bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.4%

Time: 10.3s

Alternatives: 12

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{0.6666666666666666}, \sqrt[3]{\pi \cdot 0.5}, -t\_0\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(\frac{1 - {x}^{3}}{\mathsf{fma}\left(x, 1 + x, 1\right)}\right)}, \sqrt{t\_0}, t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (+
    (fma (pow (* PI 0.5) 0.6666666666666666) (cbrt (* PI 0.5)) (- t_0))
    (fma
     (- (sqrt (asin (/ (- 1.0 (pow x 3.0)) (fma x (+ 1.0 x) 1.0)))))
     (sqrt t_0)
     t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return fma(pow((((double) M_PI) * 0.5), 0.6666666666666666), cbrt((((double) M_PI) * 0.5)), -t_0) + fma(-sqrt(asin(((1.0 - pow(x, 3.0)) / fma(x, (1.0 + x), 1.0)))), sqrt(t_0), t_0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(fma((Float64(pi * 0.5) ^ 0.6666666666666666), cbrt(Float64(pi * 0.5)), Float64(-t_0)) + fma(Float64(-sqrt(asin(Float64(Float64(1.0 - (x ^ 3.0)) / fma(x, Float64(1.0 + x), 1.0))))), sqrt(t_0), t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Power[N[(Pi * 0.5), $MachinePrecision], 0.6666666666666666], $MachinePrecision] * N[Power[N[(Pi * 0.5), $MachinePrecision], 1/3], $MachinePrecision] + (-t$95$0)), $MachinePrecision] + N[((-N[Sqrt[N[ArcSin[N[(N[(1.0 - N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] / N[(x * N[(1.0 + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]) * N[Sqrt[t$95$0], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.7%

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

   1. acos-asin 6.8%

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

   2. *-un-lft-identity 6.8%

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

   3. add-sqr-sqrt 10.2%

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

   4. prod-diff 10.1%

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

   5. add-sqr-sqrt 10.2%

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

   6. fmm-def 10.2%

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

   7. *-un-lft-identity 10.2%

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

   8. acos-asin 10.2%

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

   9. add-sqr-sqrt 10.1%

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

4. Applied egg-rr 10.1%

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

   1. acos-asin 10.1%

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

   2. add-cube-cbrt 4.9%

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

   3. fmm-def 4.9%

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

   4. cbrt-unprod 10.2%

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

   5. pow2 10.2%

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

   6. div-inv 10.2%

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

   7. metadata-eval 10.2%

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

   8. div-inv 10.2%

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

   9. metadata-eval 10.2%

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

6. Applied egg-rr 10.2%

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

   1. flip3-- 10.3%

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

   2. div-inv 10.3%

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

   3. metadata-eval 10.3%

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

   4. metadata-eval 10.3%

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

   5. +-commutative 10.3%

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

   6. distribute-rgt-out 10.3%

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

   7. +-commutative 10.3%

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

   8. fma-define 10.3%

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

8. Applied egg-rr 10.3%

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

   1. associate-*r/ 10.3%

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

   2. *-rgt-identity 10.3%

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

10. Simplified 10.3%

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

    1. pow1/3 10.3%

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

    2. pow-pow 10.3%

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

    3. metadata-eval 10.3%

       \[\leadsto \mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{\color{blue}{0.6666666666666666}}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(\frac{1 - {x}^{3}}{\mathsf{fma}\left(x, 1 + x, 1\right)}\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right) \]

12. Applied egg-rr 10.3%

    \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\pi \cdot 0.5\right)}^{0.6666666666666666}}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(\frac{1 - {x}^{3}}{\mathsf{fma}\left(x, 1 + x, 1\right)}\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right) \]
13. Add Preprocessing

