bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.5%

Time: 11.9s

Alternatives: 11

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: 10.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt[3]{t_0}, \sqrt[3]{{t_0}^{2}}, {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{t_0}\right)\right)\right)}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (+
    (acos (- 1.0 x))
    (fma
     (- (cbrt t_0))
     (cbrt (pow t_0 2.0))
     (pow (expm1 (log1p (sqrt t_0))) 2.0)))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return acos((1.0 - x)) + fma(-cbrt(t_0), cbrt(pow(t_0, 2.0)), pow(expm1(log1p(sqrt(t_0))), 2.0));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(acos(Float64(1.0 - x)) + fma(Float64(-cbrt(t_0)), cbrt((t_0 ^ 2.0)), (expm1(log1p(sqrt(t_0))) ^ 2.0)))
end

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. add-cbrt-cube 6.9%

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

   2. pow1/3 6.9%

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

   3. pow3 6.9%

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

3. Applied egg-rr 6.9%

   \[\leadsto \color{blue}{{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}^{0.3333333333333333}} \]
4. Step-by-step derivation

   1. unpow1/3 6.9%

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

   2. rem-cbrt-cube 6.9%

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

   3. acos-asin 6.9%

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

   4. div-inv 6.9%

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

   5. metadata-eval 6.9%

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

   6. add-cube-cbrt 9.9%

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

   7. prod-diff 9.9%

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

5. Applied egg-rr 10.0%

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

   1. add-sqr-sqrt 10.0%

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

   2. pow2 10.0%

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

7. Applied egg-rr 10.0%

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

   1. expm1-log1p-u 10.0%

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

9. Applied egg-rr 10.0%

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

10. Final simplification 10.0%

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

Alternative 2: 10.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ t_1 := \sin^{-1} \left(1 - x\right)\\ t_0 + \mathsf{fma}\left(-\sqrt[3]{\pi \cdot 0.5 - t_0}, \sqrt[3]{{t_1}^{2}}, {\left(\sqrt{t_1}\right)}^{2}\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	t_1 = asin(Float64(1.0 - x))
	return Float64(t_0 + fma(Float64(-cbrt(Float64(Float64(pi * 0.5) - t_0))), cbrt((t_1 ^ 2.0)), (sqrt(t_1) ^ 2.0)))
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[Power[N[(N[(Pi * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision], 1/3], $MachinePrecision]) * N[Power[N[Power[t$95$1, 2.0], $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[Sqrt[t$95$1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 6.9%

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

   1. add-cbrt-cube 6.9%

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

   2. pow1/3 6.9%

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

   3. pow3 6.9%

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

3. Applied egg-rr 6.9%

   \[\leadsto \color{blue}{{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}^{0.3333333333333333}} \]
4. Step-by-step derivation

   1. unpow1/3 6.9%

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

   2. rem-cbrt-cube 6.9%

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

   3. acos-asin 6.9%

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

   4. div-inv 6.9%

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

   5. metadata-eval 6.9%

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

   6. add-cube-cbrt 9.9%

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

   7. prod-diff 9.9%

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

5. Applied egg-rr 10.0%

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

   1. add-sqr-sqrt 10.0%

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

   2. pow2 10.0%

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

7. Applied egg-rr 10.0%

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

   1. asin-acos 10.0%

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

   2. div-inv 10.0%

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

   3. metadata-eval 10.0%

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

9. Applied egg-rr 10.0%

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

10. Final simplification 10.0%

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

Alternative 3: 10.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. add-cbrt-cube 6.9%

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

   2. pow1/3 6.9%

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

   3. pow3 6.9%

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

3. Applied egg-rr 6.9%

   \[\leadsto \color{blue}{{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}^{0.3333333333333333}} \]
4. Step-by-step derivation

