bug323 (missed optimization)

Percentage Accurate: 6.7% → 10.1%

Time: 9.6s

Alternatives: 10

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: 6.7% 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: 10.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt[3]{t\_0}\\ t_2 := {t\_1}^{2}\\ \mathsf{fma}\left(\pi, 0.5, t\_1 \cdot \left(-t\_2\right)\right) + \mathsf{fma}\left(-{\left({t\_0}^{0.16666666666666666}\right)}^{2}, t\_2, e^{\mathsf{log1p}\left(t\_0\right)} + -1\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (cbrt t_0)) (t_2 (pow t_1 2.0)))
   (+
    (fma PI 0.5 (* t_1 (- t_2)))
    (fma
     (- (pow (pow t_0 0.16666666666666666) 2.0))
     t_2
     (+ (exp (log1p t_0)) -1.0)))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = cbrt(t_0);
	double t_2 = pow(t_1, 2.0);
	return fma(((double) M_PI), 0.5, (t_1 * -t_2)) + fma(-pow(pow(t_0, 0.16666666666666666), 2.0), t_2, (exp(log1p(t_0)) + -1.0));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = cbrt(t_0)
	t_2 = t_1 ^ 2.0
	return Float64(fma(pi, 0.5, Float64(t_1 * Float64(-t_2))) + fma(Float64(-((t_0 ^ 0.16666666666666666) ^ 2.0)), t_2, Float64(exp(log1p(t_0)) + -1.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 1/3], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, N[(N[(Pi * 0.5 + N[(t$95$1 * (-t$95$2)), $MachinePrecision]), $MachinePrecision] + N[((-N[Power[N[Power[t$95$0, 0.16666666666666666], $MachinePrecision], 2.0], $MachinePrecision]) * t$95$2 + N[(N[Exp[N[Log[1 + t$95$0], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 8.0%

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

   1. acos-asin 8.0%

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

   2. div-inv 8.0%

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

   3. metadata-eval 8.0%

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

   4. add-cube-cbrt 11.6%

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

   5. prod-diff 11.6%

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

   6. pow2 11.6%

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

   7. pow2 11.6%

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

4. Applied egg-rr 11.6%

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

   1. add-sqr-sqrt 11.6%

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

   2. pow2 11.6%

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

   3. pow1/3 11.6%

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

   4. sqrt-pow1 11.6%

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

   5. metadata-eval 11.6%

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

6. Applied egg-rr 11.6%

   \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) + \mathsf{fma}\left(-\color{blue}{{\left({\sin^{-1} \left(1 - x\right)}^{0.16666666666666666}\right)}^{2}}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) \]
7. Step-by-step derivation

   1. pow1/3 11.6%

      \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) + \mathsf{fma}\left(-{\left({\sin^{-1} \left(1 - x\right)}^{0.16666666666666666}\right)}^{2}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \color{blue}{{\sin^{-1} \left(1 - x\right)}^{0.3333333333333333}} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) \]

8. Applied egg-rr 11.6%

   \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) + \mathsf{fma}\left(-{\left({\sin^{-1} \left(1 - x\right)}^{0.16666666666666666}\right)}^{2}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \color{blue}{{\sin^{-1} \left(1 - x\right)}^{0.3333333333333333}} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) \]
9. Step-by-step derivation

   1. pow1/3 11.6%

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

   2. unpow2 11.6%

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

   3. rem-3cbrt-rft 11.6%

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

   4. expm1-log1p-u 11.6%

      \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) + \mathsf{fma}\left(-{\left({\sin^{-1} \left(1 - x\right)}^{0.16666666666666666}\right)}^{2}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)}\right) \]

   5. expm1-undefine 11.6%

      \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) + \mathsf{fma}\left(-{\left({\sin^{-1} \left(1 - x\right)}^{0.16666666666666666}\right)}^{2}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \color{blue}{e^{\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)} - 1}\right) \]

10. Applied egg-rr 11.6%

    \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) + \mathsf{fma}\left(-{\left({\sin^{-1} \left(1 - x\right)}^{0.16666666666666666}\right)}^{2}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \color{blue}{e^{\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)} - 1}\right) \]

