bug323 (missed optimization)

Percentage Accurate: 6.8% → 10.3%

Time: 27.3s

Alternatives: 14

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: 6.8% 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: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 0.5}\\ t_1 := \sin^{-1} \left(1 - x\right)\\ t_2 := \sqrt{t\_1}\\ \frac{{\left(\pi \cdot 0.5\right)}^{2} - {t\_1}^{2}}{\mathsf{fma}\left(\pi, 0.5, t\_1\right)} + \mathsf{fma}\left(-t\_2, t\_2, \mathsf{fma}\left(t\_0, t\_0, -\cos^{-1} \left(1 - x\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 0.5))) (t_1 (asin (- 1.0 x))) (t_2 (sqrt t_1)))
   (+
    (/ (- (pow (* PI 0.5) 2.0) (pow t_1 2.0)) (fma PI 0.5 t_1))
    (fma (- t_2) t_2 (fma t_0 t_0 (- (acos (- 1.0 x))))))))

C

double code(double x) {
	double t_0 = sqrt((((double) M_PI) * 0.5));
	double t_1 = asin((1.0 - x));
	double t_2 = sqrt(t_1);
	return ((pow((((double) M_PI) * 0.5), 2.0) - pow(t_1, 2.0)) / fma(((double) M_PI), 0.5, t_1)) + fma(-t_2, t_2, fma(t_0, t_0, -acos((1.0 - x))));
}

Julia

function code(x)
	t_0 = sqrt(Float64(pi * 0.5))
	t_1 = asin(Float64(1.0 - x))
	t_2 = sqrt(t_1)
	return Float64(Float64(Float64((Float64(pi * 0.5) ^ 2.0) - (t_1 ^ 2.0)) / fma(pi, 0.5, t_1)) + fma(Float64(-t_2), t_2, fma(t_0, t_0, Float64(-acos(Float64(1.0 - x))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[t$95$1], $MachinePrecision]}, N[(N[(N[(N[Power[N[(Pi * 0.5), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] / N[(Pi * 0.5 + t$95$1), $MachinePrecision]), $MachinePrecision] + N[((-t$95$2) * t$95$2 + N[(t$95$0 * t$95$0 + (-N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 6.5%

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

   1. acos-asin 6.5%

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

   2. *-un-lft-identity 6.5%

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

   3. add-sqr-sqrt 9.6%

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

   4. prod-diff 9.6%

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

   5. add-sqr-sqrt 9.7%

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

   6. fma-neg 9.7%

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

   7. *-un-lft-identity 9.7%

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

   8. acos-asin 9.7%

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

   9. add-sqr-sqrt 9.6%

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

4. Applied egg-rr 9.6%

   \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right)} \]
5. Step-by-step derivation

   1. asin-acos 9.6%

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

   2. div-inv 9.6%

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

   3. metadata-eval 9.6%

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

   4. add-sqr-sqrt 9.7%

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

   5. fma-neg 9.7%

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

6. Applied egg-rr 9.7%

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

   1. acos-asin 9.7%

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

   2. flip-- 9.7%

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

   3. pow2 9.7%

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

   4. div-inv 9.7%

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

   5. metadata-eval 9.7%

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

   6. pow2 9.7%

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

   7. div-inv 9.7%

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

   8. metadata-eval 9.7%

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

   9. fma-define 9.7%

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

8. Applied egg-rr 9.7%

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

9. Final simplification 9.7%

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

Alternative 2: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\sin^{-1} \left(1 - x\right)}\\ t_1 := \sqrt{\pi \cdot 0.5}\\ t_2 := \cos^{-1} \left(1 - x\right)\\ \mathsf{fma}\left(-t\_0, t\_0, \mathsf{fma}\left(t\_1, t\_1, -t\_2\right)\right) + {\left(\sqrt[3]{t\_2}\right)}^{3} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (asin (- 1.0 x))))
        (t_1 (sqrt (* PI 0.5)))
        (t_2 (acos (- 1.0 x))))
   (+ (fma (- t_0) t_0 (fma t_1 t_1 (- t_2))) (pow (cbrt t_2) 3.0))))

C

double code(double x) {
	double t_0 = sqrt(asin((1.0 - x)));
	double t_1 = sqrt((((double) M_PI) * 0.5));
	double t_2 = acos((1.0 - x));
	return fma(-t_0, t_0, fma(t_1, t_1, -t_2)) + pow(cbrt(t_2), 3.0);
}

