bug323 (missed optimization)

Percentage Accurate: 7.0% → 10.5%

Time: 8.4s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 7.0% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt{t\_0}\\ \mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{0.6666666666666666}, \sqrt[3]{\pi \cdot 0.5}, -t\_0\right) + \mathsf{fma}\left(-t\_1, t\_1, t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (sqrt t_0)))
   (+
    (fma (pow (* PI 0.5) 0.6666666666666666) (cbrt (* PI 0.5)) (- t_0))
    (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 fma(pow((((double) M_PI) * 0.5), 0.6666666666666666), cbrt((((double) M_PI) * 0.5)), -t_0) + 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(fma((Float64(pi * 0.5) ^ 0.6666666666666666), cbrt(Float64(pi * 0.5)), Float64(-t_0)) + 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[(N[Power[N[(Pi * 0.5), $MachinePrecision], 0.6666666666666666], $MachinePrecision] * N[Power[N[(Pi * 0.5), $MachinePrecision], 1/3], $MachinePrecision] + (-t$95$0)), $MachinePrecision] + N[((-t$95$1) * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 5.9%

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

   1. acos-asin 5.9%

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

   2. *-un-lft-identity 5.9%

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

   3. add-sqr-sqrt 9.3%

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

   4. prod-diff 9.3%

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

   5. add-sqr-sqrt 9.3%

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

   6. fma-neg 9.3%

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

   7. *-un-lft-identity 9.3%

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

   8. acos-asin 9.3%

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

   9. add-sqr-sqrt 9.3%

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

4. Applied egg-rr 9.3%

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

   1. acos-asin 9.3%

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

   2. add-cube-cbrt 4.0%

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

   3. fma-neg 4.0%

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

   4. cbrt-unprod 9.3%

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

   5. pow2 9.3%

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

   6. div-inv 9.3%

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

   7. metadata-eval 9.3%

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

   8. div-inv 9.3%

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

   9. metadata-eval 9.3%

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

6. Applied egg-rr 9.3%

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

   1. pow1/3 9.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left({\left(\pi \cdot 0.5\right)}^{2}\right)}^{0.3333333333333333}}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right) \]

   2. pow-pow 9.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\pi \cdot 0.5\right)}^{\left(2 \cdot 0.3333333333333333\right)}}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right) \]

   3. metadata-eval 9.3%

      \[\leadsto \mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{\color{blue}{0.6666666666666666}}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right) \]

8. Applied egg-rr 9.3%

   \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\pi \cdot 0.5\right)}^{0.6666666666666666}}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right) \]
9. Add Preprocessing

Alternative 2: 10.5% accurate, 0.1× speedup

Math

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

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 5.9%

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

   1. acos-asin 5.9%

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

   2. *-un-lft-identity 5.9%

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

   3. add-sqr-sqrt 9.3%

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

   4. prod-diff 9.3%

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

   5. add-sqr-sqrt 9.3%

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

   6. fma-neg 9.3%

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

   7. *-un-lft-identity 9.3%

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

   8. acos-asin 9.3%

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

   9. add-sqr-sqrt 9.3%

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

4. Applied egg-rr 9.3%

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

   1. add-sqr-sqrt 9.3%

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

   2. pow2 9.3%

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

6. Applied egg-rr 9.3%

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

Alternative 3: 10.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 5.9%

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

   1. acos-asin 5.9%

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

   2. *-un-lft-identity 5.9%

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

   3. add-sqr-sqrt 9.3%

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

   4. prod-diff 9.3%

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

   5. add-sqr-sqrt 9.3%

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

   6. fma-neg 9.3%

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

   7. *-un-lft-identity 9.3%

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

   8. acos-asin 9.3%

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

   9. add-sqr-sqrt 9.3%

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

4. Applied egg-rr 9.3%

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

5. Final simplification 9.3%

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

Alternative 4: 10.4% 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 5.9%

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

   1. add-exp-log 5.9%

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

4. Applied egg-rr 5.9%

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

   1. rem-exp-log 5.9%

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

   2. acos-asin 5.9%

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

   3. add-cube-cbrt 9.3%

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

   4. cancel-sign-sub-inv 9.3%

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

   5. div-inv 9.3%

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

   6. metadata-eval 9.3%

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

   7. pow2 9.3%

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

6. Applied egg-rr 9.3%

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

7. Final simplification 9.3%

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

Alternative 5: 9.6% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.6e-17)
   (acos (cbrt x))
   (exp (log (+ (+ 1.0 (acos (- 1.0 x))) -1.0)))))

