrt(t_0), 2.0)));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(Float64(Float64(Float64((pi ^ 1.5) * (pi ^ 1.5)) * 0.125) - (t_0 ^ 3.0)) / Float64((Float64(pi * 0.5) ^ 2.0) + Float64(fma(pi, 0.5, t_0) * (sqrt(t_0) ^ 2.0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(N[Power[Pi, 1.5], $MachinePrecision] * N[Power[Pi, 1.5], $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(Pi * 0.5), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(Pi * 0.5 + t$95$0), $MachinePrecision] * N[Power[N[Sqrt[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.4%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. acos-asin 6.4%

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

   2. flip3-- 6.4%

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

   3. div-inv 6.4%

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

   4. metadata-eval 6.4%

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

   5. unpow-prod-down 6.4%

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

   6. metadata-eval 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot \color{blue}{0.125} - {\sin^{-1} \left(1 - x\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   7. pow2 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\color{blue}{{\left(\frac{\pi}{2}\right)}^{2}} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   8. div-inv 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)}}^{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   9. metadata-eval 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot \color{blue}{0.5}\right)}^{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   10. pow2 6.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \left(\color{blue}{{\sin^{-1} \left(1 - x\right)}^{2}} + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   11. div-inv 6.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \left({\sin^{-1} \left(1 - x\right)}^{2} + \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   12. metadata-eval 6.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \left({\sin^{-1} \left(1 - x\right)}^{2} + \left(\pi \cdot \color{blue}{0.5}\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

4. Applied egg-rr 6.4%

   \[\leadsto \color{blue}{\frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \left({\sin^{-1} \left(1 - x\right)}^{2} + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)}} \]
5. Step-by-step derivation

   1. unpow2 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \left(\color{blue}{\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)} + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   2. distribute-rgt-out 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \color{blue}{\sin^{-1} \left(1 - x\right) \cdot \left(\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\right)}} \]

   3. +-commutative 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\left(\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\right)}} \]

   4. fma-udef 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]

6. Simplified 6.4%

   \[\leadsto \color{blue}{\frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]
7. Step-by-step derivation

   1. sqr-pow 9.9%

      \[\leadsto \frac{\color{blue}{\left({\pi}^{\left(\frac{3}{2}\right)} \cdot {\pi}^{\left(\frac{3}{2}\right)}\right)} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   2. metadata-eval 9.9%

      \[\leadsto \frac{\left({\pi}^{\color{blue}{1.5}} \cdot {\pi}^{\left(\frac{3}{2}\right)}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   3. metadata-eval 9.9%

      \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{\color{blue}{1.5}}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

8. Applied egg-rr 9.9%

   \[\leadsto \frac{\color{blue}{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right)} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
9. Step-by-step derivation

   1. add-sqr-sqrt 9.9%

      \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \color{blue}{\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   2. pow2 9.9%

      \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

10. Applied egg-rr 9.9%

    \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

11. Final simplification 9.9%

    \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot {\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \]
12. Add Preprocessing

Alternative 3: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {t\_0}^{3}}{0.25 \cdot {\pi}^{2} + t\_0 \cdot \left(t\_0 + \pi \cdot 0.5\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/
    (- (* (* (pow PI 1.5) (pow PI 1.5)) 0.125) (pow t_0 3.0))
    (+ (* 0.25 (pow PI 2.0)) (* t_0 (+ t_0 (* PI 0.5)))))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return (((pow(((double) M_PI), 1.5) * pow(((double) M_PI), 1.5)) * 0.125) - pow(t_0, 3.0)) / ((0.25 * pow(((double) M_PI), 2.0)) + (t_0 * (t_0 + (((double) M_PI) * 0.5))));
}

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	return (((Math.pow(Math.PI, 1.5) * Math.pow(Math.PI, 1.5)) * 0.125) - Math.pow(t_0, 3.0)) / ((0.25 * Math.pow(Math.PI, 2.0)) + (t_0 * (t_0 + (Math.PI * 0.5))));
}

Python

def code(x):
	t_0 = math.asin((1.0 - x))
	return (((math.pow(math.pi, 1.5) * math.pow(math.pi, 1.5)) * 0.125) - math.pow(t_0, 3.0)) / ((0.25 * math.pow(math.pi, 2.0)) + (t_0 * (t_0 + (math.pi * 0.5))))

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(Float64(Float64(Float64((pi ^ 1.5) * (pi ^ 1.5)) * 0.125) - (t_0 ^ 3.0)) / Float64(Float64(0.25 * (pi ^ 2.0)) + Float64(t_0 * Float64(t_0 + Float64(pi * 0.5)))))
end

