bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.3%

Time: 10.4s

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: 10.3% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (asin (- 1.0 x)))))
   (fma (pow (cbrt (pow t_0 3.0)) 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, 3.0)), 2.0), -t_0, (((double) M_PI) * 0.5));
}

Julia

function code(x)
	t_0 = cbrt(asin(Float64(1.0 - x)))
	return fma((cbrt((t_0 ^ 3.0)) ^ 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[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * (-t$95$0) + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.5%

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

   1. expm1-log1p-u 6.5%

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

   2. expm1-udef 6.5%

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

   3. log1p-udef 6.5%

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

   4. add-exp-log 6.5%

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

3. Applied egg-rr 6.5%

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

   1. add-exp-log 6.5%

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

   2. log1p-udef 6.5%

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

   3. expm1-udef 6.5%

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

   4. expm1-log1p-u 6.5%

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

   5. acos-asin 6.5%

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

   6. div-inv 6.5%

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

   7. metadata-eval 6.5%

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

   8. sub-neg 6.5%

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

   9. +-commutative 6.5%

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

   10. add-cube-cbrt 10.0%

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

   11. distribute-rgt-neg-in 10.0%

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

   12. fma-def 10.0%

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

   13. pow2 10.0%

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

5. Applied egg-rr 10.0%

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

   1. add-cube-cbrt 10.0%

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

   2. pow3 10.0%

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

7. Applied egg-rr 10.0%

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

8. Final simplification 10.0%

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

Alternative 2: 10.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = cbrt(asin((1.0 - x)));
	return fma(expm1(log1p(pow(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[(Exp[N[Log[1 + N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] * (-t$95$0) + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.5%

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

   1. expm1-log1p-u 6.5%

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

   2. expm1-udef 6.5%

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

   3. log1p-udef 6.5%

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

   4. add-exp-log 6.5%

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

3. Applied egg-rr 6.5%

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

   1. add-exp-log 6.5%

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

   2. log1p-udef 6.5%

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

   3. expm1-udef 6.5%

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

   4. expm1-log1p-u 6.5%

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

   5. acos-asin 6.5%

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

   6. div-inv 6.5%

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

   7. metadata-eval 6.5%

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

   8. sub-neg 6.5%

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

   9. +-commutative 6.5%

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

   10. add-cube-cbrt 10.0%

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

   11. distribute-rgt-neg-in 10.0%

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

   12. fma-def 10.0%

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

   13. pow2 10.0%

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

5. Applied egg-rr 10.0%

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

   1. expm1-log1p-u 10.0%

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

7. Applied egg-rr 10.0%

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

8. Final simplification 10.0%

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

Alternative 3: 10.3% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (asin (- 1.0 x))))) (fma (pow 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(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((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[t$95$0, 2.0], $MachinePrecision] * (-t$95$0) + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.5%

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

   1. expm1-log1p-u 6.5%

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

   2. expm1-udef 6.5%

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

   3. log1p-udef 6.5%

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

   4. add-exp-log 6.5%

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

3. Applied egg-rr 6.5%

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

   1. add-exp-log 6.5%

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

   2. log1p-udef 6.5%

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

   3. expm1-udef 6.5%

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

   4. expm1-log1p-u 6.5%

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

   5. acos-asin 6.5%

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

   6. div-inv 6.5%

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

   7. metadata-eval 6.5%

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

   8. sub-neg 6.5%

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

   9. +-commutative 6.5%

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

   10. add-cube-cbrt 10.0%

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

   11. distribute-rgt-neg-in 10.0%

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

   12. fma-def 10.0%

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

   13. pow2 10.0%

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

5. Applied egg-rr 10.0%

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

6. Final simplification 10.0%

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

Alternative 4: 10.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = Math.cbrt(Math.asin((1.0 - x)));
	return (Math.PI * 0.5) - (t_0 * Math.pow(t_0, 2.0));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. acos-asin 6.5%

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

   2. sub-neg 6.5%

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

   3. div-inv 6.5%

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

   4. metadata-eval 6.5%

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

3. Applied egg-rr 6.5%

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

   1. sub-neg 6.5%

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

5. Simplified 6.5%

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

   1. add-cube-cbrt 10.0%

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

   2. pow2 10.0%

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

7. Applied egg-rr 10.0%

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

8. Final simplification 10.0%

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

Alternative 5: 10.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. acos-asin 6.5%