Julia

function code(x)
	t_0 = sqrt(asin(Float64(1.0 - x)))
	t_1 = sqrt(Float64(pi * 0.5))
	t_2 = acos(Float64(1.0 - x))
	return Float64(fma(Float64(-t_0), t_0, fma(t_1, t_1, Float64(-t_2))) + (cbrt(t_2) ^ 3.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[((-t$95$0) * t$95$0 + N[(t$95$1 * t$95$1 + (-t$95$2)), $MachinePrecision]), $MachinePrecision] + N[Power[N[Power[t$95$2, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 6.5%

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

   1. acos-asin 6.5%

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

   2. *-un-lft-identity 6.5%

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

   3. add-sqr-sqrt 9.6%

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

   4. prod-diff 9.6%

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

   5. add-sqr-sqrt 9.7%

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

   6. fma-neg 9.7%

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

   7. *-un-lft-identity 9.7%

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

   8. acos-asin 9.7%

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

   9. add-sqr-sqrt 9.6%

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

4. Applied egg-rr 9.6%

   \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right)} \]
5. Step-by-step derivation

   1. asin-acos 9.6%

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

   2. div-inv 9.6%

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

   3. metadata-eval 9.6%

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

   4. add-sqr-sqrt 9.7%

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

   5. fma-neg 9.7%

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

6. Applied egg-rr 9.7%

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

   1. add-cube-cbrt 9.7%

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

   2. pow3 9.7%

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

8. Applied egg-rr 9.7%

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

9. Final simplification 9.7%

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

Alternative 3: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\sin^{-1} \left(1 - x\right)}\\ t_1 := \sqrt{\pi \cdot 0.5}\\ t_2 := \cos^{-1} \left(1 - x\right)\\ \mathsf{fma}\left(-t\_0, t\_0, \mathsf{fma}\left(t\_1, t\_1, -t\_2\right)\right) + e^{\log t\_2} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (asin (- 1.0 x))))
        (t_1 (sqrt (* PI 0.5)))
        (t_2 (acos (- 1.0 x))))
   (+ (fma (- t_0) t_0 (fma t_1 t_1 (- t_2))) (exp (log t_2)))))

C

double code(double x) {
	double t_0 = sqrt(asin((1.0 - x)));
	double t_1 = sqrt((((double) M_PI) * 0.5));
	double t_2 = acos((1.0 - x));
	return fma(-t_0, t_0, fma(t_1, t_1, -t_2)) + exp(log(t_2));
}

Julia

function code(x)
	t_0 = sqrt(asin(Float64(1.0 - x)))
	t_1 = sqrt(Float64(pi * 0.5))
	t_2 = acos(Float64(1.0 - x))
	return Float64(fma(Float64(-t_0), t_0, fma(t_1, t_1, Float64(-t_2))) + exp(log(t_2)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[((-t$95$0) * t$95$0 + N[(t$95$1 * t$95$1 + (-t$95$2)), $MachinePrecision]), $MachinePrecision] + N[Exp[N[Log[t$95$2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 6.5%

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

   1. acos-asin 6.5%

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

   2. *-un-lft-identity 6.5%

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

   3. add-sqr-sqrt 9.6%

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

   4. prod-diff 9.6%

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

   5. add-sqr-sqrt 9.7%

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

   6. fma-neg 9.7%

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

   7. *-un-lft-identity 9.7%

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

   8. acos-asin 9.7%

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

   9. add-sqr-sqrt 9.6%

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

4. Applied egg-rr 9.6%

   \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right)} \]
5. Step-by-step derivation

   1. asin-acos 9.6%

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

   2. div-inv 9.6%

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

   3. metadata-eval 9.6%

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

   4. add-sqr-sqrt 9.7%

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

   5. fma-neg 9.7%

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

6. Applied egg-rr 9.7%

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

   1. add-exp-log 9.7%

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

8. Applied egg-rr 9.7%

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

9. Final simplification 9.7%

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

Alternative 4: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ t_1 := \sqrt{\sin^{-1} \left(1 - x\right)}\\ t_2 := \sqrt{\pi \cdot 0.5}\\ t\_0 + \mathsf{fma}\left(-t\_1, t\_1, \mathsf{fma}\left(t\_2, t\_2, -t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x)))
        (t_1 (sqrt (asin (- 1.0 x))))
        (t_2 (sqrt (* PI 0.5))))
   (+ t_0 (fma (- t_1) t_1 (fma t_2 t_2 (- t_0))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double t_1 = sqrt(asin((1.0 - x)));
	double t_2 = sqrt((((double) M_PI) * 0.5));
	return t_0 + fma(-t_1, t_1, fma(t_2, t_2, -t_0));
}