MATLAB

function tmp = code(x)
	t_0 = asin((1.0 - x));
	tmp = ((((pi ^ 1.5) * (pi ^ 1.5)) * 0.125) - (t_0 ^ 3.0)) / ((0.25 * (pi ^ 2.0)) + (t_0 * (t_0 + (pi * 0.5))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(N[Power[Pi, 1.5], $MachinePrecision] * N[Power[Pi, 1.5], $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(t$95$0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.4%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. acos-asin 6.4%

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

   2. flip3-- 6.4%

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

   3. div-inv 6.4%

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

   4. metadata-eval 6.4%

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

   5. unpow-prod-down 6.4%

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

   6. metadata-eval 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot \color{blue}{0.125} - {\sin^{-1} \left(1 - x\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   7. pow2 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\color{blue}{{\left(\frac{\pi}{2}\right)}^{2}} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   8. div-inv 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)}}^{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   9. metadata-eval 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot \color{blue}{0.5}\right)}^{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   10. pow2 6.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \left(\color{blue}{{\sin^{-1} \left(1 - x\right)}^{2}} + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   11. div-inv 6.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \left({\sin^{-1} \left(1 - x\right)}^{2} + \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   12. metadata-eval 6.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \left({\sin^{-1} \left(1 - x\right)}^{2} + \left(\pi \cdot \color{blue}{0.5}\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

4. Applied egg-rr 6.4%

   \[\leadsto \color{blue}{\frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \left({\sin^{-1} \left(1 - x\right)}^{2} + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)}} \]
5. Step-by-step derivation

   1. unpow2 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \left(\color{blue}{\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)} + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   2. distribute-rgt-out 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \color{blue}{\sin^{-1} \left(1 - x\right) \cdot \left(\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\right)}} \]

   3. +-commutative 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\left(\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\right)}} \]

   4. fma-udef 6.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]

6. Simplified 6.4%

   \[\leadsto \color{blue}{\frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]
7. Step-by-step derivation

   1. sqr-pow 9.9%

      \[\leadsto \frac{\color{blue}{\left({\pi}^{\left(\frac{3}{2}\right)} \cdot {\pi}^{\left(\frac{3}{2}\right)}\right)} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   2. metadata-eval 9.9%

      \[\leadsto \frac{\left({\pi}^{\color{blue}{1.5}} \cdot {\pi}^{\left(\frac{3}{2}\right)}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   3. metadata-eval 9.9%

      \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{\color{blue}{1.5}}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

8. Applied egg-rr 9.9%

   \[\leadsto \frac{\color{blue}{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right)} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

9. Taylor expanded in x around 0 9.9%

   \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\color{blue}{0.25 \cdot {\pi}^{2} + \sin^{-1} \left(1 - x\right) \cdot \left(\sin^{-1} \left(1 - x\right) + 0.5 \cdot \pi\right)}} \]

10. Final simplification 9.9%

    \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{0.25 \cdot {\pi}^{2} + \sin^{-1} \left(1 - x\right) \cdot \left(\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\right)} \]
11. Add Preprocessing

Alternative 4: 9.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17)
   (- PI (acos (- 1.0 x)))
   (- (* PI 0.5) (cbrt (pow (asin (- 1.0 x)) 3.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = ((double) M_PI) - acos((1.0 - x));
	} else {
		tmp = (((double) M_PI) * 0.5) - cbrt(pow(asin((1.0 - x)), 3.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = Math.PI - Math.acos((1.0 - x));
	} else {
		tmp = (Math.PI * 0.5) - Math.cbrt(Math.pow(Math.asin((1.0 - x)), 3.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(pi - acos(Float64(1.0 - x)));
	else
		tmp = Float64(Float64(pi * 0.5) - cbrt((asin(Float64(1.0 - x)) ^ 3.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[(Pi - N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   6. Simplified 3.9%

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

      1. add-sqr-sqrt 7.5%

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

      2. cancel-sign-sub-inv 7.5%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 6.5%

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

      5. sqr-neg 6.5%

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

      6. add-sqr-sqrt 6.5%

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

      7. add-sqr-sqrt 6.5%

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

   8. Applied egg-rr 6.5%

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

      1. +-commutative 6.5%

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

      2. asin-acos 6.5%

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

      3. div-inv 6.5%

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

      4. metadata-eval 6.5%

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

      5. associate-+l- 6.5%

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

   10. Applied egg-rr 6.5%

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

       1. associate--r- 6.5%

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

       2. +-commutative 6.5%

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

       3. associate--l+ 6.5%

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

       4. distribute-lft-out 6.5%

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

       5. metadata-eval 6.5%

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

       6. *-rgt-identity 6.5%

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

   12. Simplified 6.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 62.2%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. acos-asin 62.1%

