bug323 (missed optimization)

Percentage Accurate: 6.6% → 10.1%

Time: 29.1s

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: 6.6% 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: 10.1% 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 7.8%

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

   1. acos-asin 7.8%

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

   2. *-un-lft-identity 7.8%

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

   3. add-sqr-sqrt 11.1%

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

   4. prod-diff 11.1%

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

   5. add-sqr-sqrt 11.2%

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

   6. fma-neg 11.2%

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

   7. *-un-lft-identity 11.2%

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

   8. acos-asin 11.2%

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

   9. add-sqr-sqrt 11.1%

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

4. Applied egg-rr 11.1%

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

   1. add-sqr-sqrt 11.2%

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

   2. pow2 11.2%

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

6. Applied egg-rr 11.2%

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

7. Final simplification 11.2%

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

Alternative 2: 10.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{{\left(\pi \cdot 0.5\right)}^{2} - \left(e^{\mathsf{log1p}\left({t\_0}^{2}\right)} + -1\right)}{\mathsf{fma}\left(\pi, 0.5, t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/
    (- (pow (* PI 0.5) 2.0) (+ (exp (log1p (pow t_0 2.0))) -1.0))
    (fma PI 0.5 t_0))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.8%

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

   1. acos-asin 7.8%

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

   2. flip-- 7.8%

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

   3. pow2 7.8%

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

   4. div-inv 7.8%

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

   5. metadata-eval 7.8%

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

   6. pow2 7.8%

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

   7. div-inv 7.8%

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

   8. metadata-eval 7.8%

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

   9. fma-define 7.8%

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

4. Applied egg-rr 7.8%

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

   1. expm1-log1p-u 7.8%

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

   2. expm1-undefine 11.1%

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

6. Applied egg-rr 11.1%

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

7. Final simplification 11.1%

   \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{2} - \left(e^{\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{2}\right)} + -1\right)}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
8. Add Preprocessing

Alternative 3: 10.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.8%

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

   1. acos-asin 7.8%

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

   2. *-un-lft-identity 7.8%

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

   3. add-sqr-sqrt 11.1%

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

   4. prod-diff 11.1%

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

   5. add-sqr-sqrt 11.2%

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

   6. fma-neg 11.2%

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

   7. *-un-lft-identity 11.2%

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

   8. acos-asin 11.2%

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

   9. add-sqr-sqrt 11.1%

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

4. Applied egg-rr 11.1%

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

5. Final simplification 11.1%

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

Alternative 4: 9.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(\sqrt[3]{{\left(e^{\left(1 + t\_0\right) + -1}\right)}^{1.5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0)
     (+ (asin (- 1.0 x)) (* PI 0.5))
     (* 2.0 (log (cbrt (pow (exp (+ (+ 1.0 t_0) -1.0)) 1.5)))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = asin((1.0 - x)) + (((double) M_PI) * 0.5);
	} else {
		tmp = 2.0 * log(cbrt(pow(exp(((1.0 + t_0) + -1.0)), 1.5)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.asin((1.0 - x)) + (Math.PI * 0.5);
	} else {
		tmp = 2.0 * Math.log(Math.cbrt(Math.pow(Math.exp(((1.0 + t_0) + -1.0)), 1.5)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(asin(Float64(1.0 - x)) + Float64(pi * 0.5));
	else
		tmp = Float64(2.0 * log(cbrt((exp(Float64(Float64(1.0 + t_0) + -1.0)) ^ 1.5))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Log[N[Power[N[Power[N[Exp[N[(N[(1.0 + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 1 x)) < 0.0

   1. Initial program 3.9%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-undefine 3.9%

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

      3. log1p-undefine 3.9%

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

      4. rem-exp-log 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. add-exp-log 3.9%

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

      2. expm1-define 3.9%

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

      3. log1p-define 3.9%

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

      4. expm1-log1p-u 3.9%

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

      5. acos-asin 3.9%

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

      6. div-inv 3.9%

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

      7. metadata-eval 3.9%

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

      8. sub-neg 3.9%

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

      9. add-cube-cbrt 7.3%

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

      10. unpow2 7.3%

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

      11. *-commutative 7.3%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 6.5%

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

   6. Applied egg-rr 6.5%

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

   if 0.0 < (acos.f64 (-.f64 1 x))