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	t_1 = sqrt(asin(Float64(1.0 - x)))
	t_2 = sqrt(Float64(pi * 0.5))
	return Float64(t_0 + fma(Float64(-t_1), t_1, fma(t_2, t_2, Float64(-t_0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 + N[((-t$95$1) * t$95$1 + N[(t$95$2 * t$95$2 + (-t$95$0)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 6.5%

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

   1. acos-asin 6.5%

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

   2. *-un-lft-identity 6.5%

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

   3. add-sqr-sqrt 9.6%

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

   4. prod-diff 9.6%

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

   5. add-sqr-sqrt 9.7%

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

   6. fma-neg 9.7%

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

   7. *-un-lft-identity 9.7%

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

   8. acos-asin 9.7%

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

   9. add-sqr-sqrt 9.6%

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

4. Applied egg-rr 9.6%

   \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right)} \]
5. Step-by-step derivation

   1. asin-acos 9.6%

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

   2. div-inv 9.6%

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

   3. metadata-eval 9.6%

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

   4. add-sqr-sqrt 9.7%

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

   5. fma-neg 9.7%

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

6. Applied egg-rr 9.7%

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

7. Final simplification 9.7%

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

Alternative 5: 10.3% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (asin (- 1.0 x)))))
   (+ (acos (- 1.0 x)) (fma (- t_0) t_0 (pow t_0 2.0)))))

C

double code(double x) {
	double t_0 = sqrt(asin((1.0 - x)));
	return acos((1.0 - x)) + fma(-t_0, t_0, pow(t_0, 2.0));
}

Julia

function code(x)
	t_0 = sqrt(asin(Float64(1.0 - x)))
	return Float64(acos(Float64(1.0 - x)) + fma(Float64(-t_0), t_0, (t_0 ^ 2.0)))
end

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. acos-asin 6.5%

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

   2. *-un-lft-identity 6.5%

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

   3. add-sqr-sqrt 9.6%

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

   4. prod-diff 9.6%

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

   5. add-sqr-sqrt 9.7%

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

   6. fma-neg 9.7%

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

   7. *-un-lft-identity 9.7%

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

   8. acos-asin 9.7%

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

   9. add-sqr-sqrt 9.6%

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

4. Applied egg-rr 9.6%

   \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right)} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 9.6%

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

   2. pow2 9.6%

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

6. Applied egg-rr 9.7%

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

7. Final simplification 9.7%

   \[\leadsto \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, {\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) \]
8. Add Preprocessing

Alternative 6: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \mathsf{fma}\left({\left(\sqrt[3]{t\_0}\right)}^{2}, -{\left({t\_0}^{0.16666666666666666}\right)}^{2}, \pi \cdot 0.5\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (fma
    (pow (cbrt t_0) 2.0)
    (- (pow (pow t_0 0.16666666666666666) 2.0))
    (* PI 0.5))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return fma(pow(cbrt(t_0), 2.0), -pow(pow(t_0, 0.16666666666666666), 2.0), (((double) M_PI) * 0.5));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return fma((cbrt(t_0) ^ 2.0), Float64(-((t_0 ^ 0.16666666666666666) ^ 2.0)), Float64(pi * 0.5))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * (-N[Power[N[Power[t$95$0, 0.16666666666666666], $MachinePrecision], 2.0], $MachinePrecision]) + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.5%

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

   1. acos-asin 6.5%

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

   2. sub-neg 6.5%

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

   3. div-inv 6.5%

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

   4. metadata-eval 6.5%

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

4. Applied egg-rr 6.5%

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

   1. sub-neg 6.5%

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

6. Simplified 6.5%

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

   1. sub-neg 6.5%

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

   2. +-commutative 6.5%

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

   3. add-cube-cbrt 9.6%

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

   4. distribute-rgt-neg-in 9.6%

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

   5. fma-define 9.6%

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

   6. pow2 9.6%

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

8. Applied egg-rr 9.6%

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

   1. add-sqr-sqrt 9.6%

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

   2. pow2 9.6%

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

   3. pow1/3 9.6%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, -{\left(\sqrt{\color{blue}{{\sin^{-1} \left(1 - x\right)}^{0.3333333333333333}}}\right)}^{2}, \pi \cdot 0.5\right) \]

