bug323 (missed optimization)

Percentage Accurate: 6.8% → 10.3%

Time: 11.9s

Alternatives: 13

Speedup: 1.0×

Specification

\[0 \leq x \land x \leq 0.5\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. add-cbrt-cube 7.1%

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

   2. pow1/3 7.1%

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

   3. pow3 7.1%

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

3. Applied egg-rr 7.1%

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

   1. unpow1/3 7.1%

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

   2. rem-cbrt-cube 7.1%

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

   3. acos-asin 7.1%

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

   4. div-inv 7.1%

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

   5. metadata-eval 7.1%

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

   6. sub-neg 7.1%

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

   7. +-commutative 7.1%

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

   8. add-cube-cbrt 10.7%

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

   9. distribute-rgt-neg-in 10.7%

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

   10. fma-def 10.7%

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

   11. pow2 10.7%

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

5. Applied egg-rr 10.7%

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

   1. add-cube-cbrt 10.7%

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

   2. unpow2 10.7%

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

   3. cbrt-prod 10.7%

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

7. Applied egg-rr 10.7%

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

8. Final simplification 10.7%

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

Alternative 2: 10.3% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (asin (- 1.0 x)))))
   (fma (pow t_0 2.0) (- (pow (cbrt t_0) 3.0)) (* PI 0.5))))

C

double code(double x) {
	double t_0 = cbrt(asin((1.0 - x)));
	return fma(pow(t_0, 2.0), -pow(cbrt(t_0), 3.0), (((double) M_PI) * 0.5));
}

Julia

function code(x)
	t_0 = cbrt(asin(Float64(1.0 - x)))
	return fma((t_0 ^ 2.0), Float64(-(cbrt(t_0) ^ 3.0)), Float64(pi * 0.5))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[Power[t$95$0, 2.0], $MachinePrecision] * (-N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]) + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.1%

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

   1. add-cbrt-cube 7.1%

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

   2. pow1/3 7.1%

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

   3. pow3 7.1%

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

3. Applied egg-rr 7.1%

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

   1. unpow1/3 7.1%

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

   2. rem-cbrt-cube 7.1%

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

   3. acos-asin 7.1%

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

   4. div-inv 7.1%

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

   5. metadata-eval 7.1%

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

   6. sub-neg 7.1%

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

   7. +-commutative 7.1%

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

   8. add-cube-cbrt 10.7%

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

   9. distribute-rgt-neg-in 10.7%

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

   10. fma-def 10.7%

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

   11. pow2 10.7%

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

5. Applied egg-rr 10.7%

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

   1. add-cube-cbrt 10.7%

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

   2. pow3 10.7%

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

7. Applied egg-rr 10.7%

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

8. Final simplification 10.7%

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

Alternative 3: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sin^{-1} \left(1 - x\right)}\\ \mathsf{fma}\left({\left(\mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)\right)}^{2}, -t_0, \pi \cdot 0.5\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (asin (- 1.0 x)))))
   (fma (pow (expm1 (log1p t_0)) 2.0) (- t_0) (* PI 0.5))))

C

double code(double x) {
	double t_0 = cbrt(asin((1.0 - x)));
	return fma(pow(expm1(log1p(t_0)), 2.0), -t_0, (((double) M_PI) * 0.5));
}

Julia

function code(x)
	t_0 = cbrt(asin(Float64(1.0 - x)))
	return fma((expm1(log1p(t_0)) ^ 2.0), Float64(-t_0), Float64(pi * 0.5))
end

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. add-cbrt-cube 7.1%

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

   2. pow1/3 7.1%

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

   3. pow3 7.1%

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

3. Applied egg-rr 7.1%

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

   1. unpow1/3 7.1%

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

   2. rem-cbrt-cube 7.1%

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

   3. acos-asin 7.1%

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

   4. div-inv 7.1%

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

   5. metadata-eval 7.1%

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

   6. sub-neg 7.1%

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

   7. +-commutative 7.1%

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

   8. add-cube-cbrt 10.7%

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

   9. distribute-rgt-neg-in 10.7%

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

   10. fma-def 10.7%

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

   11. pow2 10.7%

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

5. Applied egg-rr 10.7%

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

   1. expm1-log1p-u 10.7%

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

7. Applied egg-rr 10.7%

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

8. Final simplification 10.7%

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

Alternative 4: 10.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. acos-asin 7.1%

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

   2. sub-neg 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

3. Applied egg-rr 7.1%

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

   1. sub-neg 7.1%

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

5. Simplified 7.1%

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

   1. asin-acos 7.1%

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

   2. div-inv 7.1%

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

   3. metadata-eval 7.1%

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

   4. add-sqr-sqrt 10.7%

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

   5. fma-neg 10.7%

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

7. Applied egg-rr 10.7%

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

8. Final simplification 10.7%

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

Alternative 5: 10.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. add-cbrt-cube 7.1%