   1. Initial program 67.3%

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

      1. add-log-exp 67.4%

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

      2. add-sqr-sqrt 67.5%

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

      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. add-cbrt-cube 67.7%

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

      2. add-sqr-sqrt 67.7%

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

      3. pow1 67.7%

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

      4. pow1/2 67.7%

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

      5. pow-prod-up 67.7%

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

      6. metadata-eval 67.7%

         \[\leadsto 2 \cdot \log \left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{\color{blue}{1.5}}}\right) \]

   8. Applied egg-rr 67.7%

      \[\leadsto 2 \cdot \log \color{blue}{\left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{1.5}}\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 67.3%

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

      2. expm1-undefine 67.3%

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

      3. log1p-undefine 67.3%

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

      4. rem-exp-log 67.3%

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

   10. Applied egg-rr 67.7%

       \[\leadsto 2 \cdot \log \left(\sqrt[3]{{\left(e^{\color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right) - 1}}\right)}^{1.5}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(\sqrt[3]{{\left(e^{\left(1 + \cos^{-1} \left(1 - x\right)\right) + -1}\right)}^{1.5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 6.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;2 \cdot \log \left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{1.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 1.0)
   (* 2.0 (log (cbrt (pow (exp (acos (- 1.0 x))) 1.5))))
   (+ (asin (- 1.0 x)) (* PI 0.5))))

C

double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = 2.0 * log(cbrt(pow(exp(acos((1.0 - x))), 1.5)));
	} else {
		tmp = asin((1.0 - x)) + (((double) M_PI) * 0.5);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = 2.0 * Math.log(Math.cbrt(Math.pow(Math.exp(Math.acos((1.0 - x))), 1.5)));
	} else {
		tmp = Math.asin((1.0 - x)) + (Math.PI * 0.5);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = Float64(2.0 * log(cbrt((exp(acos(Float64(1.0 - x))) ^ 1.5))));
	else
		tmp = Float64(asin(Float64(1.0 - x)) + Float64(pi * 0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[(2.0 * N[Log[N[Power[N[Power[N[Exp[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.8%

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

      1. add-log-exp 7.9%

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

      2. add-sqr-sqrt 7.9%

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

      3. log-prod 7.9%

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

   4. Applied egg-rr 7.9%

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

      1. count-2 7.9%

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

   6. Simplified 7.9%

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

      1. add-cbrt-cube 7.9%

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

      2. add-sqr-sqrt 7.9%

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

      3. pow1 7.9%

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

      4. pow1/2 7.9%

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

      5. pow-prod-up 7.9%

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

      6. metadata-eval 7.9%

         \[\leadsto 2 \cdot \log \left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{\color{blue}{1.5}}}\right) \]

   8. Applied egg-rr 7.9%

      \[\leadsto 2 \cdot \log \color{blue}{\left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{1.5}}\right)} \]

   if 1 < (-.f64 1 x)

   1. Initial program 7.8%

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

      1. expm1-log1p-u 7.8%

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

      2. expm1-undefine 7.8%

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

      3. log1p-undefine 7.8%

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

      4. rem-exp-log 7.8%

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

   4. Applied egg-rr 7.8%

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

      1. add-exp-log 7.8%

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

      2. expm1-define 7.8%

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

      3. log1p-define 7.8%

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

      4. expm1-log1p-u 7.8%

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

      5. acos-asin 7.8%

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

      6. div-inv 7.8%

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

      7. metadata-eval 7.8%

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

      8. sub-neg 7.8%

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

      9. add-cube-cbrt 11.1%

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

      10. unpow2 11.1%

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

      11. *-commutative 11.1%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 6.9%

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

   6. Applied egg-rr 6.9%

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

4. Final simplification 7.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;2 \cdot \log \left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{1.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 10.1% 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 7.8%