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

      2. sub-neg 62.1%

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

      3. div-inv 62.1%

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

      4. metadata-eval 62.1%

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

   4. Applied egg-rr 62.1%

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

      1. sub-neg 62.1%

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

   6. Simplified 62.1%

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

      1. rem-cbrt-cube 62.3%

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

   8. Applied egg-rr 62.3%

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

4. Final simplification 8.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 10.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (- (* PI 0.5) (pow (cbrt (asin (- 1.0 x))) 3.0)))

C

double code(double x) {
	return (((double) M_PI) * 0.5) - pow(cbrt(asin((1.0 - x))), 3.0);
}

Java

public static double code(double x) {
	return (Math.PI * 0.5) - Math.pow(Math.cbrt(Math.asin((1.0 - x))), 3.0);
}

Julia

function code(x)
	return Float64(Float64(pi * 0.5) - (cbrt(asin(Float64(1.0 - x))) ^ 3.0))
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.4%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. acos-asin 6.4%

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

   2. sub-neg 6.4%

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

   3. div-inv 6.4%

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

   4. metadata-eval 6.4%

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

4. Applied egg-rr 6.4%

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

   1. sub-neg 6.4%

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

6. Simplified 6.4%

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

   1. add-cube-cbrt 9.9%

      \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + {\left(\sqrt{\color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}}}\right)}^{2} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   2. pow3 9.9%

      \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + {\left(\sqrt{\color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}}}\right)}^{2} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

8. Applied egg-rr 9.8%

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

9. Final simplification 9.8%

   \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3} \]
10. Add Preprocessing

Alternative 6: 10.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (- (* PI 0.5) (pow (sqrt (asin (- 1.0 x))) 2.0)))

C

double code(double x) {
	return (((double) M_PI) * 0.5) - pow(sqrt(asin((1.0 - x))), 2.0);
}

Java

public static double code(double x) {
	return (Math.PI * 0.5) - Math.pow(Math.sqrt(Math.asin((1.0 - x))), 2.0);
}

Python

def code(x):
	return (math.pi * 0.5) - math.pow(math.sqrt(math.asin((1.0 - x))), 2.0)

Julia

function code(x)
	return Float64(Float64(pi * 0.5) - (sqrt(asin(Float64(1.0 - x))) ^ 2.0))
end

MATLAB

function tmp = code(x)
	tmp = (pi * 0.5) - (sqrt(asin((1.0 - x))) ^ 2.0);
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Sqrt[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.4%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. acos-asin 6.4%

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

   2. sub-neg 6.4%

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

   3. div-inv 6.4%

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

   4. metadata-eval 6.4%

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

4. Applied egg-rr 6.4%

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

   1. sub-neg 6.4%

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

6. Simplified 6.4%

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

   1. add-sqr-sqrt 9.9%

      \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \color{blue}{\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   2. pow2 9.9%

      \[\leadsto \frac{\left({\pi}^{1.5} \cdot {\pi}^{1.5}\right) \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{2} + \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

8. Applied egg-rr 9.9%

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

9. Final simplification 9.9%

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

Alternative 7: 9.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\pi - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x)))) (if (<= t_0 0.0) (- PI t_0) t_0)))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((double) M_PI) - t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.PI - t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.pi - t_0
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(pi - t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = pi - t_0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(Pi - t$95$0), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 1 x)) < 0.0

   1. Initial program 3.9%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   6. Simplified 3.9%

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

      1. add-sqr-sqrt 7.5%

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

      2. cancel-sign-sub-inv 7.5%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 6.5%

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

      5. sqr-neg 6.5%

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

      6. add-sqr-sqrt 6.5%

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

      7. add-sqr-sqrt 6.5%

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

   8. Applied egg-rr 6.5%

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

      1. +-commutative 6.5%

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

      2. asin-acos 6.5%

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

      3. div-inv 6.5%

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

      4. metadata-eval 6.5%

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

      5. associate-+l- 6.5%

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

   10. Applied egg-rr 6.5%

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

       1. associate--r- 6.5%

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

       2. +-commutative 6.5%

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

       3. associate--l+ 6.5%

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

       4. distribute-lft-out 6.5%

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

       5. metadata-eval 6.5%

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

       6. *-rgt-identity 6.5%

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

   12. Simplified 6.5%

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

   if 0.0 < (acos.f64 (-.f64 1 x))

   1. Initial program 62.2%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 8.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 6.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

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

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 6.4%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing

3. Final simplification 6.4%

   \[\leadsto \cos^{-1} \left(1 - x\right) \]
4. Add Preprocessing

Developer target: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024027 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :herbie-target
  (* 2.0 (asin (sqrt (/ x 2.0))))

  (acos (- 1.0 x)))