bug323 (missed optimization)

Percentage Accurate: 6.8% → 10.3%

Time: 24.6s

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

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

FPCore

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

C

double code(double x) {
	double t_0 = sqrt(asin((1.0 - x)));
	return pow(pow((acos((1.0 - x)) + fma(-t_0, t_0, pow(t_0, 2.0))), 3.0), 0.3333333333333333);
}

Julia

function code(x)
	t_0 = sqrt(asin(Float64(1.0 - x)))
	return (Float64(acos(Float64(1.0 - x)) + fma(Float64(-t_0), t_0, (t_0 ^ 2.0))) ^ 3.0) ^ 0.3333333333333333
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[Power[N[Power[N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[((-t$95$0) * t$95$0 + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]

Derivation

1. Initial program 8.8%

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

   1. add-cbrt-cube 8.8%

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

   2. pow1/3 8.8%

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

   3. pow3 8.8%

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

4. Applied egg-rr 8.8%

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

   1. acos-asin 8.8%

      \[\leadsto {\left({\color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   2. *-un-lft-identity 8.8%

      \[\leadsto {\left({\left(\color{blue}{1 \cdot \frac{\pi}{2}} - \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}^{0.3333333333333333} \]

   3. add-sqr-sqrt 12.1%

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

   4. prod-diff 12.1%

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

   5. add-sqr-sqrt 12.1%

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

   6. fma-neg 12.1%

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

   7. *-un-lft-identity 12.1%

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

   8. acos-asin 12.1%

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

   9. add-sqr-sqrt 12.1%

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

6. Applied egg-rr 12.1%

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

   1. add-sqr-sqrt 12.1%

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

   2. pow2 12.1%

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

8. Applied egg-rr 12.1%

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

9. Final simplification 12.1%

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

Alternative 2: 10.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.8%

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

   1. acos-asin 8.8%

      \[\leadsto {\left({\color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   2. *-un-lft-identity 8.8%

      \[\leadsto {\left({\left(\color{blue}{1 \cdot \frac{\pi}{2}} - \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}^{0.3333333333333333} \]

   3. add-sqr-sqrt 12.1%

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

   4. prod-diff 12.1%

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

   5. add-sqr-sqrt 12.1%

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

   6. fma-neg 12.1%

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

   7. *-un-lft-identity 12.1%

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

   8. acos-asin 12.1%

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

   9. add-sqr-sqrt 12.1%

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

4. Applied egg-rr 12.1%

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

   1. add-sqr-sqrt 12.1%

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

   2. pow2 12.1%

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

6. Applied egg-rr 12.1%

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

7. Final simplification 12.1%

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

Alternative 3: 10.4% accurate, 0.2× 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 8.8%

