bug323 (missed optimization)

Percentage Accurate: 7.1% → 10.5%

Time: 22.3s

Alternatives: 10

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: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ t_1 := \sqrt{\sin^{-1} \left(1 - x\right)}\\ t_0 + \mathsf{fma}\left(-t_1, t_1, {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi \cdot 0.5 - t_0}\right)\right)\right)}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))) (t_1 (sqrt (asin (- 1.0 x)))))
   (+
    t_0
    (fma (- t_1) t_1 (pow (expm1 (log1p (sqrt (- (* PI 0.5) t_0)))) 2.0)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double t_1 = sqrt(asin((1.0 - x)));
	return t_0 + fma(-t_1, t_1, pow(expm1(log1p(sqrt(((((double) M_PI) * 0.5) - t_0)))), 2.0));
}

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	t_1 = sqrt(asin(Float64(1.0 - x)))
	return Float64(t_0 + fma(Float64(-t_1), t_1, (expm1(log1p(sqrt(Float64(Float64(pi * 0.5) - t_0)))) ^ 2.0)))
end

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. acos-asin 6.8%

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

   2. *-un-lft-identity 6.8%

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

   3. add-sqr-sqrt 10.2%

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

   4. prod-diff 10.2%

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

   5. add-sqr-sqrt 10.2%

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

   6. fma-neg 10.2%

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

   7. *-un-lft-identity 10.2%

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

   8. acos-asin 10.2%

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

   9. add-sqr-sqrt 10.2%

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

3. Applied egg-rr 10.2%

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

   1. add-sqr-sqrt 10.2%

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

   2. pow2 10.2%

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

5. Applied egg-rr 10.2%

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

   1. expm1-log1p-u 10.2%

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

7. Applied egg-rr 10.2%

   \[\leadsto \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)\right)\right)}}^{2}\right) \]
8. Step-by-step derivation

   1. asin-acos 10.2%

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

   2. div-inv 10.2%

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

   3. metadata-eval 10.2%

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

9. Applied egg-rr 10.2%

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

10. Final simplification 10.2%

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

Alternative 2: 10.5% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (asin (- 1.0 x)))))
   (+ (acos (- 1.0 x)) (fma (- t_0) t_0 (pow (expm1 (log1p 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(expm1(log1p(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, (expm1(log1p(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[N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.8%

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

   1. acos-asin 6.8%

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

   2. *-un-lft-identity 6.8%

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

   3. add-sqr-sqrt 10.2%

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

   4. prod-diff 10.2%

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

   5. add-sqr-sqrt 10.2%

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

   6. fma-neg 10.2%

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

   7. *-un-lft-identity 10.2%

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

   8. acos-asin 10.2%

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

   9. add-sqr-sqrt 10.2%

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

3. Applied egg-rr 10.2%

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

   1. add-sqr-sqrt 10.2%

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

   2. pow2 10.2%

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

5. Applied egg-rr 10.2%

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

   1. expm1-log1p-u 10.2%

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

7. Applied egg-rr 10.2%

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

8. Final simplification 10.2%

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

Alternative 3: 10.5% 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 6.8%

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

   1. acos-asin 6.8%

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

   2. *-un-lft-identity 6.8%

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

   3. add-sqr-sqrt 10.2%

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

   4. prod-diff 10.2%

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

   5. add-sqr-sqrt 10.2%

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

   6. fma-neg 10.2%

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

   7. *-un-lft-identity 10.2%

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

   8. acos-asin 10.2%

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

   9. add-sqr-sqrt 10.2%

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

3. Applied egg-rr 10.2%

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

   1. add-sqr-sqrt 10.2%

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

   2. pow2 10.2%

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

5. Applied egg-rr 10.2%

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

6. Final simplification 10.2%

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

Alternative 4: 10.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = sqrt(t_0)
	return Float64(1.0 + Float64(Float64(acos(Float64(1.0 - x)) + fma(Float64(-t_1), t_1, t_0)) + -1.0))
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[(1.0 + N[(N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[((-t$95$1) * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 6.8%