11. Final simplification 11.6%

    \[\leadsto \mathsf{fma}\left(\pi, 0.5, \sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \left(-{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right)\right) + \mathsf{fma}\left(-{\left({\sin^{-1} \left(1 - x\right)}^{0.16666666666666666}\right)}^{2}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, e^{\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)} + -1\right) \]
12. Add Preprocessing

Alternative 2: 10.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

   1. acos-asin 8.0%

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

   2. *-un-lft-identity 8.0%

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

   3. add-sqr-sqrt 11.6%

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

   4. prod-diff 11.6%

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

   5. add-sqr-sqrt 11.6%

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

   6. fma-neg 11.6%

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

   7. *-un-lft-identity 11.6%

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

   8. acos-asin 11.6%

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

   9. add-sqr-sqrt 11.6%

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

4. Applied egg-rr 11.6%

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

   1. flip-- 11.6%

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

   2. div-inv 11.6%

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

   3. metadata-eval 11.6%

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

   4. pow2 11.6%

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

6. Applied egg-rr 11.6%

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

   1. associate-*r/ 11.6%

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

   2. *-rgt-identity 11.6%

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

   3. remove-double-neg 11.6%

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

   4. distribute-frac-neg 11.6%

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

   5. distribute-frac-neg2 11.6%

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

   6. sub-neg 11.6%

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

   7. +-commutative 11.6%

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

   8. distribute-neg-in 11.6%

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

   9. unpow2 11.6%

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

   10. sqr-neg 11.6%

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

   11. unpow2 11.6%

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

   12. remove-double-neg 11.6%

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

   13. sub-neg 11.6%

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

   14. unpow2 11.6%

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

   15. sqr-neg 11.6%

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

   16. fma-neg 11.6%

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

   17. metadata-eval 11.6%

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

   18. distribute-neg-in 11.6%

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

   19. metadata-eval 11.6%

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

   20. unsub-neg 11.6%

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

8. Simplified 11.6%

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

   1. add-cube-cbrt 11.6%

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

   2. pow3 11.6%

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

10. Applied egg-rr 11.6%

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

Alternative 3: 10.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \mathsf{fma}\left(\pi, 0.5, 0 - {\left(\sqrt[3]{t\_0}\right)}^{2} \cdot {t\_0}^{0.3333333333333333}\right) + t\_0 \cdot 0 \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (+
    (fma PI 0.5 (- 0.0 (* (pow (cbrt t_0) 2.0) (pow t_0 0.3333333333333333))))
    (* t_0 0.0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return fma(((double) M_PI), 0.5, (0.0 - (pow(cbrt(t_0), 2.0) * pow(t_0, 0.3333333333333333)))) + (t_0 * 0.0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(fma(pi, 0.5, Float64(0.0 - Float64((cbrt(t_0) ^ 2.0) * (t_0 ^ 0.3333333333333333)))) + Float64(t_0 * 0.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(Pi * 0.5 + N[(0.0 - N[(N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[t$95$0, 0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 0.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 8.0%

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

   1. acos-asin 8.0%

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

   2. div-inv 8.0%

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

   3. metadata-eval 8.0%

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

   4. add-cube-cbrt 11.6%

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

   5. prod-diff 11.6%

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

   6. pow2 11.6%

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

   7. pow2 11.6%

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

4. Applied egg-rr 11.6%

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

   1. pow1/3 11.6%

      \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) + \mathsf{fma}\left(-{\left({\sin^{-1} \left(1 - x\right)}^{0.16666666666666666}\right)}^{2}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \color{blue}{{\sin^{-1} \left(1 - x\right)}^{0.3333333333333333}} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) \]

6. Applied egg-rr 11.6%

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

   1. fma-undefine 11.6%

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

   2. distribute-lft-neg-in 11.6%

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

   3. neg-mul-1 11.6%

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

   4. metadata-eval 11.6%

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

   5. unpow2 11.6%

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

   6. rem-3cbrt-rft 6.3%

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

   7. unpow2 6.3%

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

   8. rem-3cbrt-rft 11.6%

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

   9. *-un-lft-identity 11.6%

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

   10. distribute-rgt-out 11.6%

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

   11. metadata-eval 11.6%

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

   12. metadata-eval 11.6%

       \[\leadsto \mathsf{fma}\left(\pi, 0.5, -{\sin^{-1} \left(1 - x\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{0} \]