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

   1. expm1-log1p-u 7.8%

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

   2. expm1-undefine 7.8%

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

   3. log1p-undefine 7.8%

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

   4. rem-exp-log 7.8%

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

4. Applied egg-rr 7.8%

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

   1. add-exp-log 7.8%

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

   2. expm1-define 7.8%

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

   3. log1p-define 7.8%

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

   4. expm1-log1p-u 7.8%

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

   5. acos-asin 7.8%

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

   6. div-inv 7.8%

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

   7. metadata-eval 7.8%

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

   8. sub-neg 7.8%

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

   9. add-cube-cbrt 11.1%

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

   10. unpow2 11.1%

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

   11. *-commutative 11.1%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 6.9%

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

6. Applied egg-rr 6.9%

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

   1. add-sqr-sqrt 6.9%

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

   2. sqrt-prod 6.9%

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

   3. sqr-neg 6.9%

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

   4. rem-3cbrt-rft 6.9%

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

   5. unpow2 6.9%

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

   6. rem-3cbrt-rft 6.9%

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

   7. unpow2 6.9%

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

   8. sqrt-unprod 0.0%

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

   9. add-sqr-sqrt 11.1%

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

   10. distribute-lft-neg-in 11.1%

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

   11. *-commutative 11.1%

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

   12. neg-mul-1 11.1%

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

8. Applied egg-rr 11.1%

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

9. Final simplification 11.1%

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

Alternative 7: 9.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(1 + t\_0\right) + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0)
     (+ (asin (- 1.0 x)) (* PI 0.5))
     (log (exp (+ (+ 1.0 t_0) -1.0))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = asin((1.0 - x)) + (((double) M_PI) * 0.5);
	} else {
		tmp = log(exp(((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 (t_0 <= 0.0) {
		tmp = Math.asin((1.0 - x)) + (Math.PI * 0.5);
	} else {
		tmp = Math.log(Math.exp(((1.0 + t_0) + -1.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.asin((1.0 - x)) + (math.pi * 0.5)
	else:
		tmp = math.log(math.exp(((1.0 + t_0) + -1.0)))
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(asin(Float64(1.0 - x)) + Float64(pi * 0.5));
	else
		tmp = log(exp(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 (t_0 <= 0.0)
		tmp = asin((1.0 - x)) + (pi * 0.5);
	else
		tmp = log(exp(((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[t$95$0, 0.0], N[(N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[N[(N[(1.0 + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 1 x)) < 0.0

   1. Initial program 3.9%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-undefine 3.9%

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

      3. log1p-undefine 3.9%

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

      4. rem-exp-log 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. add-exp-log 3.9%

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

      2. expm1-define 3.9%

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

      3. log1p-define 3.9%

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

      4. expm1-log1p-u 3.9%

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

      5. acos-asin 3.9%

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

      6. div-inv 3.9%

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

      7. metadata-eval 3.9%

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

      8. sub-neg 3.9%

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

      9. add-cube-cbrt 7.3%

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

      10. unpow2 7.3%

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

      11. *-commutative 7.3%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 6.5%

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

   6. Applied egg-rr 6.5%

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

   if 0.0 < (acos.f64 (-.f64 1 x))

   1. Initial program 67.3%

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

      1. add-log-exp 67.4%

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

   4. Applied egg-rr 67.4%

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

      1. expm1-log1p-u 67.3%

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

      2. expm1-undefine 67.3%

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

      3. log1p-undefine 67.3%

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

      4. rem-exp-log 67.3%

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

   6. Applied egg-rr 67.4%

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

4. Final simplification 10.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(1 + \cos^{-1} \left(1 - x\right)\right) + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 6.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = 2.0 * log(sqrt(exp(acos((1.0 - x)))));
	} else {
		tmp = asin((1.0 - x)) + (((double) M_PI) * 0.5);
	}
	return tmp;
}

Java

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

Python

def code(x):
	tmp = 0
	if (1.0 - x) <= 1.0:
		tmp = 2.0 * math.log(math.sqrt(math.exp(math.acos((1.0 - x)))))
	else:
		tmp = math.asin((1.0 - x)) + (math.pi * 0.5)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 - x) <= 1.0)
		tmp = 2.0 * log(sqrt(exp(acos((1.0 - x)))));
	else
		tmp = asin((1.0 - x)) + (pi * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.8%