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

   2. pow1/3 7.1%

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

   3. pow3 7.1%

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

3. Applied egg-rr 7.1%

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

   1. unpow1/3 7.1%

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

   2. rem-cbrt-cube 7.1%

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

   3. acos-asin 7.1%

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

   4. div-inv 7.1%

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

   5. metadata-eval 7.1%

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

   6. add-sqr-sqrt 5.3%

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

   7. fma-neg 5.3%

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

5. Applied egg-rr 5.3%

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

   1. sqrt-prod 10.7%

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

7. Applied egg-rr 10.7%

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

8. Final simplification 10.7%

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

Alternative 6: 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 7.1%

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

   1. acos-asin 7.1%

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

   2. sub-neg 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

3. Applied egg-rr 7.1%

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

   1. sub-neg 7.1%

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

5. Simplified 7.1%

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

   1. add-cube-cbrt 10.7%

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

   2. pow3 10.7%

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

7. Applied egg-rr 10.7%

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

8. Final simplification 10.7%

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

Alternative 7: 10.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. add-cbrt-cube 7.1%

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

   2. pow1/3 7.1%

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

   3. pow3 7.1%

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

3. Applied egg-rr 7.1%

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

   1. unpow1/3 7.1%

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

   2. rem-cbrt-cube 7.1%

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

   3. acos-asin 7.1%

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

   4. div-inv 7.1%

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

   5. metadata-eval 7.1%

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

   6. add-sqr-sqrt 5.3%

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

   7. fma-neg 5.3%

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

5. Applied egg-rr 5.3%

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

6. Taylor expanded in x around 0 10.7%

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

7. Final simplification 10.7%

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

Alternative 8: 9.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right) + -1\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;1 + \left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt[3]{{t_0}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (acos (- 1.0 x)) -1.0)))
   (if (<= x 5.6e-17) (+ 1.0 (fabs t_0)) (+ 1.0 (cbrt (pow t_0 3.0))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x)) + -1.0;
	double tmp;
	if (x <= 5.6e-17) {
		tmp = 1.0 + fabs(t_0);
	} else {
		tmp = 1.0 + cbrt(pow(t_0, 3.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x)) + -1.0;
	double tmp;
	if (x <= 5.6e-17) {
		tmp = 1.0 + Math.abs(t_0);
	} else {
		tmp = 1.0 + Math.cbrt(Math.pow(t_0, 3.0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(acos(Float64(1.0 - x)) + -1.0)
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = Float64(1.0 + abs(t_0));
	else
		tmp = Float64(1.0 + cbrt((t_0 ^ 3.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, 5.6e-17], N[(1.0 + N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.8%

