bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.3%

Time: 39.5s

Alternatives: 17

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: 10.3% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (exp (acos (- 1.0 x)))))
        (t_1 (cbrt (asin (- 1.0 x))))
        (t_2 (cbrt (pow t_1 3.0))))
   (+
    (+ (log (pow t_0 2.0)) (log t_0))
    (fma (- t_1) (pow t_1 2.0) (* t_2 (pow t_2 2.0))))))

C

double code(double x) {
	double t_0 = cbrt(exp(acos((1.0 - x))));
	double t_1 = cbrt(asin((1.0 - x)));
	double t_2 = cbrt(pow(t_1, 3.0));
	return (log(pow(t_0, 2.0)) + log(t_0)) + fma(-t_1, pow(t_1, 2.0), (t_2 * pow(t_2, 2.0)));
}

Julia

function code(x)
	t_0 = cbrt(exp(acos(Float64(1.0 - x))))
	t_1 = cbrt(asin(Float64(1.0 - x)))
	t_2 = cbrt((t_1 ^ 3.0))
	return Float64(Float64(log((t_0 ^ 2.0)) + log(t_0)) + fma(Float64(-t_1), (t_1 ^ 2.0), Float64(t_2 * (t_2 ^ 2.0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Exp[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Power[t$95$1, 3.0], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(N[Log[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision] + N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] + N[((-t$95$1) * N[Power[t$95$1, 2.0], $MachinePrecision] + N[(t$95$2 * N[Power[t$95$2, 2.0], $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 6.7%

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

   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. add-cube-cbrt 6.7%

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

   4. prod-diff 6.7%

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

4. Applied egg-rr 10.2%

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

   1. unpow2 10.2%

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

   2. cbrt-prod 6.7%

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

   3. unpow2 6.7%

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

   4. rem-3cbrt-rft 4.9%

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

   5. fma-neg 4.9%

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

   6. add-cube-cbrt 10.2%

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

   7. metadata-eval 10.2%

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

   8. div-inv 10.2%

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

   9. acos-asin 10.2%

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

   10. add-log-exp 10.2%

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

   11. add-cube-cbrt 10.2%

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

6. Applied egg-rr 10.2%

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

   1. add-cube-cbrt 10.2%

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

   2. pow3 10.2%

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

8. Applied egg-rr 10.2%

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

   1. add-cube-cbrt 10.2%

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

   2. pow3 10.2%

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

10. Applied egg-rr 10.2%

    \[\leadsto \left(\log \left({\left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(1 - x\right)}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \sqrt[3]{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \cdot {\left(\sqrt[3]{\color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}}}\right)}^{2}\right) \]
11. Add Preprocessing

Alternative 2: 10.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sin^{-1} \left(1 - x\right)}\\ t_1 := {t\_0}^{2}\\ t_2 := \cos^{-1} \left(1 - x\right)\\ \left(\log \left({\left(\sqrt[3]{e^{t\_2}}\right)}^{2}\right) + \left(e^{\mathsf{log1p}\left(t\_2 \cdot 0.3333333333333333\right)} + -1\right)\right) + \mathsf{fma}\left(-t\_0, t\_1, t\_1 \cdot \sqrt[3]{{t\_0}^{3}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (asin (- 1.0 x))))
        (t_1 (pow t_0 2.0))
        (t_2 (acos (- 1.0 x))))
   (+
    (+
     (log (pow (cbrt (exp t_2)) 2.0))
     (+ (exp (log1p (* t_2 0.3333333333333333))) -1.0))
    (fma (- t_0) t_1 (* t_1 (cbrt (pow t_0 3.0)))))))

C

double code(double x) {
	double t_0 = cbrt(asin((1.0 - x)));
	double t_1 = pow(t_0, 2.0);
	double t_2 = acos((1.0 - x));
	return (log(pow(cbrt(exp(t_2)), 2.0)) + (exp(log1p((t_2 * 0.3333333333333333))) + -1.0)) + fma(-t_0, t_1, (t_1 * cbrt(pow(t_0, 3.0))));
}

Julia

function code(x)
	t_0 = cbrt(asin(Float64(1.0 - x)))
	t_1 = t_0 ^ 2.0
	t_2 = acos(Float64(1.0 - x))
	return Float64(Float64(log((cbrt(exp(t_2)) ^ 2.0)) + Float64(exp(log1p(Float64(t_2 * 0.3333333333333333))) + -1.0)) + fma(Float64(-t_0), t_1, Float64(t_1 * cbrt((t_0 ^ 3.0)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, Block[{t$95$2 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Log[N[Power[N[Power[N[Exp[t$95$2], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + N[(N[Exp[N[Log[1 + N[(t$95$2 * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[((-t$95$0) * t$95$1 + N[(t$95$1 * N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 1/3], $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 6.7%