8. Applied egg-rr 11.6%

   \[\leadsto \mathsf{fma}\left(\pi, 0.5, -{\sin^{-1} \left(1 - x\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) + \color{blue}{\sin^{-1} \left(1 - x\right) \cdot 0} \]

9. Final simplification 11.6%

   \[\leadsto \mathsf{fma}\left(\pi, 0.5, 0 - {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2} \cdot {\sin^{-1} \left(1 - x\right)}^{0.3333333333333333}\right) + \sin^{-1} \left(1 - x\right) \cdot 0 \]
10. Add Preprocessing

Alternative 4: 10.2% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (pow (sqrt (- (* 0.5 (cbrt (pow PI 3.0))) (asin (- 1.0 x)))) 2.0))

C

double code(double x) {
	return pow(sqrt(((0.5 * cbrt(pow(((double) M_PI), 3.0))) - asin((1.0 - x)))), 2.0);
}

Java

public static double code(double x) {
	return Math.pow(Math.sqrt(((0.5 * Math.cbrt(Math.pow(Math.PI, 3.0))) - Math.asin((1.0 - x)))), 2.0);
}

Julia

function code(x)
	return sqrt(Float64(Float64(0.5 * cbrt((pi ^ 3.0))) - asin(Float64(1.0 - x)))) ^ 2.0
end

Wolfram

code[x_] := N[Power[N[Sqrt[N[(N[(0.5 * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] - N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 8.0%

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

   1. add-sqr-sqrt 8.0%

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

   2. pow2 8.0%

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

4. Applied egg-rr 8.0%

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

   1. acos-asin 8.0%

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

   2. sub-neg 8.0%

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

   3. div-inv 8.0%

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

   4. metadata-eval 8.0%

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

6. Applied egg-rr 8.0%

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

   1. sub-neg 8.0%

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

8. Simplified 8.0%

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

   1. add-cbrt-cube 11.6%

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

   2. pow3 11.6%

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

10. Applied egg-rr 11.6%

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

11. Final simplification 11.6%

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

Alternative 5: 6.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;{\left(\sqrt[3]{{\cos^{-1} \left(1 - x\right)}^{1.5}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 1.0)
   (pow (cbrt (pow (acos (- 1.0 x)) 1.5)) 2.0)
   (acos (- x))))

C

double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = pow(cbrt(pow(acos((1.0 - x)), 1.5)), 2.0);
	} else {
		tmp = acos(-x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = Math.pow(Math.cbrt(Math.pow(Math.acos((1.0 - x)), 1.5)), 2.0);
	} else {
		tmp = Math.acos(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = cbrt((acos(Float64(1.0 - x)) ^ 1.5)) ^ 2.0;
	else
		tmp = acos(Float64(-x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[Power[N[Power[N[Power[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision], N[ArcCos[(-x)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.0%

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

      1. add-sqr-sqrt 8.0%

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

      2. pow2 8.0%

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

   4. Applied egg-rr 8.0%

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

      1. add-cbrt-cube 8.0%

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

      2. pow1/3 8.0%

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

      3. add-sqr-sqrt 8.0%

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

      4. pow1 8.0%

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

      5. pow1/2 8.0%

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

      6. pow-prod-up 8.0%

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

      7. metadata-eval 8.0%

         \[\leadsto {\left({\left({\cos^{-1} \left(1 - x\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2} \]

   6. Applied egg-rr 8.0%

      \[\leadsto {\color{blue}{\left({\left({\cos^{-1} \left(1 - x\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2} \]