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

      1. add-log-exp 7.9%

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

      2. add-sqr-sqrt 7.9%

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

      3. log-prod 7.9%

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

   4. Applied egg-rr 7.9%

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

      1. count-2 7.9%

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

   6. Simplified 7.9%

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

   if 1 < (-.f64 1 x)

   1. Initial program 7.8%

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

      1. expm1-log1p-u 7.8%

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

      2. expm1-undefine 7.8%

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

      3. log1p-undefine 7.8%

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

      4. rem-exp-log 7.8%

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

   4. Applied egg-rr 7.8%

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

      1. add-exp-log 7.8%

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

      2. expm1-define 7.8%

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

      3. log1p-define 7.8%

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

      4. expm1-log1p-u 7.8%

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

      5. acos-asin 7.8%

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

      6. div-inv 7.8%

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

      7. metadata-eval 7.8%

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

      8. sub-neg 7.8%

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

      9. add-cube-cbrt 11.1%

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

      10. unpow2 11.1%

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

      11. *-commutative 11.1%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 6.9%

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

   6. Applied egg-rr 6.9%

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

4. Final simplification 7.9%

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

Alternative 9: 9.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0) (+ (asin (- 1.0 x)) (* PI 0.5)) (log (exp t_0)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = asin((1.0 - x)) + (((double) M_PI) * 0.5);
	} else {
		tmp = log(exp(t_0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.asin((1.0 - x)) + (Math.PI * 0.5);
	} else {
		tmp = Math.log(Math.exp(t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.asin((1.0 - x)) + (math.pi * 0.5)
	else:
		tmp = math.log(math.exp(t_0))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 1 x)) < 0.0

   1. Initial program 3.9%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-undefine 3.9%

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

      3. log1p-undefine 3.9%

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

      4. rem-exp-log 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. add-exp-log 3.9%

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

      2. expm1-define 3.9%

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

      3. log1p-define 3.9%

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

      4. expm1-log1p-u 3.9%

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

      5. acos-asin 3.9%

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

      6. div-inv 3.9%

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

      7. metadata-eval 3.9%

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

      8. sub-neg 3.9%

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

      9. add-cube-cbrt 7.3%

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

      10. unpow2 7.3%

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

      11. *-commutative 7.3%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 6.5%

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

   6. Applied egg-rr 6.5%

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

   if 0.0 < (acos.f64 (-.f64 1 x))

   1. Initial program 67.3%

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

      1. add-log-exp 67.4%

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

   4. Applied egg-rr 67.4%

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

4. Final simplification 10.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 6.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.8%

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

      1. add-log-exp 7.9%

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

      2. add-sqr-sqrt 7.9%

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

      3. log-prod 7.9%

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

   4. Applied egg-rr 7.9%

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

      1. count-2 7.9%

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

   6. Simplified 7.9%

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

      1. pow1/2 7.9%

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

      2. pow-exp 7.9%

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

   8. Applied egg-rr 7.9%

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

   if 1 < (-.f64 1 x)

   1. Initial program 7.8%

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

      1. expm1-log1p-u 7.8%

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

      2. expm1-undefine 7.8%

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

      3. log1p-undefine 7.8%

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

      4. rem-exp-log 7.8%

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

   4. Applied egg-rr 7.8%

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

      1. add-exp-log 7.8%

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

      2. expm1-define 7.8%

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

      3. log1p-define 7.8%

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

      4. expm1-log1p-u 7.8%

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

      5. acos-asin 7.8%

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

      6. div-inv 7.8%

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

      7. metadata-eval 7.8%

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

      8. sub-neg 7.8%

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

      9. add-cube-cbrt 11.1%

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

      10. unpow2 11.1%

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

      11. *-commutative 11.1%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 6.9%