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

   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. add-cube-cbrt 6.7%

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

   4. prod-diff 6.7%

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

4. Applied egg-rr 10.2%

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

   1. unpow2 10.2%

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

   2. cbrt-prod 6.7%

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

   3. unpow2 6.7%

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

   4. rem-3cbrt-rft 4.9%

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

   5. fma-neg 4.9%

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

   6. add-cube-cbrt 10.2%

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

   7. metadata-eval 10.2%

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

   8. div-inv 10.2%

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

   9. acos-asin 10.2%

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

   10. add-log-exp 10.2%

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

   11. add-cube-cbrt 10.2%

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

6. Applied egg-rr 10.2%

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

   1. add-cube-cbrt 10.2%

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

   2. pow3 10.2%

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

8. Applied egg-rr 10.2%

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

   1. expm1-log1p-u 10.2%

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

   2. expm1-undefine 10.2%

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

   3. pow1/3 10.2%

      \[\leadsto \left(\log \left({\left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)}^{2}\right) + \left(e^{\mathsf{log1p}\left(\log \color{blue}{\left({\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}\right)} - 1\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(1 - x\right)}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \sqrt[3]{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) \]

   4. log-pow 10.2%

      \[\leadsto \left(\log \left({\left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)}^{2}\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{0.3333333333333333 \cdot \log \left(e^{\cos^{-1} \left(1 - x\right)}\right)}\right)} - 1\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(1 - x\right)}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \sqrt[3]{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) \]

   5. add-log-exp 10.2%

      \[\leadsto \left(\log \left({\left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)}^{2}\right) + \left(e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \color{blue}{\cos^{-1} \left(1 - x\right)}\right)} - 1\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(1 - x\right)}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \sqrt[3]{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) \]

10. Applied egg-rr 10.2%

    \[\leadsto \left(\log \left({\left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)}^{2}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(1 - x\right)\right)} - 1\right)}\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(1 - x\right)}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \sqrt[3]{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) \]

11. Final simplification 10.2%

    \[\leadsto \left(\log \left({\left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)}^{2}\right) + \left(e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right) \cdot 0.3333333333333333\right)} + -1\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(1 - x\right)}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}}\right) \]
12. Add Preprocessing

Alternative 3: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sin^{-1} \left(1 - x\right)}\\ t_1 := \cos^{-1} \left(1 - x\right)\\ \left(\log \left({\left(\sqrt[3]{e^{t\_1}}\right)}^{2}\right) + t\_1 \cdot 0.3333333333333333\right) + \mathsf{fma}\left(-t\_0, {t\_0}^{2}, t\_0 \cdot {\left(\sqrt[3]{{t\_0}^{3}}\right)}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (asin (- 1.0 x)))) (t_1 (acos (- 1.0 x))))
   (+
    (+ (log (pow (cbrt (exp t_1)) 2.0)) (* t_1 0.3333333333333333))
    (fma (- t_0) (pow t_0 2.0) (* t_0 (pow (cbrt (pow t_0 3.0)) 2.0))))))

C

double code(double x) {
	double t_0 = cbrt(asin((1.0 - x)));
	double t_1 = acos((1.0 - x));
	return (log(pow(cbrt(exp(t_1)), 2.0)) + (t_1 * 0.3333333333333333)) + fma(-t_0, pow(t_0, 2.0), (t_0 * pow(cbrt(pow(t_0, 3.0)), 2.0)));
}

Julia

function code(x)
	t_0 = cbrt(asin(Float64(1.0 - x)))
	t_1 = acos(Float64(1.0 - x))
	return Float64(Float64(log((cbrt(exp(t_1)) ^ 2.0)) + Float64(t_1 * 0.3333333333333333)) + fma(Float64(-t_0), (t_0 ^ 2.0), Float64(t_0 * (cbrt((t_0 ^ 3.0)) ^ 2.0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Log[N[Power[N[Power[N[Exp[t$95$1], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + N[(t$95$1 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[((-t$95$0) * N[Power[t$95$0, 2.0], $MachinePrecision] + N[(t$95$0 * N[Power[N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $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 6.7%