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

   2. sub-neg 6.5%

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

   3. div-inv 6.5%

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

   4. metadata-eval 6.5%

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

3. Applied egg-rr 6.5%

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

   1. sub-neg 6.5%

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

5. Simplified 6.5%

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

   1. add-cube-cbrt 10.0%

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

   2. pow3 10.0%

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

7. Applied egg-rr 10.0%

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

8. Final simplification 10.0%

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

Alternative 6: 9.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi - t_0\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{1 - {t_0}^{2}}{1 - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= x 5.5e-17)
     (- PI t_0)
     (+ -1.0 (/ (- 1.0 (pow t_0 2.0)) (- 1.0 t_0))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = ((double) M_PI) - t_0;
	} else {
		tmp = -1.0 + ((1.0 - pow(t_0, 2.0)) / (1.0 - t_0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = Math.PI - t_0;
	} else {
		tmp = -1.0 + ((1.0 - Math.pow(t_0, 2.0)) / (1.0 - t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.pi - t_0
	else:
		tmp = -1.0 + ((1.0 - math.pow(t_0, 2.0)) / (1.0 - t_0))
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(pi - t_0);
	else
		tmp = Float64(-1.0 + Float64(Float64(1.0 - (t_0 ^ 2.0)) / Float64(1.0 - t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = pi - t_0;
	else
		tmp = -1.0 + ((1.0 - (t_0 ^ 2.0)) / (1.0 - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(Pi - t$95$0), $MachinePrecision], N[(-1.0 + N[(N[(1.0 - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-udef 3.9%

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

      3. log1p-udef 3.9%

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

      4. add-exp-log 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. add-exp-log 3.9%

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

      2. log1p-udef 3.9%

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

      3. expm1-udef 3.9%

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

      4. expm1-log1p-u 3.9%

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

      5. acos-asin 3.9%

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

      6. div-inv 3.9%

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

      7. metadata-eval 3.9%

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

      8. sub-neg 3.9%

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

      9. +-commutative 3.9%

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

      10. add-cube-cbrt 7.5%

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

      11. distribute-rgt-neg-in 7.5%

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

      12. fma-def 7.5%

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

      13. pow2 7.5%

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

   5. Applied egg-rr 7.5%

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

      1. add-cube-cbrt 7.5%

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

      2. pow3 7.5%

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

   7. Applied egg-rr 7.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.5%

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

      3. sqr-neg 6.5%

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

      4. sqrt-prod 6.5%

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

      5. add-sqr-sqrt 6.5%

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

      6. fma-def 6.5%

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

      7. rem-cbrt-cube 6.5%

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

      8. pow2 6.5%

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

      9. add-cube-cbrt 6.5%

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

      10. +-commutative 6.5%

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

      11. asin-acos 6.5%

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

      12. div-inv 6.5%

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

      13. metadata-eval 6.5%

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

   9. Applied egg-rr 6.5%

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

       1. distribute-lft-out 6.5%

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

       2. metadata-eval 6.5%

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

       3. *-rgt-identity 6.5%

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

   11. Simplified 6.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 56.0%

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

      1. expm1-log1p-u 55.9%

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

      2. expm1-udef 55.8%

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

      3. log1p-udef 56.3%

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

      4. add-exp-log 56.3%

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

   3. Applied egg-rr 56.3%

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

      1. flip-+ 56.3%

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

      2. div-inv 56.3%

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

      3. fma-neg 56.0%

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

      4. metadata-eval 56.0%

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

      5. pow2 56.0%

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

      6. metadata-eval 56.0%

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

   5. Applied egg-rr 56.0%

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

      1. fma-udef 56.3%

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

      2. +-commutative 56.3%

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

      3. sub-neg 56.3%

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

      4. metadata-eval 56.3%

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

      5. distribute-neg-in 56.3%

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

      6. +-commutative 56.3%

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

      7. associate-*r/ 56.3%

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

      8. *-rgt-identity 56.3%

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

      9. +-commutative 56.3%

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

      10. distribute-neg-in 56.3%

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

      11. metadata-eval 56.3%

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

      12. sub-neg 56.3%

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

   7. Simplified 56.3%

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

4. Final simplification 9.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{1 - {\cos^{-1} \left(1 - x\right)}^{2}}{1 - \cos^{-1} \left(1 - x\right)}\\ \end{array} \]