C

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

Java

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 5.6e-17], N[ArcCos[N[Power[x, 1/3], $MachinePrecision]], $MachinePrecision], N[Exp[N[Log[N[(N[(1.0 + N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

   3. Taylor expanded in x around inf 6.5%

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

      1. neg-mul-1 6.5%

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

   5. Simplified 6.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.5%

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

      3. sqr-neg 6.5%

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

      4. sqrt-unprod 6.5%

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

      5. add-exp-log 6.5%

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

      6. add-sqr-sqrt 6.5%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 0.0%

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

      9. sqr-neg 0.0%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 0.0%

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

      12. rec-exp 0.0%

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

      13. add-exp-log 0.0%

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

      14. add-cube-cbrt 0.0%

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

      15. cbrt-div 0.0%

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

      16. metadata-eval 0.0%

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

      17. cbrt-div 0.0%

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

      18. metadata-eval 0.0%

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

      19. cbrt-div 0.0%

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

      20. metadata-eval 0.0%

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

      21. un-div-inv 0.0%

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

   7. Applied egg-rr 6.5%

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

      1. unpow2 6.5%

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

      2. associate-/l* 6.5%

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

      3. *-inverses 6.5%

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

      4. *-rgt-identity 6.5%

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

   9. Simplified 6.5%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 55.0%

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

      1. add-exp-log 55.0%

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

   4. Applied egg-rr 55.0%

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

      1. expm1-log1p-u 55.0%

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

      2. expm1-undefine 55.0%

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

      3. log1p-undefine 55.0%

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

      4. rem-exp-log 55.0%

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

   6. Applied egg-rr 55.0%

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

4. Final simplification 8.4%

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

Alternative 6: 9.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(x \cdot \left({\left(\frac{1}{{x}^{0.3333333333333333}}\right)}^{3} + -1\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (acos (* x (+ (pow (/ 1.0 (pow x 0.3333333333333333)) 3.0) -1.0))))

C

double code(double x) {
	return acos((x * (pow((1.0 / pow(x, 0.3333333333333333)), 3.0) + -1.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((x * (((1.0d0 / (x ** 0.3333333333333333d0)) ** 3.0d0) + (-1.0d0))))
end function

Java

public static double code(double x) {
	return Math.acos((x * (Math.pow((1.0 / Math.pow(x, 0.3333333333333333)), 3.0) + -1.0)));
}

Python

def code(x):
	return math.acos((x * (math.pow((1.0 / math.pow(x, 0.3333333333333333)), 3.0) + -1.0)))

Julia

function code(x)
	return acos(Float64(x * Float64((Float64(1.0 / (x ^ 0.3333333333333333)) ^ 3.0) + -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = acos((x * (((1.0 / (x ^ 0.3333333333333333)) ^ 3.0) + -1.0)));
end

Wolfram

code[x_] := N[ArcCos[N[(x * N[(N[Power[N[(1.0 / N[Power[x, 0.3333333333333333], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 5.9%

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

3. Taylor expanded in x around inf 6.4%

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

   1. add-cube-cbrt 4.9%

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

   2. pow3 4.9%

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

   3. cbrt-div 4.1%

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

   4. metadata-eval 4.1%

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

5. Applied egg-rr 4.1%

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

   1. pow1/3 8.4%

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

7. Applied egg-rr 8.4%

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

8. Final simplification 8.4%

   \[\leadsto \cos^{-1} \left(x \cdot \left({\left(\frac{1}{{x}^{0.3333333333333333}}\right)}^{3} + -1\right)\right) \]
9. Add Preprocessing

Alternative 7: 9.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.6e-17) (acos (cbrt x)) (+ (+ 1.0 (acos (- 1.0 x))) -1.0)))

C

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

Java

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 5.6e-17], N[ArcCos[N[Power[x, 1/3], $MachinePrecision]], $MachinePrecision], N[(N[(1.0 + N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