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

   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. add-cube-cbrt 6.7%

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

   4. prod-diff 6.7%

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

4. Applied egg-rr 10.2%

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

   1. unpow2 10.2%

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

   2. cbrt-prod 6.7%

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

   3. unpow2 6.7%

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

   4. rem-3cbrt-rft 4.9%

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

   5. fma-neg 4.9%

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

   6. add-cube-cbrt 10.2%

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

   7. metadata-eval 10.2%

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

   8. div-inv 10.2%

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

   9. acos-asin 10.2%

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

   10. add-log-exp 10.2%

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

   11. add-cube-cbrt 10.2%

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

6. Applied egg-rr 10.2%

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

   1. add-cube-cbrt 10.2%

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

   2. pow3 10.2%

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

8. Applied egg-rr 10.2%

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

9. Taylor expanded in x around -inf 10.2%

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

    1. mul-1-neg 10.2%

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

    2. sub-neg 10.2%

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

    3. unpow1/3 10.2%

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

    4. exp-prod 10.2%

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

    5. *-commutative 10.2%

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

    6. rem-log-exp 10.2%

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

    7. *-commutative 10.2%

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

11. Simplified 10.2%

    \[\leadsto \left(\log \left({\left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)}^{2}\right) + \color{blue}{\cos^{-1} \left(1 - x\right) \cdot 0.3333333333333333}\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(1 - x\right)}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot {\left(\sqrt[3]{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}}\right)}^{2}\right) \]
12. Add Preprocessing

Alternative 4: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sin^{-1} \left(1 - x\right)}\\ t_1 := \cos^{-1} \left(1 - x\right)\\ \mathsf{fma}\left(-t\_0, {t\_0}^{2}, t\_0 \cdot {\left(\sqrt[3]{{t\_0}^{3}}\right)}^{2}\right) + \left(\log \left(\sqrt[3]{e^{t\_1}}\right) + t\_1 \cdot 0.6666666666666666\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (asin (- 1.0 x)))) (t_1 (acos (- 1.0 x))))
   (+
    (fma (- t_0) (pow t_0 2.0) (* t_0 (pow (cbrt (pow t_0 3.0)) 2.0)))
    (+ (log (cbrt (exp t_1))) (* t_1 0.6666666666666666)))))

C

double code(double x) {
	double t_0 = cbrt(asin((1.0 - x)));
	double t_1 = acos((1.0 - x));
	return fma(-t_0, pow(t_0, 2.0), (t_0 * pow(cbrt(pow(t_0, 3.0)), 2.0))) + (log(cbrt(exp(t_1))) + (t_1 * 0.6666666666666666));
}

Julia

function code(x)
	t_0 = cbrt(asin(Float64(1.0 - x)))
	t_1 = acos(Float64(1.0 - x))
	return Float64(fma(Float64(-t_0), (t_0 ^ 2.0), Float64(t_0 * (cbrt((t_0 ^ 3.0)) ^ 2.0))) + Float64(log(cbrt(exp(t_1))) + Float64(t_1 * 0.6666666666666666)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[((-t$95$0) * N[Power[t$95$0, 2.0], $MachinePrecision] + N[(t$95$0 * N[Power[N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[N[Power[N[Exp[t$95$1], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] + N[(t$95$1 * 0.6666666666666666), $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 6.7%

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

   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. add-cube-cbrt 6.7%

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

   4. prod-diff 6.7%

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

4. Applied egg-rr 10.2%

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

   1. unpow2 10.2%

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

   2. cbrt-prod 6.7%

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

   3. unpow2 6.7%

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

   4. rem-3cbrt-rft 4.9%

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

   5. fma-neg 4.9%

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

   6. add-cube-cbrt 10.2%

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

   7. metadata-eval 10.2%

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

   8. div-inv 10.2%

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

   9. acos-asin 10.2%

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

   10. add-log-exp 10.2%

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

   11. add-cube-cbrt 10.2%

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

6. Applied egg-rr 10.2%

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

   1. add-cube-cbrt 10.2%

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

   2. pow3 10.2%

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

8. Applied egg-rr 10.2%

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

   1. unpow2 10.2%

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

   2. log-prod 10.2%

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

   3. pow1/3 10.2%

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

   4. log-pow 10.2%

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

   5. add-log-exp 10.2%

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

   6. pow1/3 10.2%

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

   7. log-pow 10.2%

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

   8. add-log-exp 10.2%

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

10. Applied egg-rr 10.2%

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

    1. distribute-rgt-out 10.2%

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

    2. metadata-eval 10.2%

       \[\leadsto \left(\cos^{-1} \left(1 - x\right) \cdot \color{blue}{0.6666666666666666} + \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(1 - x\right)}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot {\left(\sqrt[3]{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}}\right)}^{2}\right) \]

