bug323 (missed optimization)

Percentage Accurate: 6.7% → 10.3%

Time: 6.5s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{4}}, \sqrt[3]{{\pi}^{2} \cdot 0.25}, -{t\_0}^{2}\right)}{\mathsf{fma}\left(\pi, 0.5, t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/
    (fma
     (cbrt (pow (* PI 0.5) 4.0))
     (cbrt (* (pow PI 2.0) 0.25))
     (- (pow t_0 2.0)))
    (fma PI 0.5 t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return fma(cbrt(pow((((double) M_PI) * 0.5), 4.0)), cbrt((pow(((double) M_PI), 2.0) * 0.25)), -pow(t_0, 2.0)) / fma(((double) M_PI), 0.5, t_0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(fma(cbrt((Float64(pi * 0.5) ^ 4.0)), cbrt(Float64((pi ^ 2.0) * 0.25)), Float64(-(t_0 ^ 2.0))) / fma(pi, 0.5, t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Power[N[Power[N[(Pi * 0.5), $MachinePrecision], 4.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.25), $MachinePrecision], 1/3], $MachinePrecision] + (-N[Power[t$95$0, 2.0], $MachinePrecision])), $MachinePrecision] / N[(Pi * 0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.7%

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

   1. acos-asin 6.7%

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

   2. flip-- 6.7%

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

   3. pow2 6.7%

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

   4. div-inv 6.7%

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

   5. metadata-eval 6.7%

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

   6. pow2 6.7%

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

   7. div-inv 6.7%

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

   8. metadata-eval 6.7%

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

   9. fma-define 6.7%

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

4. Applied egg-rr 6.7%

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

   1. add-cbrt-cube 10.1%

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

   2. pow3 10.1%

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

   3. pow-pow 10.1%

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

   4. metadata-eval 10.1%

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

6. Applied egg-rr 10.1%

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

   1. add-cbrt-cube 6.7%

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

   2. pow3 6.7%

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

8. Applied egg-rr 6.7%

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

   1. pow1/3 4.9%

      \[\leadsto \frac{\color{blue}{{\left({\left(\pi \cdot 0.5\right)}^{6}\right)}^{0.3333333333333333}} - {\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{3}}\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   2. pow-pow 4.9%

      \[\leadsto \frac{\color{blue}{{\left(\pi \cdot 0.5\right)}^{\left(6 \cdot 0.3333333333333333\right)}} - {\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{3}}\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   3. metadata-eval 4.9%

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

   4. add-cube-cbrt 4.9%

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

   5. rem-cbrt-cube 4.9%

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

   6. fmm-def 4.9%

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

   7. cbrt-unprod 4.9%

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

   8. pow-prod-up 10.2%

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

   9. metadata-eval 10.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{\color{blue}{4}}}, \sqrt[3]{{\left(\pi \cdot 0.5\right)}^{2}}, -{\sin^{-1} \left(1 - x\right)}^{2}\right)}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   10. pow1 10.2%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{4}}, \sqrt[3]{\color{blue}{{\left({\left(\pi \cdot 0.5\right)}^{2}\right)}^{1}}}, -{\sin^{-1} \left(1 - x\right)}^{2}\right)}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   11. pow1 10.2%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{4}}, \sqrt[3]{\color{blue}{{\left(\pi \cdot 0.5\right)}^{2}}}, -{\sin^{-1} \left(1 - x\right)}^{2}\right)}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   12. unpow-prod-down 10.2%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{4}}, \sqrt[3]{\color{blue}{{\pi}^{2} \cdot {0.5}^{2}}}, -{\sin^{-1} \left(1 - x\right)}^{2}\right)}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   13. metadata-eval 10.2%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{4}}, \sqrt[3]{{\pi}^{2} \cdot \color{blue}{0.25}}, -{\sin^{-1} \left(1 - x\right)}^{2}\right)}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