Alternative 7: 9.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi - t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-1 + e^{\mathsf{log1p}\left(0.5 \cdot t_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= x 5.5e-17) (- PI t_0) (* 2.0 (+ -1.0 (exp (log1p (* 0.5 t_0))))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = ((double) M_PI) - t_0;
	} else {
		tmp = 2.0 * (-1.0 + exp(log1p((0.5 * t_0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = Math.PI - t_0;
	} else {
		tmp = 2.0 * (-1.0 + Math.exp(Math.log1p((0.5 * t_0))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.pi - t_0
	else:
		tmp = 2.0 * (-1.0 + math.exp(math.log1p((0.5 * t_0))))
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(pi - t_0);
	else
		tmp = Float64(2.0 * Float64(-1.0 + exp(log1p(Float64(0.5 * t_0)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(Pi - t$95$0), $MachinePrecision], N[(2.0 * N[(-1.0 + N[Exp[N[Log[1 + N[(0.5 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-udef 3.9%

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

      3. log1p-udef 3.9%

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

      4. add-exp-log 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. add-exp-log 3.9%

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

      2. log1p-udef 3.9%

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

      3. expm1-udef 3.9%

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

      4. expm1-log1p-u 3.9%

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

      5. acos-asin 3.9%

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

      6. div-inv 3.9%

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

      7. metadata-eval 3.9%

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

      8. sub-neg 3.9%

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

      9. +-commutative 3.9%

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

      10. add-cube-cbrt 7.5%

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

      11. distribute-rgt-neg-in 7.5%

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

      12. fma-def 7.5%

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

      13. pow2 7.5%

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

   5. Applied egg-rr 7.5%

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

      1. add-cube-cbrt 7.5%

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

      2. pow3 7.5%

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

   7. Applied egg-rr 7.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.5%

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

      3. sqr-neg 6.5%

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

      4. sqrt-prod 6.5%

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

      5. add-sqr-sqrt 6.5%

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

      6. fma-def 6.5%

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

      7. rem-cbrt-cube 6.5%

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

      8. pow2 6.5%

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

      9. add-cube-cbrt 6.5%

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

      10. +-commutative 6.5%

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

      11. asin-acos 6.5%

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

      12. div-inv 6.5%

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

      13. metadata-eval 6.5%

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

   9. Applied egg-rr 6.5%

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

       1. distribute-lft-out 6.5%

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

       2. metadata-eval 6.5%

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

       3. *-rgt-identity 6.5%

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

   11. Simplified 6.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 56.0%

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

      1. expm1-log1p-u 55.9%

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

      2. expm1-udef 55.8%

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

      3. log1p-udef 56.3%

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

      4. add-exp-log 56.3%

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

   3. Applied egg-rr 56.3%

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

      1. add-exp-log 56.3%

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

      2. log1p-udef 55.8%

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

      3. expm1-udef 55.9%

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

      4. expm1-log1p-u 56.0%

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

      5. add-log-exp 56.0%

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

      6. add-sqr-sqrt 56.0%

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

      7. log-prod 56.0%

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

   5. Applied egg-rr 56.0%

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

      1. count-2 56.0%

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

   7. Simplified 56.0%

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

      1. expm1-log1p-u 56.0%

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

      2. expm1-udef 56.0%

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

      3. pow1/2 56.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\log \color{blue}{\left({\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{0.5}\right)}\right)} - 1\right) \]

      4. log-pow 56.3%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{0.5 \cdot \log \left(e^{\cos^{-1} \left(1 - x\right)}\right)}\right)} - 1\right) \]

      5. add-log-exp 56.3%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(0.5 \cdot \color{blue}{\cos^{-1} \left(1 - x\right)}\right)} - 1\right) \]

   9. Applied egg-rr 56.3%

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

4. Final simplification 9.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-1 + e^{\mathsf{log1p}\left(0.5 \cdot \cos^{-1} \left(1 - x\right)\right)}\right)\\ \end{array} \]

Alternative 8: 9.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = ((double) M_PI) - t_0;
	} else {
		tmp = exp(log((-1.0 + (1.0 + t_0))));
	}
	return tmp;
}