12. Simplified 10.2%

    \[\leadsto \left(\color{blue}{\cos^{-1} \left(1 - x\right) \cdot 0.6666666666666666} + \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(1 - x\right)}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot {\left(\sqrt[3]{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}}\right)}^{2}\right) \]

13. Final simplification 10.2%

    \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(1 - x\right)}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot {\left(\sqrt[3]{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}}\right)}^{2}\right) + \left(\log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right) + \cos^{-1} \left(1 - x\right) \cdot 0.6666666666666666\right) \]
14. Add Preprocessing

Alternative 5: 10.3% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (asin (- 1.0 x)))) (t_1 (pow t_0 2.0)))
   (+ (acos (- 1.0 x)) (fma (- t_0) t_1 (* t_0 t_1)))))

C

double code(double x) {
	double t_0 = cbrt(asin((1.0 - x)));
	double t_1 = pow(t_0, 2.0);
	return acos((1.0 - x)) + fma(-t_0, t_1, (t_0 * t_1));
}

Julia

function code(x)
	t_0 = cbrt(asin(Float64(1.0 - x)))
	t_1 = t_0 ^ 2.0
	return Float64(acos(Float64(1.0 - x)) + fma(Float64(-t_0), t_1, Float64(t_0 * t_1)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[((-t$95$0) * t$95$1 + N[(t$95$0 * t$95$1), $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 6.7%

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

   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. add-cube-cbrt 6.7%

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

   4. prod-diff 6.7%

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

4. Applied egg-rr 10.2%

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

   1. unpow2 10.2%

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

   2. cbrt-prod 6.7%

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

   3. unpow2 6.7%

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

   4. rem-3cbrt-rft 4.9%

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

   5. fma-neg 4.9%

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

   6. add-cube-cbrt 10.2%

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

   7. metadata-eval 10.2%

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

   8. div-inv 10.2%

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

   9. acos-asin 10.2%

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

   10. add-log-exp 10.2%

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

   11. *-un-lft-identity 10.2%

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

   12. log-prod 10.2%

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

6. Applied egg-rr 10.2%

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

7. Final simplification 10.2%

   \[\leadsto \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(1 - x\right)}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) \]
8. Add Preprocessing

Alternative 6: 10.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

   1. acos-asin 6.7%

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

   2. *-un-lft-identity 6.7%

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

   3. add-sqr-sqrt 10.1%

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

   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. fma-neg 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 7: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt{t\_0}\\ \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-t\_1, \mathsf{expm1}\left(\mathsf{log1p}\left(t\_1\right)\right), t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (sqrt t_0)))
   (+ (acos (- 1.0 x)) (fma (- t_1) (expm1 (log1p t_1)) t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = sqrt(t_0);
	return acos((1.0 - x)) + fma(-t_1, expm1(log1p(t_1)), t_0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = sqrt(t_0)
	return Float64(acos(Float64(1.0 - x)) + fma(Float64(-t_1), expm1(log1p(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[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[((-t$95$1) * N[(Exp[N[Log[1 + t$95$1], $MachinePrecision]] - 1), $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 6.7%

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

   2. *-un-lft-identity 6.7%

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

   3. add-sqr-sqrt 10.1%

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

   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. fma-neg 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. expm1-log1p-u 10.2%

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

6. Applied egg-rr 10.2%

   \[\leadsto \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)\right)}, \sin^{-1} \left(1 - x\right)\right) \]
7. Add Preprocessing

Alternative 8: 10.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 + N[((-N[Sqrt[N[(N[(Pi * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]) * N[Sqrt[t$95$1], $MachinePrecision] + t$95$1), $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 6.7%

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

   2. *-un-lft-identity 6.7%

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

   3. add-sqr-sqrt 10.1%

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

   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. fma-neg 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. asin-acos 10.1%

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

   2. div-inv 10.1%

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

   3. metadata-eval 10.1%

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

6. Applied egg-rr 10.1%

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

Alternative 9: 10.3% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (sqrt t_0)))
   (+ (acos (- 1.0 x)) (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 acos((1.0 - x)) + 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(acos(Float64(1.0 - x)) + 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[ArcCos[N[(1.0 - x), $MachinePrecision]], $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 6.7%