   4. sqrt-pow1 9.6%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, -{\color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{2}, \pi \cdot 0.5\right) \]

   5. metadata-eval 9.6%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, -{\left({\sin^{-1} \left(1 - x\right)}^{\color{blue}{0.16666666666666666}}\right)}^{2}, \pi \cdot 0.5\right) \]

10. Applied egg-rr 9.6%

    \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, -\color{blue}{{\left({\sin^{-1} \left(1 - x\right)}^{0.16666666666666666}\right)}^{2}}, \pi \cdot 0.5\right) \]

11. Final simplification 9.6%

    \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, -{\left({\sin^{-1} \left(1 - x\right)}^{0.16666666666666666}\right)}^{2}, \pi \cdot 0.5\right) \]
12. Add Preprocessing

Alternative 7: 10.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \pi \cdot 0.5 - {\left(\sqrt[3]{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\right)}^{1.5}}\right)}^{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* PI 0.5) (pow (cbrt (pow (expm1 (log1p (asin (- 1.0 x)))) 1.5)) 2.0)))

C

double code(double x) {
	return (((double) M_PI) * 0.5) - pow(cbrt(pow(expm1(log1p(asin((1.0 - x)))), 1.5)), 2.0);
}

Java

public static double code(double x) {
	return (Math.PI * 0.5) - Math.pow(Math.cbrt(Math.pow(Math.expm1(Math.log1p(Math.asin((1.0 - x)))), 1.5)), 2.0);
}

Julia

function code(x)
	return Float64(Float64(pi * 0.5) - (cbrt((expm1(log1p(asin(Float64(1.0 - x)))) ^ 1.5)) ^ 2.0))
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Power[N[Power[N[(Exp[N[Log[1 + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

   1. acos-asin 6.5%

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

   2. sub-neg 6.5%

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

   3. div-inv 6.5%

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

   4. metadata-eval 6.5%

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

4. Applied egg-rr 6.5%

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

   1. sub-neg 6.5%

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

6. Simplified 6.5%

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

   1. add-sqr-sqrt 9.6%

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

   2. pow2 9.6%

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

8. Applied egg-rr 9.6%

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

   1. add-cbrt-cube 9.6%

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

   2. pow1/3 9.6%

      \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left({\left(\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}}^{2} \]

   3. add-sqr-sqrt 9.6%

      \[\leadsto \pi \cdot 0.5 - {\left({\left(\color{blue}{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}^{2} \]

   4. pow1 9.6%

      \[\leadsto \pi \cdot 0.5 - {\left({\left(\color{blue}{{\sin^{-1} \left(1 - x\right)}^{1}} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}^{2} \]

   5. pow1/2 9.6%

      \[\leadsto \pi \cdot 0.5 - {\left({\left({\sin^{-1} \left(1 - x\right)}^{1} \cdot \color{blue}{{\sin^{-1} \left(1 - x\right)}^{0.5}}\right)}^{0.3333333333333333}\right)}^{2} \]

   6. pow-prod-up 9.6%

      \[\leadsto \pi \cdot 0.5 - {\left({\color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)}^{2} \]

   7. metadata-eval 9.6%

      \[\leadsto \pi \cdot 0.5 - {\left({\left({\sin^{-1} \left(1 - x\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2} \]

10. Applied egg-rr 9.6%

    \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left({\left({\sin^{-1} \left(1 - x\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2} \]
11. Step-by-step derivation

    1. unpow1/3 9.6%

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

12. Simplified 9.6%

    \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}}^{2} \]
13. Step-by-step derivation

    1. expm1-log1p-u 9.6%

       \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\right)}}^{1.5}}\right)}^{2} \]

    2. expm1-undefine 9.6%

       \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{{\color{blue}{\left(e^{\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)} - 1\right)}}^{1.5}}\right)}^{2} \]

14. Applied egg-rr 9.6%

    \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{{\color{blue}{\left(e^{\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)} - 1\right)}}^{1.5}}\right)}^{2} \]
15. Step-by-step derivation

    1. expm1-define 9.6%

       \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\right)}}^{1.5}}\right)}^{2} \]

16. Simplified 9.6%

    \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\right)}}^{1.5}}\right)}^{2} \]