Java

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

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.pi - t_0
	else:
		tmp = math.exp(math.log((-1.0 + (1.0 + t_0))))
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(pi - t_0);
	else
		tmp = exp(log(Float64(-1.0 + Float64(1.0 + t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = pi - t_0;
	else
		tmp = exp(log((-1.0 + (1.0 + t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-udef 3.9%

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

      3. log1p-udef 3.9%

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

      4. add-exp-log 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. add-exp-log 3.9%

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

      2. log1p-udef 3.9%

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

      3. expm1-udef 3.9%

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

      4. expm1-log1p-u 3.9%

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

      5. acos-asin 3.9%

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

      6. div-inv 3.9%

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

      7. metadata-eval 3.9%

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

      8. sub-neg 3.9%

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

      9. +-commutative 3.9%

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

      10. add-cube-cbrt 7.5%

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

      11. distribute-rgt-neg-in 7.5%

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

      12. fma-def 7.5%

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

      13. pow2 7.5%

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

   5. Applied egg-rr 7.5%

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

      1. add-cube-cbrt 7.5%

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

      2. pow3 7.5%

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

   7. Applied egg-rr 7.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.5%

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

      3. sqr-neg 6.5%

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

      4. sqrt-prod 6.5%

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

      5. add-sqr-sqrt 6.5%

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

      6. fma-def 6.5%

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

      7. rem-cbrt-cube 6.5%

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

      8. pow2 6.5%

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

      9. add-cube-cbrt 6.5%

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

      10. +-commutative 6.5%

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

      11. asin-acos 6.5%

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

      12. div-inv 6.5%

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

      13. metadata-eval 6.5%

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

   9. Applied egg-rr 6.5%

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

       1. distribute-lft-out 6.5%

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

       2. metadata-eval 6.5%

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

       3. *-rgt-identity 6.5%

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

   11. Simplified 6.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 56.0%

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

      1. add-exp-log 56.0%

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

   3. Applied egg-rr 56.0%

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

      1. expm1-log1p-u 55.9%

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

      2. expm1-udef 55.8%

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

      3. log1p-udef 56.3%

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

      4. add-exp-log 56.3%

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

   5. Applied egg-rr 56.3%

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

4. Final simplification 9.1%

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

Alternative 9: 9.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-udef 3.9%

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

      3. log1p-udef 3.9%

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

      4. add-exp-log 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. add-exp-log 3.9%

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

      2. log1p-udef 3.9%

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

      3. expm1-udef 3.9%

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

      4. expm1-log1p-u 3.9%

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

      5. acos-asin 3.9%

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

      6. div-inv 3.9%

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

      7. metadata-eval 3.9%

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

      8. sub-neg 3.9%

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

      9. +-commutative 3.9%

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

      10. add-cube-cbrt 7.5%

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

      11. distribute-rgt-neg-in 7.5%

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

      12. fma-def 7.5%

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

      13. pow2 7.5%

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

   5. Applied egg-rr 7.5%

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

      1. add-cube-cbrt 7.5%

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

      2. pow3 7.5%

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

   7. Applied egg-rr 7.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.5%

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

      3. sqr-neg 6.5%

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

      4. sqrt-prod 6.5%

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

      5. add-sqr-sqrt 6.5%

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

      6. fma-def 6.5%

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

      7. rem-cbrt-cube 6.5%

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

      8. pow2 6.5%

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

      9. add-cube-cbrt 6.5%

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

      10. +-commutative 6.5%

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

      11. asin-acos 6.5%

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

      12. div-inv 6.5%

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

      13. metadata-eval 6.5%

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

   9. Applied egg-rr 6.5%

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

       1. distribute-lft-out 6.5%

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

       2. metadata-eval 6.5%

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

       3. *-rgt-identity 6.5%

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

   11. Simplified 6.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 56.0%

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

      1. expm1-log1p-u 55.9%

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

      2. expm1-udef 55.8%

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

      3. log1p-udef 56.3%

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

      4. add-exp-log 56.3%

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

   3. Applied egg-rr 56.3%

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

4. Final simplification 9.1%

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

Alternative 10: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. expm1-log1p-u 6.5%

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

   2. expm1-udef 6.5%

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

   3. log1p-udef 6.5%

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

   4. add-exp-log 6.5%

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

3. Applied egg-rr 6.5%

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

4. Final simplification 6.5%

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

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 6.5%

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

2. Final simplification 6.5%

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

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 2023238 
(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)))