Alternative 2: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt{t\_0}\\ \mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{0.6666666666666666}, \sqrt[3]{\pi \cdot 0.5}, -t\_0\right) + \mathsf{fma}\left(-t\_1, t\_1, t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (sqrt t_0)))
   (+
    (fma (pow (* PI 0.5) 0.6666666666666666) (cbrt (* PI 0.5)) (- t_0))
    (fma (- t_1) t_1 t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = sqrt(t_0);
	return fma(pow((((double) M_PI) * 0.5), 0.6666666666666666), cbrt((((double) M_PI) * 0.5)), -t_0) + fma(-t_1, t_1, t_0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = sqrt(t_0)
	return Float64(fma((Float64(pi * 0.5) ^ 0.6666666666666666), cbrt(Float64(pi * 0.5)), Float64(-t_0)) + fma(Float64(-t_1), t_1, t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, N[(N[(N[Power[N[(Pi * 0.5), $MachinePrecision], 0.6666666666666666], $MachinePrecision] * N[Power[N[(Pi * 0.5), $MachinePrecision], 1/3], $MachinePrecision] + (-t$95$0)), $MachinePrecision] + N[((-t$95$1) * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 6.7%

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

   1. acos-asin 6.8%

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

   2. *-un-lft-identity 6.8%

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

   3. add-sqr-sqrt 10.2%

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

   4. prod-diff 10.1%

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

   5. add-sqr-sqrt 10.2%

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

   6. fmm-def 10.2%

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

   7. *-un-lft-identity 10.2%

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

   8. acos-asin 10.2%

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

   9. add-sqr-sqrt 10.1%

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

4. Applied egg-rr 10.1%

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

   1. acos-asin 10.1%

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

   2. add-cube-cbrt 4.9%

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

   3. fmm-def 4.9%

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

   4. cbrt-unprod 10.2%

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

   5. pow2 10.2%

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

   6. div-inv 10.2%

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

   7. metadata-eval 10.2%

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

   8. div-inv 10.2%

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

   9. metadata-eval 10.2%

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

6. Applied egg-rr 10.2%

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

   1. pow1/3 10.3%

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

   2. pow-pow 10.3%

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

   3. metadata-eval 10.3%

      \[\leadsto \mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{\color{blue}{0.6666666666666666}}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(\frac{1 - {x}^{3}}{\mathsf{fma}\left(x, 1 + x, 1\right)}\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right) \]

8. Applied egg-rr 10.2%

   \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\pi \cdot 0.5\right)}^{0.6666666666666666}}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right) \]
9. Add Preprocessing

Alternative 3: 10.4% 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 6.7%

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

   1. acos-asin 6.8%

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

   2. *-un-lft-identity 6.8%

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

   3. add-sqr-sqrt 10.2%

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

   4. prod-diff 10.1%

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

   5. add-sqr-sqrt 10.2%

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

   6. fmm-def 10.2%

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

   7. *-un-lft-identity 10.2%

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

   8. acos-asin 10.2%

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

   9. add-sqr-sqrt 10.1%

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

4. Applied egg-rr 10.1%

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

   1. add-sqr-sqrt 10.2%

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

   2. pow2 10.2%

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

6. Applied egg-rr 10.2%

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

Alternative 4: 10.4% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (-
  (* PI 0.5)
  (*
   (sqrt (asin (/ (- 1.0 (pow x 3.0)) (fma x (+ 1.0 x) 1.0))))
   (sqrt (asin (- 1.0 x))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

   1. acos-asin 6.8%

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

   2. *-un-lft-identity 6.8%

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

   3. add-sqr-sqrt 10.2%

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

   4. prod-diff 10.1%

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

   5. add-sqr-sqrt 10.2%

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

   6. fmm-def 10.2%

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

   7. *-un-lft-identity 10.2%

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

   8. acos-asin 10.2%

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

   9. add-sqr-sqrt 10.1%

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

4. Applied egg-rr 10.1%

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

   1. acos-asin 10.1%

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

   2. add-cube-cbrt 4.9%

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

   3. fmm-def 4.9%

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

   4. cbrt-unprod 10.2%

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

   5. pow2 10.2%

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

   6. div-inv 10.2%

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

   7. metadata-eval 10.2%

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

   8. div-inv 10.2%

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

   9. metadata-eval 10.2%

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

6. Applied egg-rr 10.2%

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

   1. flip3-- 10.3%

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

   2. div-inv 10.3%

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

   3. metadata-eval 10.3%

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

   4. metadata-eval 10.3%

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

   5. +-commutative 10.3%

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

   6. distribute-rgt-out 10.3%

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

   7. +-commutative 10.3%

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

   8. fma-define 10.3%

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

8. Applied egg-rr 10.3%

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

   1. associate-*r/ 10.3%

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

   2. *-rgt-identity 10.3%

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

10. Simplified 10.3%

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

    1. expm1-log1p-u 10.3%

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

    2. expm1-undefine 6.8%

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

12. Applied egg-rr 10.2%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(\frac{1 - {x}^{3}}{\mathsf{fma}\left(x, 1 + x, 1\right)}\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right)\right)} - 1} \]
13. Step-by-step derivation