17. Final simplification 9.6%

    \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\right)}^{1.5}}\right)}^{2} \]
18. Add Preprocessing

Alternative 8: 10.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \pi \cdot 0.5 - {\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}^{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* PI 0.5) (pow (cbrt (pow (asin (- 1.0 x)) 1.5)) 2.0)))

C

double code(double x) {
	return (((double) M_PI) * 0.5) - pow(cbrt(pow(asin((1.0 - x)), 1.5)), 2.0);
}

Java

public static double code(double x) {
	return (Math.PI * 0.5) - Math.pow(Math.cbrt(Math.pow(Math.asin((1.0 - x)), 1.5)), 2.0);
}

Julia

function code(x)
	return Float64(Float64(pi * 0.5) - (cbrt((asin(Float64(1.0 - x)) ^ 1.5)) ^ 2.0))
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Power[N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

   1. acos-asin 6.5%

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

   2. sub-neg 6.5%

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

   3. div-inv 6.5%

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

   4. metadata-eval 6.5%

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

4. Applied egg-rr 6.5%

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

   1. sub-neg 6.5%

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

6. Simplified 6.5%

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

   1. add-sqr-sqrt 9.6%

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

   2. pow2 9.6%

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

8. Applied egg-rr 9.6%

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

   1. add-cbrt-cube 9.6%

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

   2. pow1/3 9.6%

      \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left({\left(\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}}^{2} \]

   3. add-sqr-sqrt 9.6%

      \[\leadsto \pi \cdot 0.5 - {\left({\left(\color{blue}{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}^{2} \]

   4. pow1 9.6%

      \[\leadsto \pi \cdot 0.5 - {\left({\left(\color{blue}{{\sin^{-1} \left(1 - x\right)}^{1}} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}^{2} \]

   5. pow1/2 9.6%

      \[\leadsto \pi \cdot 0.5 - {\left({\left({\sin^{-1} \left(1 - x\right)}^{1} \cdot \color{blue}{{\sin^{-1} \left(1 - x\right)}^{0.5}}\right)}^{0.3333333333333333}\right)}^{2} \]

   6. pow-prod-up 9.6%

      \[\leadsto \pi \cdot 0.5 - {\left({\color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)}^{2} \]

   7. metadata-eval 9.6%

      \[\leadsto \pi \cdot 0.5 - {\left({\left({\sin^{-1} \left(1 - x\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2} \]

10. Applied egg-rr 9.6%

    \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left({\left({\sin^{-1} \left(1 - x\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2} \]
11. Step-by-step derivation

    1. unpow1/3 9.6%

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

12. Simplified 9.6%

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

13. Final simplification 9.6%

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

Alternative 9: 6.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\pi \cdot 0.5, \sin^{-1} \left(1 - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 1.0)
   (log (exp (acos (- 1.0 x))))
   (hypot (* PI 0.5) (asin (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = log(exp(acos((1.0 - x))));
	} else {
		tmp = hypot((((double) M_PI) * 0.5), asin((1.0 - x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = Math.log(Math.exp(Math.acos((1.0 - x))));
	} else {
		tmp = Math.hypot((Math.PI * 0.5), Math.asin((1.0 - x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 - x) <= 1.0:
		tmp = math.log(math.exp(math.acos((1.0 - x))))
	else:
		tmp = math.hypot((math.pi * 0.5), math.asin((1.0 - x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = log(exp(acos(Float64(1.0 - x))));
	else
		tmp = hypot(Float64(pi * 0.5), asin(Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 - x) <= 1.0)
		tmp = log(exp(acos((1.0 - x))));
	else
		tmp = hypot((pi * 0.5), asin((1.0 - x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[Log[N[Exp[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(Pi * 0.5), $MachinePrecision] ^ 2 + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.5%

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

      1. add-log-exp 6.5%

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

   4. Applied egg-rr 6.5%

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

   if 1 < (-.f64 1 x)

   1. Initial program 6.5%

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

      1. acos-asin 6.5%

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

      2. sub-neg 6.5%

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

      3. div-inv 6.5%

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

      4. metadata-eval 6.5%

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

   4. Applied egg-rr 6.5%

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

      1. sub-neg 6.5%

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

   6. Simplified 6.5%

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

      1. add-sqr-sqrt 6.5%

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

      2. sqrt-unprod 6.5%

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

      3. add-sqr-sqrt 6.5%

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

      4. cancel-sign-sub-inv 6.5%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 4.4%