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

   1. acos-asin 8.8%

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

   2. add-sqr-sqrt 7.1%

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

   3. fma-neg 7.1%

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

   4. div-inv 7.1%

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

   5. metadata-eval 7.1%

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

   6. div-inv 7.1%

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

   7. metadata-eval 7.1%

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

4. Applied egg-rr 7.1%

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

   1. sqrt-prod 12.1%

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

6. Applied egg-rr 12.1%

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

7. Final simplification 12.1%

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

Alternative 4: 10.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.8%

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

   1. acos-asin 8.8%

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

   2. sub-neg 8.8%

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

   3. div-inv 8.8%

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

   4. metadata-eval 8.8%

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

4. Applied egg-rr 8.8%

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

   1. sub-neg 8.8%

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

6. Simplified 8.8%

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

   1. asin-acos 8.8%

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

   2. div-inv 8.8%

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

   3. metadata-eval 8.8%

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

   4. *-commutative 8.8%

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

   5. add-sqr-sqrt 12.1%

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

   6. associate-*r* 12.1%

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

   7. fma-neg 12.1%

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

8. Applied egg-rr 12.1%

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

9. Final simplification 12.1%

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

Alternative 5: 9.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{hypot}\left(\pi \cdot 0.5, \sin^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(e^{\cos^{-1} \left(1 - x\right) \cdot 0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.6e-17)
   (hypot (* PI 0.5) (asin (- 1.0 x)))
   (* 2.0 (log (exp (* (acos (- 1.0 x)) 0.5))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = hypot((((double) M_PI) * 0.5), asin((1.0 - x)));
	} else {
		tmp = 2.0 * log(exp((acos((1.0 - x)) * 0.5)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = Math.hypot((Math.PI * 0.5), Math.asin((1.0 - x)));
	} else {
		tmp = 2.0 * Math.log(Math.exp((Math.acos((1.0 - x)) * 0.5)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.6e-17:
		tmp = math.hypot((math.pi * 0.5), math.asin((1.0 - x)))
	else:
		tmp = 2.0 * math.log(math.exp((math.acos((1.0 - x)) * 0.5)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = hypot(Float64(pi * 0.5), asin(Float64(1.0 - x)));
	else
		tmp = Float64(2.0 * log(exp(Float64(acos(Float64(1.0 - x)) * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.6e-17)
		tmp = hypot((pi * 0.5), asin((1.0 - x)));
	else
		tmp = 2.0 * log(exp((acos((1.0 - x)) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.6e-17], N[Sqrt[N[(Pi * 0.5), $MachinePrecision] ^ 2 + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision], N[(2.0 * N[Log[N[Exp[N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   6. Simplified 3.9%

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

      1. add-sqr-sqrt 3.9%

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

      2. sqrt-unprod 3.9%

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

      3. add-sqr-sqrt 3.9%

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

      4. cancel-sign-sub-inv 3.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 3.9%

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

      7. sqr-neg 3.9%

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

      8. add-sqr-sqrt 3.9%

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

      9. add-sqr-sqrt 3.9%

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

      10. difference-of-squares 3.9%

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

   8. Applied egg-rr 6.5%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 67.7%

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

      1. add-log-exp 67.7%

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

      2. add-sqr-sqrt 67.6%

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

      3. log-prod 67.6%

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

   4. Applied egg-rr 67.6%

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

      1. count-2 67.6%

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

   6. Simplified 67.6%

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

      1. pow1/2 67.6%

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

      2. pow-exp 67.8%

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

   8. Applied egg-rr 67.8%

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

4. Final simplification 11.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{hypot}\left(\pi \cdot 0.5, \sin^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(e^{\cos^{-1} \left(1 - x\right) \cdot 0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 9.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{hypot}\left(\pi \cdot 0.5, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \mathsf{expm1}\left(\mathsf{log1p}\left(t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (if (<= x 5.6e-17)
     (hypot (* PI 0.5) t_0)
     (- (* PI 0.5) (expm1 (log1p t_0))))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double tmp;
	if (x <= 5.6e-17) {
		tmp = hypot((((double) M_PI) * 0.5), t_0);
	} else {
		tmp = (((double) M_PI) * 0.5) - expm1(log1p(t_0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	double tmp;
	if (x <= 5.6e-17) {
		tmp = Math.hypot((Math.PI * 0.5), t_0);
	} else {
		tmp = (Math.PI * 0.5) - Math.expm1(Math.log1p(t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.asin((1.0 - x))
	tmp = 0
	if x <= 5.6e-17:
		tmp = math.hypot((math.pi * 0.5), t_0)
	else:
		tmp = (math.pi * 0.5) - math.expm1(math.log1p(t_0))
	return tmp

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = hypot(Float64(pi * 0.5), t_0);
	else
		tmp = Float64(Float64(pi * 0.5) - expm1(log1p(t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.6e-17], N[Sqrt[N[(Pi * 0.5), $MachinePrecision] ^ 2 + t$95$0 ^ 2], $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] - N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   6. Simplified 3.9%

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

      1. add-sqr-sqrt 3.9%

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

      2. sqrt-unprod 3.9%

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

      3. add-sqr-sqrt 3.9%

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

      4. cancel-sign-sub-inv 3.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 3.9%