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

   1. expm1-log1p-u 6.8%

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

   2. expm1-udef 6.8%

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

   3. log1p-udef 6.8%

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

   4. rem-exp-log 6.8%

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

3. Applied egg-rr 6.8%

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

   1. associate--l+ 6.8%

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

   2. add-exp-log 6.8%

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

   3. expm1-udef 6.8%

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

   4. +-commutative 6.8%

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

   5. expm1-udef 6.8%

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

   6. add-exp-log 6.8%

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

   7. sub-neg 6.8%

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

   8. metadata-eval 6.8%

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

5. Applied egg-rr 6.8%

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

   1. acos-asin 6.8%

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

   2. *-un-lft-identity 6.8%

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

   3. add-sqr-sqrt 10.2%

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

   4. prod-diff 10.2%

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

   5. add-sqr-sqrt 10.2%

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

   6. fma-neg 10.2%

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

   7. *-un-lft-identity 10.2%

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

   8. acos-asin 10.2%

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

   9. add-sqr-sqrt 10.2%

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

7. Applied egg-rr 10.2%

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

8. Final simplification 10.2%

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

Alternative 5: 10.5% 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 6.8%

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

   1. acos-asin 6.8%

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

   2. *-un-lft-identity 6.8%

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

   3. add-sqr-sqrt 10.2%

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

   4. prod-diff 10.2%

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

   5. add-sqr-sqrt 10.2%

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

   6. fma-neg 10.2%

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

   7. *-un-lft-identity 10.2%

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

   8. acos-asin 10.2%

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

   9. add-sqr-sqrt 10.2%

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

3. Applied egg-rr 10.2%

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

4. Final simplification 10.2%

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

Alternative 6: 10.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. expm1-log1p-u 6.8%

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

   2. expm1-udef 6.8%

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

   3. log1p-udef 6.8%

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

   4. rem-exp-log 6.8%

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

3. Applied egg-rr 6.8%

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

   1. associate--l+ 6.8%

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

   2. add-exp-log 6.8%

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

   3. expm1-udef 6.8%

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

   4. add-exp-log 6.8%

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

   5. log1p-udef 6.8%

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

   6. log1p-expm1-u 6.8%

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

   7. add-exp-log 6.8%

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

   8. acos-asin 6.8%

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

   9. add-cube-cbrt 10.1%

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

   10. cancel-sign-sub-inv 10.1%

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

   11. div-inv 10.1%

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

   12. metadata-eval 10.1%

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

   13. pow2 10.1%

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

5. Applied egg-rr 10.1%

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

   1. unpow2 10.1%

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

   2. cbrt-unprod 10.2%

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

   3. pow2 10.2%

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

7. Applied egg-rr 10.2%

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

8. Final simplification 10.2%

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

Alternative 7: 7.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \cos^{-1} \left(1 - x\right)\\ t_2 := t_1 + -1\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\sqrt[3]{{t_1}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(-1 + \left(t_1 + \left(t_0 + t_0\right)\right)\right) \cdot t_2}{1 - t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (acos (- 1.0 x))) (t_2 (+ t_1 -1.0)))
   (if (<= (- 1.0 x) 1.0)
     (cbrt (pow t_1 3.0))
     (/ (- 1.0 (* (+ -1.0 (+ t_1 (+ t_0 t_0))) t_2)) (- 1.0 t_2)))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = acos((1.0 - x));
	double t_2 = t_1 + -1.0;
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = cbrt(pow(t_1, 3.0));
	} else {
		tmp = (1.0 - ((-1.0 + (t_1 + (t_0 + t_0))) * t_2)) / (1.0 - t_2);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	double t_1 = Math.acos((1.0 - x));
	double t_2 = t_1 + -1.0;
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = Math.cbrt(Math.pow(t_1, 3.0));
	} else {
		tmp = (1.0 - ((-1.0 + (t_1 + (t_0 + t_0))) * t_2)) / (1.0 - t_2);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = acos(Float64(1.0 - x))
	t_2 = Float64(t_1 + -1.0)
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = cbrt((t_1 ^ 3.0));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(-1.0 + Float64(t_1 + Float64(t_0 + t_0))) * t_2)) / Float64(1.0 - t_2));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + -1.0), $MachinePrecision]}, If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[Power[N[Power[t$95$1, 3.0], $MachinePrecision], 1/3], $MachinePrecision], N[(N[(1.0 - N[(N[(-1.0 + N[(t$95$1 + N[(t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.8%