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

   2. *-un-lft-identity 6.7%

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

   3. add-sqr-sqrt 10.1%

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

   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. fma-neg 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. Add Preprocessing

Alternative 10: 9.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;t\_0 + \pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \sqrt[3]{{t\_0}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (if (<= x 5.5e-17) (+ t_0 (* PI 0.5)) (- (* PI 0.5) (cbrt (pow t_0 3.0))))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = t_0 + (((double) M_PI) * 0.5);
	} else {
		tmp = (((double) M_PI) * 0.5) - cbrt(pow(t_0, 3.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = t_0 + (Math.PI * 0.5);
	} else {
		tmp = (Math.PI * 0.5) - Math.cbrt(Math.pow(t_0, 3.0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(t_0 + Float64(pi * 0.5));
	else
		tmp = Float64(Float64(pi * 0.5) - cbrt((t_0 ^ 3.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(t$95$0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-undefine 3.9%

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

      3. log1p-undefine 3.9%

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

      4. rem-exp-log 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. add-exp-log 3.9%

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

      2. expm1-define 3.9%

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

      3. log1p-define 3.9%

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

      4. expm1-log1p-u 3.9%

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

      5. acos-asin 3.9%

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

      6. div-inv 3.9%

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

      7. metadata-eval 3.9%

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

      8. add-cube-cbrt 7.5%

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

      9. unpow2 7.5%

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

      10. *-commutative 7.5%

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

      11. sub-neg 7.5%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 6.5%

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

   6. Applied egg-rr 6.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 59.7%

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

      1. acos-asin 59.7%

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

      2. sub-neg 59.7%

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

      3. div-inv 59.7%

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

      4. metadata-eval 59.7%

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

   4. Applied egg-rr 59.7%

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

      1. sub-neg 59.7%

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

   6. Simplified 59.7%

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

      1. add-cbrt-cube 59.8%

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

      2. pow3 59.9%

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

   8. Applied egg-rr 59.9%

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

4. Final simplification 9.2%

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

Alternative 11: 9.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;t\_0 + \pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \mathsf{expm1}\left(\mathsf{log1p}\left(t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (if (<= x 5.5e-17) (+ t_0 (* PI 0.5)) (- (* PI 0.5) (expm1 (log1p t_0))))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = t_0 + (((double) M_PI) * 0.5);
	} else {
		tmp = (((double) M_PI) * 0.5) - expm1(log1p(t_0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = t_0 + (Math.PI * 0.5);
	} else {
		tmp = (Math.PI * 0.5) - Math.expm1(Math.log1p(t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.asin((1.0 - x))
	tmp = 0
	if x <= 5.5e-17:
		tmp = t_0 + (math.pi * 0.5)
	else:
		tmp = (math.pi * 0.5) - math.expm1(math.log1p(t_0))
	return tmp

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(t_0 + Float64(pi * 0.5));
	else
		tmp = Float64(Float64(pi * 0.5) - expm1(log1p(t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(t$95$0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] - N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-undefine 3.9%

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

      3. log1p-undefine 3.9%

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

      4. rem-exp-log 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. add-exp-log 3.9%

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

      2. expm1-define 3.9%

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

      3. log1p-define 3.9%

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

      4. expm1-log1p-u 3.9%

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

      5. acos-asin 3.9%

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

      6. div-inv 3.9%

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

      7. metadata-eval 3.9%

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

      8. add-cube-cbrt 7.5%

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

      9. unpow2 7.5%

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

      10. *-commutative 7.5%

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

      11. sub-neg 7.5%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 6.5%

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

   6. Applied egg-rr 6.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 59.7%

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

      1. acos-asin 59.7%

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

      2. sub-neg 59.7%

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

      3. div-inv 59.7%

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

      4. metadata-eval 59.7%

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

   4. Applied egg-rr 59.7%

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

      1. sub-neg 59.7%

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

   6. Simplified 59.7%

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

      1. expm1-log1p-u 59.8%

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

   8. Applied egg-rr 59.8%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 10.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