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

      7. sqr-neg 3.9%

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

      8. add-sqr-sqrt 3.9%

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

      9. add-sqr-sqrt 3.9%

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

      10. difference-of-squares 3.9%

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

   8. Applied egg-rr 6.5%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 67.7%

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

      1. acos-asin 67.7%

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

      2. sub-neg 67.7%

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

      3. div-inv 67.7%

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

      4. metadata-eval 67.7%

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

   4. Applied egg-rr 67.7%

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

      1. sub-neg 67.7%

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

   6. Simplified 67.7%

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

      1. expm1-log1p-u 67.8%

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

      2. expm1-undefine 67.7%

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

   8. Applied egg-rr 67.7%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(e^{\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-define 67.8%

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

   10. Simplified 67.8%

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

4. Final simplification 11.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{hypot}\left(\pi \cdot 0.5, \sin^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 9.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{hypot}\left(\pi \cdot 0.5, \sin^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.6e-17)
   (hypot (* PI 0.5) (asin (- 1.0 x)))
   (pow (pow (acos (- 1.0 x)) 3.0) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = hypot((((double) M_PI) * 0.5), asin((1.0 - x)));
	} else {
		tmp = pow(pow(acos((1.0 - x)), 3.0), 0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = Math.hypot((Math.PI * 0.5), Math.asin((1.0 - x)));
	} else {
		tmp = Math.pow(Math.pow(Math.acos((1.0 - x)), 3.0), 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.6e-17:
		tmp = math.hypot((math.pi * 0.5), math.asin((1.0 - x)))
	else:
		tmp = math.pow(math.pow(math.acos((1.0 - x)), 3.0), 0.3333333333333333)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = hypot(Float64(pi * 0.5), asin(Float64(1.0 - x)));
	else
		tmp = (acos(Float64(1.0 - x)) ^ 3.0) ^ 0.3333333333333333;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.6e-17)
		tmp = hypot((pi * 0.5), asin((1.0 - x)));
	else
		tmp = (acos((1.0 - x)) ^ 3.0) ^ 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.6e-17], N[Sqrt[N[(Pi * 0.5), $MachinePrecision] ^ 2 + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision], N[Power[N[Power[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   6. Simplified 3.9%

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

      1. add-sqr-sqrt 3.9%

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

      2. sqrt-unprod 3.9%

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

      3. add-sqr-sqrt 3.9%

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

      4. cancel-sign-sub-inv 3.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 3.9%

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

      7. sqr-neg 3.9%

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

      8. add-sqr-sqrt 3.9%

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

      9. add-sqr-sqrt 3.9%

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

      10. difference-of-squares 3.9%

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

   8. Applied egg-rr 6.5%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 67.7%

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

      1. add-cbrt-cube 67.7%

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

      2. pow1/3 67.7%

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

      3. pow3 67.7%

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

   4. Applied egg-rr 67.7%

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

4. Final simplification 11.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{hypot}\left(\pi \cdot 0.5, \sin^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 8.8%