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

      1. add-cbrt-cube 6.8%

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

      2. pow3 6.8%

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

   3. Applied egg-rr 6.8%

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

   if 1 < (-.f64 1 x)

   1. Initial program 6.8%

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

      1. expm1-log1p-u 6.8%

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

      2. expm1-udef 6.8%

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

      3. log1p-udef 6.8%

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

      4. rem-exp-log 6.8%

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

   3. Applied egg-rr 6.8%

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

      1. associate--l+ 6.8%

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

      2. add-exp-log 6.8%

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

      3. expm1-udef 6.8%

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

      4. flip-+ 6.8%

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

      5. metadata-eval 6.8%

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

      6. expm1-udef 6.8%

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

      7. add-exp-log 6.8%

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

      8. expm1-udef 6.8%

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

      9. add-exp-log 6.8%

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

      10. sub-neg 6.8%

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

      11. metadata-eval 6.8%

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

      12. sub-neg 6.8%

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

      13. metadata-eval 6.8%

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

   5. Applied egg-rr 6.8%

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

      1. acos-asin 6.8%

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

      2. *-un-lft-identity 6.8%

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

      3. add-sqr-sqrt 10.2%

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

      4. prod-diff 10.2%

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

      5. add-sqr-sqrt 10.2%

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

      6. fma-neg 10.2%

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

      7. *-un-lft-identity 10.2%

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

      8. acos-asin 10.2%

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

      9. add-sqr-sqrt 10.2%

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

   7. Applied egg-rr 10.2%

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

      1. fma-udef 10.2%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 6.8%

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

      4. sqr-neg 6.8%

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

      5. add-sqr-sqrt 6.8%

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

      6. add-sqr-sqrt 6.8%

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

   9. Applied egg-rr 6.8%

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

4. Final simplification 6.8%

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

Alternative 8: 9.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 5.6e-17], N[(N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $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. Step-by-step derivation

      1. expm1-log1p-u 3.9%

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

      2. expm1-udef 3.9%

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

      3. log1p-udef 3.9%

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

      4. rem-exp-log 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. associate--l+ 3.9%

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

      2. add-exp-log 3.9%

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

      3. expm1-udef 3.9%

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

      4. add-exp-log 3.9%

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

      5. log1p-udef 3.9%

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

      6. log1p-expm1-u 3.9%

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

      7. add-exp-log 3.9%

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

      8. acos-asin 3.9%

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

      9. add-sqr-sqrt 7.4%

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

      10. cancel-sign-sub-inv 7.4%

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

      11. div-inv 7.4%

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

      12. metadata-eval 7.4%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 6.5%

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

      15. sqr-neg 6.5%

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

      16. add-sqr-sqrt 6.5%

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

      17. add-sqr-sqrt 6.5%

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

   5. Applied egg-rr 6.5%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 62.0%

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

      1. add-cbrt-cube 62.1%

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

      2. pow3 62.1%

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

   3. Applied egg-rr 62.1%

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

4. Final simplification 9.3%

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

Alternative 9: 9.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 5.6e-17], N[(N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision], N[ArcCos[N[(1.0 - x), $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. Step-by-step derivation

      1. expm1-log1p-u 3.9%

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

      2. expm1-udef 3.9%

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

      3. log1p-udef 3.9%

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

      4. rem-exp-log 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. associate--l+ 3.9%

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

      2. add-exp-log 3.9%

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

      3. expm1-udef 3.9%

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

      4. add-exp-log 3.9%

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

      5. log1p-udef 3.9%

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

      6. log1p-expm1-u 3.9%

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

      7. add-exp-log 3.9%

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

      8. acos-asin 3.9%

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

      9. add-sqr-sqrt 7.4%

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

      10. cancel-sign-sub-inv 7.4%

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

      11. div-inv 7.4%

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

      12. metadata-eval 7.4%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 6.5%

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

      15. sqr-neg 6.5%

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

      16. add-sqr-sqrt 6.5%

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

      17. add-sqr-sqrt 6.5%

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

   5. Applied egg-rr 6.5%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 62.0%

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

4. Final simplification 9.3%

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

Alternative 10: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Final simplification 6.8%

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

Developer target: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023318 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

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

  (acos (- 1.0 x)))