10. Applied egg-rr 10.2%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{4}}, \sqrt[3]{{\pi}^{2} \cdot 0.25}, -{\sin^{-1} \left(1 - x\right)}^{2}\right)}}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
11. Add Preprocessing

Alternative 2: 10.2% 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 6.7%

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

   1. acos-asin 6.7%

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

   2. *-un-lft-identity 6.7%

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

   3. add-sqr-sqrt 10.1%

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

   4. prod-diff 10.1%

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

   5. add-sqr-sqrt 10.1%

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

   6. fmm-def 10.1%

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

   7. *-un-lft-identity 10.1%

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

   8. acos-asin 10.1%

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

   9. add-sqr-sqrt 10.0%

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

4. Applied egg-rr 10.0%

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

   1. add-sqr-sqrt 10.1%

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

   2. pow2 10.1%

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

6. Applied egg-rr 10.1%

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

Alternative 3: 10.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{6}} - {t\_0}^{2}}{\mathsf{fma}\left(\pi, 0.5, t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/ (- (cbrt (pow (* PI 0.5) 6.0)) (pow t_0 2.0)) (fma PI 0.5 t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return (cbrt(pow((((double) M_PI) * 0.5), 6.0)) - pow(t_0, 2.0)) / fma(((double) M_PI), 0.5, t_0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(Float64(cbrt((Float64(pi * 0.5) ^ 6.0)) - (t_0 ^ 2.0)) / fma(pi, 0.5, t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Power[N[Power[N[(Pi * 0.5), $MachinePrecision], 6.0], $MachinePrecision], 1/3], $MachinePrecision] - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[(Pi * 0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.7%

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

   1. acos-asin 6.7%

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

   2. flip-- 6.7%

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

   3. pow2 6.7%

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

   4. div-inv 6.7%

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

   5. metadata-eval 6.7%

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

   6. pow2 6.7%

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

   7. div-inv 6.7%

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

   8. metadata-eval 6.7%

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

   9. fma-define 6.7%

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

4. Applied egg-rr 6.7%

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

   1. add-cbrt-cube 10.1%

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

   2. pow3 10.1%

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

   3. pow-pow 10.1%

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

   4. metadata-eval 10.1%

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

6. Applied egg-rr 10.1%

   \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{6}}} - {\sin^{-1} \left(1 - x\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
7. Add Preprocessing

Alternative 4: 10.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

   1. acos-asin 6.7%

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

   2. sub-neg 6.7%

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

   3. div-inv 6.7%

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

   4. metadata-eval 6.7%

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

4. Applied egg-rr 6.7%

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

   1. sub-neg 6.7%

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

6. Simplified 6.7%

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

   1. add-cbrt-cube 10.1%

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

   2. pow3 10.1%

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

8. Applied egg-rr 10.1%

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

9. Final simplification 10.1%

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

Alternative 5: 10.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

   1. acos-asin 6.7%

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

   2. sub-neg 6.7%

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

   3. div-inv 6.7%

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

   4. metadata-eval 6.7%

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

4. Applied egg-rr 6.7%

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

   1. sub-neg 6.7%

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

6. Simplified 6.7%

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

   1. add-cube-cbrt 10.0%

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

   2. pow3 10.0%

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

8. Applied egg-rr 10.0%

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

Alternative 6: 9.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17) (acos (- x)) (acos (- 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-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 <= 5.5d-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 <= 5.5e-17) {
		tmp = Math.acos(-x);
	} else {
		tmp = Math.acos((1.0 - x));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[ArcCos[(-x)], $MachinePrecision], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

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

   if 5.50000000000000001e-17 < x

   1. Initial program 55.5%

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

Alternative 7: 6.9% accurate, 1.0× speedup

Math

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

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(Float64(-x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

3. Taylor expanded in x around inf 6.9%

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

   1. neg-mul-1 6.9%

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

5. Simplified 6.9%

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

Alternative 8: 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 6.7%

   \[\cos^{-1} \left(1 - x\right) \]
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 2024191 
(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)))