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

      7. sqr-neg 4.4%

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

      8. add-sqr-sqrt 4.4%

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

      9. add-sqr-sqrt 4.4%

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

      10. difference-of-squares 4.4%

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

   8. Applied egg-rr 6.7%

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

4. Final simplification 6.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\pi \cdot 0.5, \sin^{-1} \left(1 - x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 6.5%

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

   1. acos-asin 6.5%

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

   2. sub-neg 6.5%

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

   3. div-inv 6.5%

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

   4. metadata-eval 6.5%

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

4. Applied egg-rr 6.5%

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

   1. sub-neg 6.5%

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

6. Simplified 6.5%

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

   1. add-cube-cbrt 9.6%

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

   2. pow3 9.6%

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

8. Applied egg-rr 9.6%

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

9. Final simplification 9.6%

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

Alternative 11: 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 6.5%

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

   1. acos-asin 6.5%

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

   2. sub-neg 6.5%

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

   3. div-inv 6.5%

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

   4. metadata-eval 6.5%

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

4. Applied egg-rr 6.5%

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

   1. sub-neg 6.5%

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

6. Simplified 6.5%

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

   1. add-sqr-sqrt 9.6%

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

   2. pow2 9.6%

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

8. Applied egg-rr 9.6%

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

9. Final simplification 9.6%

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

Alternative 12: 6.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\pi \cdot 0.5, \sin^{-1} \left(1 - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 1.0) (acos (- 1.0 x)) (hypot (* PI 0.5) (asin (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = acos((1.0 - x));
	} else {
		tmp = hypot((((double) M_PI) * 0.5), asin((1.0 - x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = Math.acos((1.0 - x));
	} else {
		tmp = Math.hypot((Math.PI * 0.5), Math.asin((1.0 - x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 - x) <= 1.0:
		tmp = math.acos((1.0 - x))
	else:
		tmp = math.hypot((math.pi * 0.5), math.asin((1.0 - x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = acos(Float64(1.0 - x));
	else
		tmp = hypot(Float64(pi * 0.5), asin(Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 - x) <= 1.0)
		tmp = acos((1.0 - x));
	else
		tmp = hypot((pi * 0.5), asin((1.0 - x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(Pi * 0.5), $MachinePrecision] ^ 2 + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.5%

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

   if 1 < (-.f64 1 x)

   1. Initial program 6.5%

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

      1. acos-asin 6.5%

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

      2. sub-neg 6.5%

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

      3. div-inv 6.5%

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

      4. metadata-eval 6.5%

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

   4. Applied egg-rr 6.5%

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

      1. sub-neg 6.5%

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

   6. Simplified 6.5%

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

      1. add-sqr-sqrt 6.5%

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

      2. sqrt-unprod 6.5%

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

      3. add-sqr-sqrt 6.5%

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

      4. cancel-sign-sub-inv 6.5%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 4.4%

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

      7. sqr-neg 4.4%

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

      8. add-sqr-sqrt 4.4%

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

      9. add-sqr-sqrt 4.4%

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

      10. difference-of-squares 4.4%

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

   8. Applied egg-rr 6.7%

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

4. Final simplification 6.5%

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

Alternative 13: 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. add-log-exp 3.9%

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

   4. Applied egg-rr 3.9%

      \[\leadsto \color{blue}{\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)} \]
   5. Step-by-step derivation

      1. rem-log-exp 3.9%

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

      2. acos-asin 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

      5. add-sqr-sqrt 7.3%

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

      6. cancel-sign-sub-inv 7.3%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 6.4%

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

      9. sqr-neg 6.4%

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

      10. add-sqr-sqrt 6.4%

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

      11. add-sqr-sqrt 6.4%

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

      12. asin-acos 6.4%

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

      13. div-inv 6.4%

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

      14. metadata-eval 6.4%

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

      15. associate-+r- 6.4%

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

   6. Applied egg-rr 6.4%

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

      1. fma-undefine 6.4%

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

      2. distribute-lft-out 6.4%

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

      3. metadata-eval 6.4%

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

   8. Simplified 6.4%

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

   9. Taylor expanded in x around 0 6.4%

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

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

   1. Initial program 54.0%

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

4. Final simplification 8.8%

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

Alternative 14: 6.8% 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 6.5%

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

3. Final simplification 6.5%

   \[\leadsto \cos^{-1} \left(1 - x\right) \]
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 2024043 
(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)))