   1. unpow1/3 6.9%

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

   2. rem-cbrt-cube 6.9%

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

   3. acos-asin 6.9%

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

   4. div-inv 6.9%

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

   5. metadata-eval 6.9%

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

   6. add-cube-cbrt 9.9%

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

   7. prod-diff 9.9%

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

5. Applied egg-rr 10.0%

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

   1. add-sqr-sqrt 10.0%

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

   2. pow2 10.0%

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

7. Applied egg-rr 10.0%

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

8. Final simplification 10.0%

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

Alternative 4: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ t_1 := \sin^{-1} \left(1 - x\right)\\ t_0 + \mathsf{fma}\left(-\sqrt[3]{\pi \cdot 0.5 - t_0}, \sqrt[3]{{t_1}^{2}}, t_1\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))) (t_1 (asin (- 1.0 x))))
   (+ t_0 (fma (- (cbrt (- (* PI 0.5) t_0))) (cbrt (pow t_1 2.0)) 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(-cbrt(((((double) M_PI) * 0.5) - t_0)), cbrt(pow(t_1, 2.0)), 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(-cbrt(Float64(Float64(pi * 0.5) - t_0))), cbrt((t_1 ^ 2.0)), 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[Power[N[(N[(Pi * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision], 1/3], $MachinePrecision]) * N[Power[N[Power[t$95$1, 2.0], $MachinePrecision], 1/3], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 6.9%

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

   1. add-cbrt-cube 6.9%

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

   2. pow1/3 6.9%

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

   3. pow3 6.9%

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

3. Applied egg-rr 6.9%

   \[\leadsto \color{blue}{{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}^{0.3333333333333333}} \]
4. Step-by-step derivation

   1. unpow1/3 6.9%

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

   2. rem-cbrt-cube 6.9%

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

   3. acos-asin 6.9%

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

   4. div-inv 6.9%

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

   5. metadata-eval 6.9%

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

   6. add-cube-cbrt 9.9%

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

   7. prod-diff 9.9%

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

5. Applied egg-rr 10.0%

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

   1. asin-acos 10.0%

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

   2. div-inv 10.0%

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

   3. metadata-eval 10.0%

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

7. Applied egg-rr 10.0%

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

8. Final simplification 10.0%

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

Alternative 5: 10.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. add-cbrt-cube 6.9%

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

   2. pow1/3 6.9%

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

   3. pow3 6.9%

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

3. Applied egg-rr 6.9%

   \[\leadsto \color{blue}{{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}^{0.3333333333333333}} \]
4. Step-by-step derivation

   1. unpow1/3 6.9%

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

   2. rem-cbrt-cube 6.9%

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

   3. acos-asin 6.9%

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

   4. div-inv 6.9%

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

   5. metadata-eval 6.9%

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

   6. add-cube-cbrt 9.9%

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

   7. prod-diff 9.9%

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

5. Applied egg-rr 10.0%

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

6. Final simplification 10.0%

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

Alternative 6: 10.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := N[Log[N[Exp[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]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 6.9%