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

      1. add-cbrt-cube 3.8%

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

      2. pow1/3 3.8%

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

      3. pow3 3.8%

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

   3. Applied egg-rr 3.8%

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

      1. unpow1/3 3.8%

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

      2. rem-cbrt-cube 3.8%

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

      3. expm1-log1p-u 3.8%

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

      4. expm1-udef 3.8%

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

      5. log1p-udef 3.8%

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

      6. *-rgt-identity 3.8%

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

      7. add-exp-log 3.8%

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

      8. *-rgt-identity 3.8%

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

      9. associate--l+ 3.8%

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

      10. +-commutative 3.8%

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

      11. sub-neg 3.8%

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

      12. metadata-eval 3.8%

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

   5. Applied egg-rr 3.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.6%

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

      3. pow2 6.6%

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

   7. Applied egg-rr 6.6%

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

      1. unpow2 6.6%

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

      2. rem-sqrt-square 6.6%

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

   9. Simplified 6.6%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 73.4%

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

      1. add-cbrt-cube 73.2%

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

      2. pow1/3 73.4%

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

      3. pow3 73.4%

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

   3. Applied egg-rr 73.4%

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

      1. unpow1/3 73.3%

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

      2. rem-cbrt-cube 73.4%

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

      3. expm1-log1p-u 73.3%

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

      4. expm1-udef 73.4%

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

      5. log1p-udef 73.4%

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

      6. *-rgt-identity 73.4%

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

      7. add-exp-log 73.4%

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

      8. *-rgt-identity 73.4%

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

      9. associate--l+ 73.4%

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

      10. +-commutative 73.4%

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

      11. sub-neg 73.4%

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

      12. metadata-eval 73.4%

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

   5. Applied egg-rr 73.4%

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

      1. add-cbrt-cube 73.7%

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

      2. pow3 73.7%

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

   7. Applied egg-rr 73.7%

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

4. Final simplification 9.7%

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

Alternative 9: 9.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right) + -1\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;1 + \left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (acos (- 1.0 x)) -1.0)))
   (if (<= x 5.6e-17) (+ 1.0 (fabs t_0)) (exp (log1p t_0)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x)) + -1.0;
	double tmp;
	if (x <= 5.6e-17) {
		tmp = 1.0 + fabs(t_0);
	} else {
		tmp = exp(log1p(t_0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x)) + -1.0;
	double tmp;
	if (x <= 5.6e-17) {
		tmp = 1.0 + Math.abs(t_0);
	} else {
		tmp = Math.exp(Math.log1p(t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x)) + -1.0
	tmp = 0
	if x <= 5.6e-17:
		tmp = 1.0 + math.fabs(t_0)
	else:
		tmp = math.exp(math.log1p(t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(acos(Float64(1.0 - x)) + -1.0)
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = Float64(1.0 + abs(t_0));
	else
		tmp = exp(log1p(t_0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, 5.6e-17], N[(1.0 + N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision], N[Exp[N[Log[1 + t$95$0], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.8%

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

      1. add-cbrt-cube 3.8%

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

      2. pow1/3 3.8%

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

      3. pow3 3.8%

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

   3. Applied egg-rr 3.8%

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

      1. unpow1/3 3.8%

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

      2. rem-cbrt-cube 3.8%

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

      3. expm1-log1p-u 3.8%

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

      4. expm1-udef 3.8%

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

      5. log1p-udef 3.8%

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

      6. *-rgt-identity 3.8%

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

      7. add-exp-log 3.8%

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

      8. *-rgt-identity 3.8%

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

      9. associate--l+ 3.8%

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

      10. +-commutative 3.8%

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

      11. sub-neg 3.8%

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

      12. metadata-eval 3.8%

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

   5. Applied egg-rr 3.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.6%

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

      3. pow2 6.6%

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

   7. Applied egg-rr 6.6%

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

      1. unpow2 6.6%

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

      2. rem-sqrt-square 6.6%

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

   9. Simplified 6.6%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 73.4%

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

      1. add-exp-log 73.4%

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

   3. Applied egg-rr 73.4%

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

      1. log1p-expm1-u 73.5%

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

      2. expm1-udef 73.5%

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

      3. add-exp-log 73.5%

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

      4. sub-neg 73.5%

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

      5. metadata-eval 73.5%

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

   5. Applied egg-rr 73.5%

      \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right) + -1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;1 + \left|\cos^{-1} \left(1 - x\right) + -1\right|\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right) + -1\right)}\\ \end{array} \]

Alternative 10: 9.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right) + -1\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;1 + \left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;1 + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (acos (- 1.0 x)) -1.0)))
   (if (<= x 5.6e-17) (+ 1.0 (fabs t_0)) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = acos((1.0 - x)) + -1.0;
	double tmp;
	if (x <= 5.6e-17) {
		tmp = 1.0 + fabs(t_0);
	} else {
		tmp = 1.0 + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = acos((1.0d0 - x)) + (-1.0d0)
    if (x <= 5.6d-17) then
        tmp = 1.0d0 + abs(t_0)
    else
        tmp = 1.0d0 + t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x)) + -1.0;
	double tmp;
	if (x <= 5.6e-17) {
		tmp = 1.0 + Math.abs(t_0);
	} else {
		tmp = 1.0 + t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x)) + -1.0
	tmp = 0
	if x <= 5.6e-17:
		tmp = 1.0 + math.fabs(t_0)
	else:
		tmp = 1.0 + t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(acos(Float64(1.0 - x)) + -1.0)
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = Float64(1.0 + abs(t_0));
	else
		tmp = Float64(1.0 + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x)) + -1.0;
	tmp = 0.0;
	if (x <= 5.6e-17)
		tmp = 1.0 + abs(t_0);
	else
		tmp = 1.0 + t_0;
	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[x, 5.6e-17], N[(1.0 + N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision], N[(1.0 + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.8%