    1. sub-neg 10.2%

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

    2. metadata-eval 10.2%

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

    3. +-commutative 10.2%

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

    4. log1p-undefine 10.2%

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

    5. rem-exp-log 10.2%

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

14. Simplified 10.2%

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

Alternative 5: 6.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 1

   1. Initial program 6.7%

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

      1. add-log-exp 6.7%

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

   4. Applied egg-rr 6.7%

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

      1. acos-asin 6.8%

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

      2. sub-neg 6.8%

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

      3. div-inv 6.8%

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

      4. metadata-eval 6.8%

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

   6. Applied egg-rr 6.8%

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

      1. sub-neg 6.8%

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

   8. Simplified 6.8%

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

   if 1 < (-.f64 #s(literal 1 binary64) x)

   1. Initial program 6.7%

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

   3. Taylor expanded in x around inf 6.8%

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

      1. neg-mul-1 6.8%

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

   5. Simplified 6.8%

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

Alternative 6: 10.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\pi}^{3} \cdot 0.125} - \sin^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (cbrt (* (pow PI 3.0) 0.125)) (asin (- 1.0 x))))

C

double code(double x) {
	return cbrt((pow(((double) M_PI), 3.0) * 0.125)) - asin((1.0 - x));
}

Java

public static double code(double x) {
	return Math.cbrt((Math.pow(Math.PI, 3.0) * 0.125)) - Math.asin((1.0 - x));
}

Julia

function code(x)
	return Float64(cbrt(Float64((pi ^ 3.0) * 0.125)) - asin(Float64(1.0 - x)))
end

Wolfram

code[x_] := N[(N[Power[N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.125), $MachinePrecision], 1/3], $MachinePrecision] - N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.7%

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

   1. acos-asin 6.8%

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

   2. add-cube-cbrt 4.9%

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

   3. *-un-lft-identity 4.9%

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

   4. prod-diff 4.9%

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

   5. cbrt-unprod 4.9%

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

   6. pow2 4.9%

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

   7. div-inv 4.9%

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

   8. metadata-eval 4.9%

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

   9. div-inv 4.9%

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

   10. metadata-eval 4.9%

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

4. Applied egg-rr 4.9%

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

   1. fma-undefine 4.9%

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

   2. *-rgt-identity 4.9%

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

   3. *-rgt-identity 4.9%

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

   4. +-commutative 4.9%

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

   5. sub-neg 4.9%

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

   6. +-inverses 4.9%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{2}}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right) \cdot 1\right) + \color{blue}{0} \]

   7. +-rgt-identity 4.9%

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

   8. fmm-undef 4.9%

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

   9. *-rgt-identity 4.9%

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

6. Simplified 4.9%

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

   1. cbrt-unprod 10.2%

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

   2. unpow2 10.2%

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

   3. pow3 10.2%

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

   4. unpow-prod-down 10.2%

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

   5. metadata-eval 10.2%

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

8. Applied egg-rr 10.2%

   \[\leadsto \color{blue}{\sqrt[3]{{\pi}^{3} \cdot 0.125}} - \sin^{-1} \left(1 - x\right) \]
9. Add Preprocessing

Alternative 7: 6.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\left(1 + \cos^{-1} \left(1 - x\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 1.0) (+ (+ 1.0 (acos (- 1.0 x))) -1.0) (acos (- x))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((1.0d0 - x) <= 1.0d0) then
        tmp = (1.0d0 + acos((1.0d0 - x))) + (-1.0d0)
    else
        tmp = acos(-x)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 1

   1. Initial program 6.7%

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

      1. expm1-log1p-u 6.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      2. expm1-undefine 6.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      3. log1p-undefine 6.8%