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

   1. add-log-exp 6.9%

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

3. Applied egg-rr 6.9%

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

   1. acos-asin 6.9%

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

   2. sub-neg 6.9%

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

   3. div-inv 6.9%

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

   4. metadata-eval 6.9%

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

5. Applied egg-rr 6.9%

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

   1. sub-neg 6.9%

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

7. Simplified 6.9%

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

   1. add-cube-cbrt 9.9%

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

   2. pow3 9.9%

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

9. Applied egg-rr 10.0%

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

10. Final simplification 10.0%

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

Alternative 7: 10.4% 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 6.9%

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

   1. acos-asin 6.9%

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

   2. sub-neg 6.9%

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

   3. div-inv 6.9%

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

   4. metadata-eval 6.9%

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

3. Applied egg-rr 6.9%

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

   1. sub-neg 6.9%

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

5. Simplified 6.9%

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

   1. add-cube-cbrt 9.9%

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

   2. pow3 9.9%

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

7. Applied egg-rr 9.9%

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

8. Final simplification 9.9%

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

Alternative 8: 6.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right) + -1\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\frac{1 - {t_0}^{2}}{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (acos (- 1.0 x)) -1.0)))
   (if (<= (- 1.0 x) 1.0)
     (/ (- 1.0 (pow t_0 2.0)) (- 1.0 t_0))
     (+ (asin (- 1.0 x)) (* PI 0.5)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x)) + -1.0;
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = (1.0 - pow(t_0, 2.0)) / (1.0 - t_0);
	} else {
		tmp = asin((1.0 - x)) + (((double) M_PI) * 0.5);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x)) + -1.0;
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = (1.0 - Math.pow(t_0, 2.0)) / (1.0 - t_0);
	} else {
		tmp = Math.asin((1.0 - x)) + (Math.PI * 0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x)) + -1.0
	tmp = 0
	if (1.0 - x) <= 1.0:
		tmp = (1.0 - math.pow(t_0, 2.0)) / (1.0 - t_0)
	else:
		tmp = math.asin((1.0 - x)) + (math.pi * 0.5)
	return tmp

Julia

function code(x)
	t_0 = Float64(acos(Float64(1.0 - x)) + -1.0)
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = Float64(Float64(1.0 - (t_0 ^ 2.0)) / Float64(1.0 - t_0));
	else
		tmp = Float64(asin(Float64(1.0 - x)) + Float64(pi * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x)) + -1.0;
	tmp = 0.0;
	if ((1.0 - x) <= 1.0)
		tmp = (1.0 - (t_0 ^ 2.0)) / (1.0 - t_0);
	else
		tmp = asin((1.0 - x)) + (pi * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.9%

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

      1. add-cbrt-cube 6.9%

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

      2. pow1/3 6.9%

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

      3. pow3 6.9%

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

   3. Applied egg-rr 6.9%

      \[\leadsto \color{blue}{{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}^{0.3333333333333333}} \]
   4. Step-by-step derivation

      1. unpow1/3 6.9%

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

      2. rem-cbrt-cube 6.9%

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

      3. expm1-log1p-u 6.9%

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

      4. expm1-udef 6.9%

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

      5. log1p-udef 6.9%

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

      6. add-exp-log 6.9%

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

      7. associate--l+ 6.9%

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

      8. +-commutative 6.9%

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

      9. sub-neg 6.9%

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

      10. metadata-eval 6.9%

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

   5. Applied egg-rr 6.9%

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

      1. +-commutative 6.9%

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

      2. flip-+ 6.9%

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

      3. metadata-eval 6.9%

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

      4. pow2 6.9%

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

   7. Applied egg-rr 6.9%

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

   if 1 < (-.f64 1 x)

   1. Initial program 6.9%

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

      1. add-cbrt-cube 6.9%

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

      2. pow1/3 6.9%

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

      3. pow3 6.9%

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

   3. Applied egg-rr 6.9%

      \[\leadsto \color{blue}{{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}^{0.3333333333333333}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 6.9%

         \[\leadsto {\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

      2. expm1-udef 6.9%

         \[\leadsto {\left({\color{blue}{\left(e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1\right)}}^{3}\right)}^{0.3333333333333333} \]

      3. log1p-udef 6.9%

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

      4. add-exp-log 6.9%

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

   5. Applied egg-rr 6.9%

      \[\leadsto {\left({\color{blue}{\left(\left(1 + \cos^{-1} \left(1 - x\right)\right) - 1\right)}}^{3}\right)}^{0.3333333333333333} \]
   6. Step-by-step derivation

      1. unpow1/3 6.9%

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

      2. rem-cbrt-cube 6.9%

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

      3. +-commutative 6.9%

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

      4. associate--l+ 6.9%

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

      5. metadata-eval 6.9%

         \[\leadsto \cos^{-1} \left(1 - x\right) + \color{blue}{0} \]