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

   6. Applied egg-rr 6.9%

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

4. Final simplification 7.9%

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

Alternative 11: 9.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(0.3333333333333333 \cdot \left(t\_0 \cdot 0.5\right)\right) \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0)
     (+ (asin (- 1.0 x)) (* PI 0.5))
     (* 2.0 (* (* 0.3333333333333333 (* t_0 0.5)) 3.0)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = asin((1.0 - x)) + (((double) M_PI) * 0.5);
	} else {
		tmp = 2.0 * ((0.3333333333333333 * (t_0 * 0.5)) * 3.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.asin((1.0 - x)) + (Math.PI * 0.5);
	} else {
		tmp = 2.0 * ((0.3333333333333333 * (t_0 * 0.5)) * 3.0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.asin((1.0 - x)) + (math.pi * 0.5)
	else:
		tmp = 2.0 * ((0.3333333333333333 * (t_0 * 0.5)) * 3.0)
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(asin(Float64(1.0 - x)) + Float64(pi * 0.5));
	else
		tmp = Float64(2.0 * Float64(Float64(0.3333333333333333 * Float64(t_0 * 0.5)) * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = asin((1.0 - x)) + (pi * 0.5);
	else
		tmp = 2.0 * ((0.3333333333333333 * (t_0 * 0.5)) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(0.3333333333333333 * N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 1 x)) < 0.0

   1. Initial program 3.9%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-undefine 3.9%

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

      3. log1p-undefine 3.9%

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

      4. rem-exp-log 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. add-exp-log 3.9%

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

      2. expm1-define 3.9%

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

      3. log1p-define 3.9%

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

      4. expm1-log1p-u 3.9%

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

      5. acos-asin 3.9%

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

      6. div-inv 3.9%

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

      7. metadata-eval 3.9%

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

      8. sub-neg 3.9%

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

      9. add-cube-cbrt 7.3%

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

      10. unpow2 7.3%

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

      11. *-commutative 7.3%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 6.5%

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

   6. Applied egg-rr 6.5%

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

   if 0.0 < (acos.f64 (-.f64 1 x))

   1. Initial program 67.3%

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

      1. add-log-exp 67.4%

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

      2. add-sqr-sqrt 67.5%

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

      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. add-cube-cbrt 66.7%

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

      2. pow3 66.7%

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

   8. Applied egg-rr 66.7%

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

      1. log-pow 66.7%

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

      2. *-commutative 66.7%

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

      3. pow1/3 67.9%

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

      4. log-pow 67.6%

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

      5. pow1/2 67.6%

         \[\leadsto 2 \cdot \left(\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{0.5}\right)}\right) \cdot 3\right) \]

      6. log-pow 67.4%

         \[\leadsto 2 \cdot \left(\left(0.3333333333333333 \cdot \color{blue}{\left(0.5 \cdot \log \left(e^{\cos^{-1} \left(1 - x\right)}\right)\right)}\right) \cdot 3\right) \]

      7. rem-log-exp 67.3%

         \[\leadsto 2 \cdot \left(\left(0.3333333333333333 \cdot \left(0.5 \cdot \color{blue}{\cos^{-1} \left(1 - x\right)}\right)\right) \cdot 3\right) \]

   10. Applied egg-rr 67.3%

       \[\leadsto 2 \cdot \color{blue}{\left(\left(0.3333333333333333 \cdot \left(0.5 \cdot \cos^{-1} \left(1 - x\right)\right)\right) \cdot 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(0.3333333333333333 \cdot \left(\cos^{-1} \left(1 - x\right) \cdot 0.5\right)\right) \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 6.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.8%

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

   if 1 < (-.f64 1 x)

   1. Initial program 7.8%

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

      1. expm1-log1p-u 7.8%

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

      2. expm1-undefine 7.8%

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

      3. log1p-undefine 7.8%

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

      4. rem-exp-log 7.8%

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

   4. Applied egg-rr 7.8%

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

      1. add-exp-log 7.8%

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

      2. expm1-define 7.8%

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

      3. log1p-define 7.8%

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

      4. expm1-log1p-u 7.8%

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

      5. acos-asin 7.8%

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

      6. div-inv 7.8%

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

      7. metadata-eval 7.8%

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

      8. sub-neg 7.8%

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

      9. add-cube-cbrt 11.1%

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

      10. unpow2 11.1%

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

      11. *-commutative 11.1%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 6.9%

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

   6. Applied egg-rr 6.9%

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

4. Final simplification 7.8%

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

Alternative 13: 6.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.8%

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

3. Final simplification 7.8%

   \[\leadsto \cos^{-1} \left(1 - x\right) \]
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 2024055 
(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)))