   8. Simplified 8.0%

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

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

   1. Initial program 8.0%

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

   3. Taylor expanded in x around inf 7.1%

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

      1. neg-mul-1 7.1%

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

   5. Simplified 7.1%

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

Alternative 6: 9.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{0.3333333333333333 \cdot \left(3 \cdot \log \cos^{-1} \left(1 - x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.6e-17)
   (acos (- x))
   (exp (* 0.3333333333333333 (* 3.0 (log (acos (- 1.0 x))))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = acos(-x);
	} else {
		tmp = exp((0.3333333333333333 * (3.0 * log(acos((1.0 - x))))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.6d-17) then
        tmp = acos(-x)
    else
        tmp = exp((0.3333333333333333d0 * (3.0d0 * log(acos((1.0d0 - x))))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = Math.acos(-x);
	} else {
		tmp = Math.exp((0.3333333333333333 * (3.0 * Math.log(Math.acos((1.0 - x))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.6e-17:
		tmp = math.acos(-x)
	else:
		tmp = math.exp((0.3333333333333333 * (3.0 * math.log(math.acos((1.0 - x))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = acos(Float64(-x));
	else
		tmp = exp(Float64(0.3333333333333333 * Float64(3.0 * log(acos(Float64(1.0 - x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.6e-17)
		tmp = acos(-x);
	else
		tmp = exp((0.3333333333333333 * (3.0 * log(acos((1.0 - x))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.6e-17], N[ArcCos[(-x)], $MachinePrecision], N[Exp[N[(0.3333333333333333 * N[(3.0 * N[Log[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.8%

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

   3. Taylor expanded in x around inf 6.6%

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

      1. neg-mul-1 6.6%

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

   5. Simplified 6.6%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 67.3%

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

      1. add-cbrt-cube 67.1%

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

      2. pow1/3 67.3%

         \[\leadsto \color{blue}{{\left(\left(\cos^{-1} \left(1 - x\right) \cdot \cos^{-1} \left(1 - x\right)\right) \cdot \cos^{-1} \left(1 - x\right)\right)}^{0.3333333333333333}} \]

      3. pow-to-exp 67.4%

         \[\leadsto \color{blue}{e^{\log \left(\left(\cos^{-1} \left(1 - x\right) \cdot \cos^{-1} \left(1 - x\right)\right) \cdot \cos^{-1} \left(1 - x\right)\right) \cdot 0.3333333333333333}} \]

      4. pow3 67.4%

         \[\leadsto e^{\log \color{blue}{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)} \cdot 0.3333333333333333} \]

      5. log-pow 67.3%

         \[\leadsto e^{\color{blue}{\left(3 \cdot \log \cos^{-1} \left(1 - x\right)\right)} \cdot 0.3333333333333333} \]

   4. Applied egg-rr 67.3%

      \[\leadsto \color{blue}{e^{\left(3 \cdot \log \cos^{-1} \left(1 - x\right)\right) \cdot 0.3333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{0.3333333333333333 \cdot \left(3 \cdot \log \cos^{-1} \left(1 - x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 6.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = exp(log(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 = exp(log(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.exp(Math.log(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.exp(math.log(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 = exp(log(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 = exp(log(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[Exp[N[Log[N[ArcCos[N[(1.0 - x), $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 8.0%

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

      1. add-exp-log 8.0%

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

   4. Applied egg-rr 8.0%

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

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

   1. Initial program 8.0%

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

   3. Taylor expanded in x around inf 7.1%

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

      1. neg-mul-1 7.1%

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

   5. Simplified 7.1%

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

Alternative 8: 9.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.6e-17) (acos (- x)) (acos (- 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = acos(-x);
	} else {
		tmp = acos((1.0 - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.6d-17) then
        tmp = acos(-x)
    else
        tmp = acos((1.0d0 - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = Math.acos(-x);
	} else {
		tmp = Math.acos((1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.6e-17:
		tmp = math.acos(-x)
	else:
		tmp = math.acos((1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = acos(Float64(-x));
	else
		tmp = acos(Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.6e-17)
		tmp = acos(-x);
	else
		tmp = acos((1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.6e-17], N[ArcCos[(-x)], $MachinePrecision], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.8%

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

   3. Taylor expanded in x around inf 6.6%

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

      1. neg-mul-1 6.6%

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

   5. Simplified 6.6%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 67.3%

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

Alternative 9: 6.9% accurate, 1.0× speedup

Math

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

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(Float64(-x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

3. Taylor expanded in x around inf 7.1%

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

   1. neg-mul-1 7.1%

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

5. Simplified 7.1%

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

Alternative 10: 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 8.0%

   \[\cos^{-1} \left(1 - x\right) \]
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 2024145 
(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)))