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

      1. add-cbrt-cube 3.8%

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

      2. pow1/3 3.8%

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

      3. pow3 3.8%

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

   3. Applied egg-rr 3.8%

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

      1. unpow1/3 3.8%

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

      2. rem-cbrt-cube 3.8%

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

      3. expm1-log1p-u 3.8%

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

      4. expm1-udef 3.8%

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

      5. log1p-udef 3.8%

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

      6. *-rgt-identity 3.8%

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

      7. add-exp-log 3.8%

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

      8. *-rgt-identity 3.8%

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

      9. associate--l+ 3.8%

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

      10. +-commutative 3.8%

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

      11. sub-neg 3.8%

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

      12. metadata-eval 3.8%

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

   5. Applied egg-rr 3.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.6%

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

      3. pow2 6.6%

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

   7. Applied egg-rr 6.6%

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

      1. unpow2 6.6%

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

      2. rem-sqrt-square 6.6%

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

   9. Simplified 6.6%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 73.4%

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

      1. add-cbrt-cube 73.2%

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

      2. pow1/3 73.4%

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

      3. pow3 73.4%

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

   3. Applied egg-rr 73.4%

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

      1. unpow1/3 73.3%

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

      2. rem-cbrt-cube 73.4%

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

      3. expm1-log1p-u 73.3%

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

      4. expm1-udef 73.4%

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

      5. log1p-udef 73.4%

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

      6. *-rgt-identity 73.4%

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

      7. add-exp-log 73.4%

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

      8. *-rgt-identity 73.4%

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

      9. associate--l+ 73.4%

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

      10. +-commutative 73.4%

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

      11. sub-neg 73.4%

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

      12. metadata-eval 73.4%

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

   5. Applied egg-rr 73.4%

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

4. Final simplification 9.7%

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

Alternative 11: 6.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= (- 1.0 x) 1.0) (+ 1.0 (+ t_0 -1.0)) (- PI t_0))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = 1.0 + (t_0 + -1.0);
	} else {
		tmp = ((double) M_PI) - t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = 1.0 + (t_0 + -1.0);
	} else {
		tmp = Math.PI - t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if (1.0 - x) <= 1.0:
		tmp = 1.0 + (t_0 + -1.0)
	else:
		tmp = math.pi - t_0
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = Float64(1.0 + Float64(t_0 + -1.0));
	else
		tmp = Float64(pi - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if ((1.0 - x) <= 1.0)
		tmp = 1.0 + (t_0 + -1.0);
	else
		tmp = pi - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.1%

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

      1. add-cbrt-cube 7.1%

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

      2. pow1/3 7.1%

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

      3. pow3 7.1%

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

   3. Applied egg-rr 7.1%

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

      1. unpow1/3 7.1%

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

      2. rem-cbrt-cube 7.1%

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

      3. expm1-log1p-u 7.1%

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

      4. expm1-udef 7.1%

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

      5. log1p-udef 7.1%

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

      6. *-rgt-identity 7.1%

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

      7. add-exp-log 7.1%

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

      8. *-rgt-identity 7.1%

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

      9. associate--l+ 7.1%

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

      10. +-commutative 7.1%

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

      11. sub-neg 7.1%

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

      12. metadata-eval 7.1%

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

   5. Applied egg-rr 7.1%

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

   if 1 < (-.f64 1 x)

   1. Initial program 7.1%

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

      1. add-cbrt-cube 7.1%

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

      2. pow1/3 7.1%

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

      3. pow3 7.1%

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

   3. Applied egg-rr 7.1%

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

      1. unpow1/3 7.1%

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

      2. rem-cbrt-cube 7.1%

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

      3. acos-asin 7.1%

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

      4. div-inv 7.1%

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

      5. metadata-eval 7.1%

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

      6. sub-neg 7.1%

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

      7. +-commutative 7.1%

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

      8. add-cube-cbrt 10.7%

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

      9. distribute-rgt-neg-in 10.7%

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

      10. fma-def 10.7%

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

      11. pow2 10.7%

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

   5. Applied egg-rr 10.7%

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

      1. add-cube-cbrt 10.7%

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

      2. unpow2 10.7%

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

      3. cbrt-prod 10.7%

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

   7. Applied egg-rr 10.7%

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

      1. cbrt-unprod 10.7%

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

      2. unpow2 10.7%

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

      3. add-cube-cbrt 10.7%

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

      4. fma-def 10.7%

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

      5. distribute-rgt-neg-in 10.7%

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

      6. unpow2 10.7%

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

      7. add-cube-cbrt 7.1%

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

   9. Applied egg-rr 6.9%

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

       1. associate--r- 6.9%

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

       2. +-commutative 6.9%

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

       3. associate--l+ 6.9%

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

       4. distribute-lft-out 6.9%

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

       5. metadata-eval 6.9%

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

       6. *-rgt-identity 6.9%

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

   11. Simplified 6.9%

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

4. Final simplification 7.1%

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

Alternative 12: 6.8% 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 7.1%

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

   1. add-cbrt-cube 7.1%

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

   2. pow1/3 7.1%

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

   3. pow3 7.1%

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

3. Applied egg-rr 7.1%

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

   1. unpow1/3 7.1%

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

   2. rem-cbrt-cube 7.1%

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

   3. expm1-log1p-u 7.1%

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

   4. expm1-udef 7.1%

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

   5. log1p-udef 7.1%

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

   6. *-rgt-identity 7.1%

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

   7. add-exp-log 7.1%

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

   8. *-rgt-identity 7.1%

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

   9. associate--l+ 7.1%

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

   10. +-commutative 7.1%

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

   11. sub-neg 7.1%

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

   12. metadata-eval 7.1%

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

5. Applied egg-rr 7.1%

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

6. Final simplification 7.1%

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

Alternative 13: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Final simplification 7.1%

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

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 2023173 
(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)))