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

      4. rem-exp-log 6.8%

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

   4. Applied egg-rr 6.8%

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

   if 1 < (-.f64 #s(literal 1 binary64) x)

   1. Initial program 6.7%

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

   3. Taylor expanded in x around inf 6.8%

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

      1. neg-mul-1 6.8%

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

   5. Simplified 6.8%

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

4. Final simplification 6.8%

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

Alternative 8: 6.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 1

   1. Initial program 6.7%

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

      1. acos-asin 6.8%

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

      2. sub-neg 6.8%

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

      3. div-inv 6.8%

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

      4. metadata-eval 6.8%

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

   4. Applied egg-rr 6.8%

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

      1. sub-neg 6.8%

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

   6. Simplified 6.8%

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

   if 1 < (-.f64 #s(literal 1 binary64) x)

   1. Initial program 6.7%

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

   3. Taylor expanded in x around inf 6.8%

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

      1. neg-mul-1 6.8%

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

   5. Simplified 6.8%

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

Alternative 9: 6.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;1 + \left(\cos^{-1} \left(1 - x\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 1.0) (+ 1.0 (+ (acos (- 1.0 x)) -1.0)) (acos (- x))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((1.0d0 - x) <= 1.0d0) then
        tmp = 1.0d0 + (acos((1.0d0 - x)) + (-1.0d0))
    else
        tmp = acos(-x)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 1

   1. Initial program 6.7%

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

      1. expm1-log1p-u 6.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      2. expm1-undefine 6.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      3. log1p-undefine 6.8%

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

      4. rem-exp-log 6.8%

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

   4. Applied egg-rr 6.8%

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

      1. associate--l+ 6.7%

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

      2. +-commutative 6.7%

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

      3. sub-neg 6.7%

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

      4. metadata-eval 6.7%

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

   6. Applied egg-rr 6.7%

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

   if 1 < (-.f64 #s(literal 1 binary64) x)

   1. Initial program 6.7%

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

   3. Taylor expanded in x around inf 6.8%

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

      1. neg-mul-1 6.8%

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

   5. Simplified 6.8%

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

4. Final simplification 6.7%

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

Alternative 10: 6.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 1.0) (acos (- 1.0 x)) (acos (- x))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((1.0d0 - x) <= 1.0d0) then
        tmp = acos((1.0d0 - x))
    else
        tmp = acos(-x)
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if (1.0 - x) <= 1.0:
		tmp = math.acos((1.0 - x))
	else:
		tmp = math.acos(-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 = acos(Float64(-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 = acos(-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[ArcCos[(-x)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 1

   1. Initial program 6.7%

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

   if 1 < (-.f64 #s(literal 1 binary64) x)

   1. Initial program 6.7%

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

   3. Taylor expanded in x around inf 6.8%

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

      1. neg-mul-1 6.8%

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

   5. Simplified 6.8%

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

Alternative 11: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return acos(x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

3. Taylor expanded in x around inf 6.8%

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

   1. neg-mul-1 6.8%

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

5. Simplified 6.8%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 6.8%

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

   3. sqr-neg 6.8%

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

   4. sqrt-unprod 6.8%

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

   5. add-sqr-sqrt 6.8%

      \[\leadsto \cos^{-1} \color{blue}{x} \]

   6. *-un-lft-identity 6.8%

      \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]

7. Applied egg-rr 6.8%

   \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]
8. Step-by-step derivation

   1. *-lft-identity 6.8%

      \[\leadsto \color{blue}{\cos^{-1} x} \]

9. Simplified 6.8%

   \[\leadsto \color{blue}{\cos^{-1} x} \]
10. Add Preprocessing

Alternative 12: 3.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return acos(1.0)
end

MATLAB

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

Wolfram

code[x_] := N[ArcCos[1.0], $MachinePrecision]

Derivation

1. Initial program 6.7%

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

3. Taylor expanded in x around 0 3.8%

   \[\leadsto \cos^{-1} \color{blue}{1} \]
4. Add Preprocessing

Developer Target 1: 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 2024165 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :alt
  (! :herbie-platform default (* 2 (asin (sqrt (/ x 2)))))

  (acos (- 1.0 x)))