   1. acos-asin 6.7%

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

   2. sub-neg 6.7%

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

   3. div-inv 6.7%

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

   4. metadata-eval 6.7%

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

4. Applied egg-rr 6.7%

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

   1. sub-neg 6.7%

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

6. Simplified 6.7%

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

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

8. Applied egg-rr 10.1%

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

Alternative 13: 10.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

   1. acos-asin 6.7%

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

   2. sub-neg 6.7%

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

   3. div-inv 6.7%

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

   4. metadata-eval 6.7%

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

4. Applied egg-rr 6.7%

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

   1. sub-neg 6.7%

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

6. Simplified 6.7%

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

   1. add-cube-cbrt 10.2%

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

   2. pow3 10.2%

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

8. Applied egg-rr 10.1%

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

Alternative 14: 9.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.asin((1.0 - x)) + (math.pi * 0.5)
	else:
		tmp = (1.0 + math.acos((1.0 - x))) + -1.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = asin((1.0 - x)) + (pi * 0.5);
	else
		tmp = (1.0 + acos((1.0 - x))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-undefine 3.9%

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

      3. log1p-undefine 3.9%

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

      4. rem-exp-log 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. add-exp-log 3.9%

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

      2. expm1-define 3.9%

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

      3. log1p-define 3.9%

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

      4. expm1-log1p-u 3.9%

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

      5. acos-asin 3.9%

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

      6. div-inv 3.9%

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

      7. metadata-eval 3.9%

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

      8. add-cube-cbrt 7.5%

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

      9. unpow2 7.5%

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

      10. *-commutative 7.5%

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

      11. sub-neg 7.5%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 6.5%

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

   6. Applied egg-rr 6.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 59.7%

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

      1. expm1-log1p-u 59.7%

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

      2. expm1-undefine 59.7%

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

      3. log1p-undefine 59.7%

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

      4. rem-exp-log 59.7%

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

   4. Applied egg-rr 59.7%

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

4. Final simplification 9.2%

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

Alternative 15: 9.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;t\_0 + \pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (if (<= x 5.5e-17) (+ t_0 (* PI 0.5)) (- (* PI 0.5) t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = t_0 + (((double) M_PI) * 0.5);
	} else {
		tmp = (((double) M_PI) * 0.5) - t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = t_0 + (Math.PI * 0.5);
	} else {
		tmp = (Math.PI * 0.5) - t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.asin((1.0 - x))
	tmp = 0
	if x <= 5.5e-17:
		tmp = t_0 + (math.pi * 0.5)
	else:
		tmp = (math.pi * 0.5) - t_0
	return tmp

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(t_0 + Float64(pi * 0.5));
	else
		tmp = Float64(Float64(pi * 0.5) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = asin((1.0 - x));
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = t_0 + (pi * 0.5);
	else
		tmp = (pi * 0.5) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(t$95$0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-undefine 3.9%

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

      3. log1p-undefine 3.9%

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

      4. rem-exp-log 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. add-exp-log 3.9%

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

      2. expm1-define 3.9%

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

      3. log1p-define 3.9%

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

      4. expm1-log1p-u 3.9%

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

      5. acos-asin 3.9%

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

      6. div-inv 3.9%

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

      7. metadata-eval 3.9%

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

      8. add-cube-cbrt 7.5%

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

      9. unpow2 7.5%

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

      10. *-commutative 7.5%

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

      11. sub-neg 7.5%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 6.5%

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

   6. Applied egg-rr 6.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 59.7%

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

      1. acos-asin 59.7%

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

      2. sub-neg 59.7%

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

      3. div-inv 59.7%

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

      4. metadata-eval 59.7%

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

   4. Applied egg-rr 59.7%

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

      1. sub-neg 59.7%

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

   6. Simplified 59.7%

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

4. Final simplification 9.2%

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

Alternative 16: 9.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.asin((1.0 - x)) + (math.pi * 0.5)
	else:
		tmp = math.acos((1.0 - x))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = asin((1.0 - x)) + (pi * 0.5);
	else
		tmp = acos((1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-undefine 3.9%

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

      3. log1p-undefine 3.9%

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

      4. rem-exp-log 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. add-exp-log 3.9%

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

      2. expm1-define 3.9%

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

      3. log1p-define 3.9%

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

      4. expm1-log1p-u 3.9%

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

      5. acos-asin 3.9%

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

      6. div-inv 3.9%

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

      7. metadata-eval 3.9%

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

      8. add-cube-cbrt 7.5%

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

      9. unpow2 7.5%

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

      10. *-commutative 7.5%

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

      11. sub-neg 7.5%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 6.5%

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

   6. Applied egg-rr 6.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 59.7%

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

4. Final simplification 9.2%

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

Alternative 17: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

Developer target: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024103 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :alt
  (* 2.0 (asin (sqrt (/ x 2.0))))

  (acos (- 1.0 x)))