   3. Taylor expanded in x around inf 6.5%

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

      1. neg-mul-1 6.5%

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

   5. Simplified 6.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.5%

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

      3. sqr-neg 6.5%

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

      4. sqrt-unprod 6.5%

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

      5. add-exp-log 6.5%

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

      6. add-sqr-sqrt 6.5%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 0.0%

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

      9. sqr-neg 0.0%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 0.0%

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

      12. rec-exp 0.0%

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

      13. add-exp-log 0.0%

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

      14. add-cube-cbrt 0.0%

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

      15. cbrt-div 0.0%

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

      16. metadata-eval 0.0%

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

      17. cbrt-div 0.0%

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

      18. metadata-eval 0.0%

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

      19. cbrt-div 0.0%

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

      20. metadata-eval 0.0%

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

      21. un-div-inv 0.0%

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

   7. Applied egg-rr 6.5%

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

      1. unpow2 6.5%

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

      2. associate-/l* 6.5%

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

      3. *-inverses 6.5%

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

      4. *-rgt-identity 6.5%

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

   9. Simplified 6.5%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 55.0%

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

      1. expm1-log1p-u 55.0%

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

      2. expm1-undefine 55.0%

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

      3. log1p-undefine 55.0%

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

      4. rem-exp-log 55.0%

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

   4. Applied egg-rr 55.0%

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

4. Final simplification 8.4%

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

Alternative 8: 9.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 5.6e-17], N[ArcCos[x], $MachinePrecision], N[(N[(1.0 + N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

   3. Taylor expanded in x around inf 6.5%

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

      1. neg-mul-1 6.5%

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

   5. Simplified 6.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.5%

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

      3. sqr-neg 6.5%

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

      4. sqrt-unprod 6.5%

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

      5. add-sqr-sqrt 6.5%

         \[\leadsto \cos^{-1} \color{blue}{x} \]

      6. *-un-lft-identity 6.5%

         \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]

   7. Applied egg-rr 6.5%

      \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]
   8. Step-by-step derivation

      1. *-lft-identity 6.5%

         \[\leadsto \color{blue}{\cos^{-1} x} \]

   9. Simplified 6.5%

      \[\leadsto \color{blue}{\cos^{-1} x} \]

   if 5.5999999999999998e-17 < x

   1. Initial program 55.0%

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

      1. expm1-log1p-u 55.0%

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

      2. expm1-undefine 55.0%

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

      3. log1p-undefine 55.0%

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

      4. rem-exp-log 55.0%

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

   4. Applied egg-rr 55.0%

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

4. Final simplification 8.4%

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

Alternative 9: 9.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 5.6e-17], N[ArcCos[x], $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. Add Preprocessing

   3. Taylor expanded in x around inf 6.5%

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

      1. neg-mul-1 6.5%

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

   5. Simplified 6.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.5%

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

      3. sqr-neg 6.5%

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

      4. sqrt-unprod 6.5%

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

      5. add-sqr-sqrt 6.5%

         \[\leadsto \cos^{-1} \color{blue}{x} \]

      6. *-un-lft-identity 6.5%

         \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]

   7. Applied egg-rr 6.5%

      \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]
   8. Step-by-step derivation

      1. *-lft-identity 6.5%

         \[\leadsto \color{blue}{\cos^{-1} x} \]

   9. Simplified 6.5%

      \[\leadsto \color{blue}{\cos^{-1} x} \]

   if 5.5999999999999998e-17 < x

   1. Initial program 55.0%

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

Alternative 10: 7.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return acos(x)
end

MATLAB

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

Wolfram

code[x_] := N[ArcCos[x], $MachinePrecision]

Derivation

1. Initial program 5.9%

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

3. Taylor expanded in x around inf 6.7%

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

   1. neg-mul-1 6.7%

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

5. Simplified 6.7%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 6.7%

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

   3. sqr-neg 6.7%

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

   4. sqrt-unprod 6.7%

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

   5. add-sqr-sqrt 6.7%

      \[\leadsto \cos^{-1} \color{blue}{x} \]

   6. *-un-lft-identity 6.7%

      \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]

7. Applied egg-rr 6.7%

   \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]
8. Step-by-step derivation

   1. *-lft-identity 6.7%

      \[\leadsto \color{blue}{\cos^{-1} x} \]

9. Simplified 6.7%

   \[\leadsto \color{blue}{\cos^{-1} x} \]
10. Add Preprocessing

Alternative 11: 3.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return acos(1.0)
end

MATLAB

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

Wolfram

code[x_] := N[ArcCos[1.0], $MachinePrecision]

Derivation

1. Initial program 5.9%

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

3. Taylor expanded in x around 0 3.8%

   \[\leadsto \cos^{-1} \color{blue}{1} \]
4. Add Preprocessing

Developer Target 1: 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 2024117 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :alt
  (! :herbie-platform default (* 2 (asin (sqrt (/ x 2)))))

  (acos (- 1.0 x)))