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

   1. acos-asin 8.8%

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

   2. sub-neg 8.8%

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

   3. div-inv 8.8%

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

   4. metadata-eval 8.8%

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

4. Applied egg-rr 8.8%

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

   1. sub-neg 8.8%

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

6. Simplified 8.8%

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

   1. add-cube-cbrt 12.0%

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

   2. pow3 12.0%

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

8. Applied egg-rr 12.0%

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

9. Final simplification 12.0%

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

Alternative 9: 10.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.8%

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

   1. acos-asin 8.8%

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

   2. sub-neg 8.8%

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

   3. div-inv 8.8%

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

   4. metadata-eval 8.8%

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

4. Applied egg-rr 8.8%

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

   1. sub-neg 8.8%

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

6. Simplified 8.8%

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

   1. add-sqr-sqrt 12.1%

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

   2. pow2 12.1%

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

8. Applied egg-rr 12.1%

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

9. Final simplification 12.1%

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

Alternative 10: 9.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{hypot}\left(\pi \cdot 0.5, \sin^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \cos^{-1} \left(1 - x\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.6e-17)
   (hypot (* PI 0.5) (asin (- 1.0 x)))
   (+ (+ 1.0 (acos (- 1.0 x))) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = hypot((((double) M_PI) * 0.5), asin((1.0 - x)));
	} else {
		tmp = (1.0 + acos((1.0 - x))) + -1.0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = Math.hypot((Math.PI * 0.5), Math.asin((1.0 - x)));
	} else {
		tmp = (1.0 + Math.acos((1.0 - x))) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.6e-17:
		tmp = math.hypot((math.pi * 0.5), math.asin((1.0 - x)))
	else:
		tmp = (1.0 + math.acos((1.0 - x))) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = hypot(Float64(pi * 0.5), asin(Float64(1.0 - x)));
	else
		tmp = Float64(Float64(1.0 + acos(Float64(1.0 - x))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.6e-17)
		tmp = hypot((pi * 0.5), asin((1.0 - x)));
	else
		tmp = (1.0 + acos((1.0 - x))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.6e-17], N[Sqrt[N[(Pi * 0.5), $MachinePrecision] ^ 2 + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision], N[(N[(1.0 + N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   6. Simplified 3.9%

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

      1. add-sqr-sqrt 3.9%

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

      2. sqrt-unprod 3.9%

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

      3. add-sqr-sqrt 3.9%

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

      4. cancel-sign-sub-inv 3.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 3.9%

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

      7. sqr-neg 3.9%

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

      8. add-sqr-sqrt 3.9%

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

      9. add-sqr-sqrt 3.9%

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

      10. difference-of-squares 3.9%

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

   8. Applied egg-rr 6.5%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 67.7%

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

      1. expm1-log1p-u 67.7%

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

      2. expm1-undefine 67.5%

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

      3. log1p-undefine 67.7%

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

      4. rem-exp-log 67.7%

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

   4. Applied egg-rr 67.7%

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

4. Final simplification 11.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{hypot}\left(\pi \cdot 0.5, \sin^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \cos^{-1} \left(1 - x\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 9.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\pi - t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t\_0\right) + -1\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (x <= 5.6e-17)
		tmp = pi - t_0;
	else
		tmp = (1.0 + t_0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

      1. add-log-exp 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. rem-log-exp 3.9%

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

      2. acos-asin 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

      5. add-sqr-sqrt 7.4%

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

      6. cancel-sign-sub-inv 7.4%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 6.5%

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

      9. sqr-neg 6.5%

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

      10. add-sqr-sqrt 6.5%

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

      11. add-sqr-sqrt 6.5%

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

      12. asin-acos 6.5%

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

      13. div-inv 6.5%

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

      14. metadata-eval 6.5%

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

      15. associate-+r- 6.5%

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

   6. Applied egg-rr 6.5%

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

      1. fma-undefine 6.5%

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

      2. distribute-lft-out 6.5%

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

      3. metadata-eval 6.5%

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

      4. *-rgt-identity 6.5%

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

   8. Simplified 6.5%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 67.7%

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

      1. expm1-log1p-u 67.7%

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

      2. expm1-undefine 67.5%

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

      3. log1p-undefine 67.7%

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

      4. rem-exp-log 67.7%

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

   4. Applied egg-rr 67.7%

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

4. Final simplification 11.3%

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

Alternative 12: 9.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\pi - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

      1. add-log-exp 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. rem-log-exp 3.9%

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

      2. acos-asin 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

      5. add-sqr-sqrt 7.4%

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

      6. cancel-sign-sub-inv 7.4%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 6.5%

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

      9. sqr-neg 6.5%

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

      10. add-sqr-sqrt 6.5%

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

      11. add-sqr-sqrt 6.5%

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

      12. asin-acos 6.5%

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

      13. div-inv 6.5%

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

      14. metadata-eval 6.5%

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

      15. associate-+r- 6.5%

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

   6. Applied egg-rr 6.5%

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

      1. fma-undefine 6.5%

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

      2. distribute-lft-out 6.5%

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

      3. metadata-eval 6.5%

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

      4. *-rgt-identity 6.5%

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

   8. Simplified 6.5%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 67.7%

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

4. Final simplification 11.3%

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

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 8.8%

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

3. Final simplification 8.8%

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

Developer target: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024066 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

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

  (acos (- 1.0 x)))