      6. +-rgt-identity 6.9%

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

      7. acos-asin 6.9%

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

      8. div-inv 6.9%

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

      9. metadata-eval 6.9%

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

      10. sub-neg 6.9%

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

      11. add-cbrt-cube 5.0%

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

      12. unpow2 5.0%

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

      13. cbrt-prod 9.9%

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

      14. distribute-rgt-neg-in 9.9%

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

      15. add-sqr-sqrt 0.0%

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

      16. sqrt-unprod 6.6%

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

   7. Applied egg-rr 6.6%

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

4. Final simplification 6.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\frac{1 - {\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2}}{1 - \left(\cos^{-1} \left(1 - x\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\\ \end{array} \]

Alternative 9: 6.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.9%

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

      1. add-cbrt-cube 6.9%

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

      2. pow1/3 6.9%

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

      3. pow3 6.9%

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

   3. Applied egg-rr 6.9%

      \[\leadsto \color{blue}{{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}^{0.3333333333333333}} \]
   4. Step-by-step derivation

      1. unpow1/3 6.9%

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

      2. rem-cbrt-cube 6.9%

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

      3. expm1-log1p-u 6.9%

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

      4. expm1-udef 6.9%

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

      5. log1p-udef 6.9%

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

      6. add-exp-log 6.9%

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

      7. associate--l+ 6.9%

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

      8. +-commutative 6.9%

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

      9. sub-neg 6.9%

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

      10. metadata-eval 6.9%

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

   5. Applied egg-rr 6.9%

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

   if 1 < (-.f64 1 x)

   1. Initial program 6.9%

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

      1. add-cbrt-cube 6.9%

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

      2. pow1/3 6.9%

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

      3. pow3 6.9%

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

   3. Applied egg-rr 6.9%

      \[\leadsto \color{blue}{{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}^{0.3333333333333333}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 6.9%

         \[\leadsto {\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

      2. expm1-udef 6.9%

         \[\leadsto {\left({\color{blue}{\left(e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1\right)}}^{3}\right)}^{0.3333333333333333} \]

      3. log1p-udef 6.9%

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

      4. add-exp-log 6.9%

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

   5. Applied egg-rr 6.9%

      \[\leadsto {\left({\color{blue}{\left(\left(1 + \cos^{-1} \left(1 - x\right)\right) - 1\right)}}^{3}\right)}^{0.3333333333333333} \]
   6. Step-by-step derivation

      1. unpow1/3 6.9%

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

      2. rem-cbrt-cube 6.9%

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

      3. +-commutative 6.9%

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

      4. associate--l+ 6.9%

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

      5. metadata-eval 6.9%

         \[\leadsto \cos^{-1} \left(1 - x\right) + \color{blue}{0} \]

      6. +-rgt-identity 6.9%

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

      7. acos-asin 6.9%

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

      8. div-inv 6.9%

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

      9. metadata-eval 6.9%

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

      10. sub-neg 6.9%

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

      11. add-cbrt-cube 5.0%

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

      12. unpow2 5.0%

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

      13. cbrt-prod 9.9%

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

      14. distribute-rgt-neg-in 9.9%

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

      15. add-sqr-sqrt 0.0%

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

      16. sqrt-unprod 6.6%

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

   7. Applied egg-rr 6.6%

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

4. Final simplification 6.9%

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

Alternative 10: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. add-cbrt-cube 6.9%

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

   2. pow1/3 6.9%

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

   3. pow3 6.9%

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

3. Applied egg-rr 6.9%

   \[\leadsto \color{blue}{{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}^{0.3333333333333333}} \]
4. Step-by-step derivation

   1. unpow1/3 6.9%

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

   2. rem-cbrt-cube 6.9%

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

   3. expm1-log1p-u 6.9%

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

   4. expm1-udef 6.9%

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

   5. log1p-udef 6.9%

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

   6. add-exp-log 6.9%

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

   7. associate--l+ 6.9%

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

   8. +-commutative 6.9%

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

   9. sub-neg 6.9%

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

   10. metadata-eval 6.9%

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

5. Applied egg-rr 6.9%

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

6. Final simplification 6.9%

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

Alternative 11: 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 6.9%

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

2. Final simplification 6.9%

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

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 2023203 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

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

  (acos (- 1.0 x)))