bug323 (missed optimization)

Percentage Accurate: 6.8% → 10.3%

Time: 19.5s

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: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.3% accurate, 0.1× speedup

Math

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

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 6.3%

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

   1. acos-asin 6.3%

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

   2. *-un-lft-identity 6.3%

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

   3. add-sqr-sqrt 9.7%

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

   4. prod-diff 9.7%

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

   5. add-sqr-sqrt 9.7%

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

   6. fma-neg 9.7%

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

   7. *-un-lft-identity 9.7%

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

   8. acos-asin 9.7%

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

   9. add-sqr-sqrt 9.7%

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

4. Applied egg-rr 9.7%

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

   1. add-sqr-sqrt 9.7%

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

   2. pow2 9.7%

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

6. Applied egg-rr 9.7%

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

Alternative 2: 10.3% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 0.5))))
   (- (* PI 0.5) (fma t_0 t_0 (- (acos (- 1.0 x)))))))

C

double code(double x) {
	double t_0 = sqrt((((double) M_PI) * 0.5));
	return (((double) M_PI) * 0.5) - fma(t_0, t_0, -acos((1.0 - x)));
}

Julia

function code(x)
	t_0 = sqrt(Float64(pi * 0.5))
	return Float64(Float64(pi * 0.5) - fma(t_0, t_0, Float64(-acos(Float64(1.0 - x)))))
end

Wolfram

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

Derivation

1. Initial program 6.3%

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

   1. acos-asin 6.3%

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

   2. sub-neg 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

4. Applied egg-rr 6.3%

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

   1. sub-neg 6.3%

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

6. Simplified 6.3%

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

   1. add-sqr-sqrt 9.7%

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

   2. pow2 9.7%

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

8. Applied egg-rr 9.7%

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

   1. unpow2 9.7%

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

   2. add-sqr-sqrt 6.3%

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

   3. asin-acos 6.3%

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

   4. div-inv 6.3%

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

   5. metadata-eval 6.3%

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

   6. add-sqr-sqrt 9.7%

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

   7. acos-asin 9.7%

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

   8. div-inv 9.7%

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

   9. metadata-eval 9.7%

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

   10. add-cube-cbrt 9.6%

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

   11. add-cube-cbrt 9.7%

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

   12. add-sqr-sqrt 9.6%

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

   13. unpow2 9.6%

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

   14. fma-neg 9.6%

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

10. Applied egg-rr 9.7%

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

Alternative 3: 10.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

   1. acos-asin 6.3%

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

   2. sub-neg 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

4. Applied egg-rr 6.3%

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

   1. sub-neg 6.3%

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

6. Simplified 6.3%

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

   1. add-sqr-sqrt 9.7%

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

   2. pow2 9.7%

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

8. Applied egg-rr 9.7%

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

   1. unpow2 9.7%

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

   2. add-sqr-sqrt 6.3%

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

   3. asin-acos 6.3%

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

   4. div-inv 6.3%

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

   5. metadata-eval 6.3%

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

   6. *-commutative 6.3%

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

   7. add-sqr-sqrt 9.7%

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

   8. associate-*r* 9.7%

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

   9. acos-asin 9.7%

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

   10. div-inv 9.7%

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

   11. metadata-eval 9.7%

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

   12. add-cube-cbrt 9.6%

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

   13. add-cube-cbrt 9.7%

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

   14. add-sqr-sqrt 9.6%

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

   15. unpow2 9.6%

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

   16. fma-neg 9.6%

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

10. Applied egg-rr 9.7%

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

Alternative 4: 9.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi - t\_0\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= x 5.5e-17) (- PI t_0) (log (exp t_0)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = ((double) M_PI) - t_0;
	} else {
		tmp = log(exp(t_0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = Math.PI - t_0;
	} else {
		tmp = Math.log(Math.exp(t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.pi - t_0
	else:
		tmp = math.log(math.exp(t_0))
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(pi - t_0);
	else
		tmp = log(exp(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = pi - t_0;
	else
		tmp = log(exp(t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(Pi - t$95$0), $MachinePrecision], N[Log[N[Exp[t$95$0], $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. Step-by-step derivation

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   6. Simplified 3.9%

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

      1. add-sqr-sqrt 7.4%

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

      2. cancel-sign-sub-inv 7.4%

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

      3. distribute-lft-neg-in 7.4%

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

      4. add-sqr-sqrt 3.9%

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

      5. rem-3cbrt-rft 7.4%

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

      6. unpow2 7.4%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 6.5%

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

   8. Applied egg-rr 6.5%

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

      1. +-commutative 6.5%

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

      2. asin-acos 6.5%

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

      3. div-inv 6.5%

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

      4. metadata-eval 6.5%

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

      5. associate-+l- 6.5%

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

   10. Applied egg-rr 6.5%

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

       1. sub-neg 6.5%

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

       2. neg-sub0 6.5%

          \[\leadsto \pi \cdot 0.5 + \color{blue}{\left(0 - \left(\cos^{-1} \left(1 - x\right) - \pi \cdot 0.5\right)\right)} \]

       3. associate-+l- 6.5%

          \[\leadsto \pi \cdot 0.5 + \color{blue}{\left(\left(0 - \cos^{-1} \left(1 - x\right)\right) + \pi \cdot 0.5\right)} \]

       4. neg-sub0 6.5%

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

       5. +-commutative 6.5%

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

       6. sub-neg 6.5%

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

       7. associate--l+ 6.5%

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

       8. distribute-lft-out 6.5%

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

       9. metadata-eval 6.5%

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

       10. *-rgt-identity 6.5%

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

   12. Simplified 6.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 52.4%

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

      1. add-log-exp 52.4%

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

   4. Applied egg-rr 52.4%

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

Alternative 5: 10.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

   1. acos-asin 6.3%

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

   2. sub-neg 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

4. Applied egg-rr 6.3%

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

   1. sub-neg 6.3%

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

6. Simplified 6.3%

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

   1. add-sqr-sqrt 9.7%

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

   2. pow2 9.7%

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

8. Applied egg-rr 9.7%

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

Alternative 6: 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 6.3%

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

   1. acos-asin 6.3%

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

   2. sub-neg 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

4. Applied egg-rr 6.3%

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

   1. sub-neg 6.3%

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

6. Simplified 6.3%

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

   1. add-cube-cbrt 9.6%

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

   2. pow3 9.6%

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

8. Applied egg-rr 9.6%

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

Alternative 7: 9.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x)))) (if (<= x 5.5e-17) (- PI t_0) t_0)))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = ((double) M_PI) - t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = Math.PI - t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.pi - t_0
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(pi - t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = pi - t_0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(Pi - t$95$0), $MachinePrecision], t$95$0]]

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. Step-by-step derivation

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   6. Simplified 3.9%

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

      1. add-sqr-sqrt 7.4%

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

      2. cancel-sign-sub-inv 7.4%

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

      3. distribute-lft-neg-in 7.4%

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

      4. add-sqr-sqrt 3.9%

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

      5. rem-3cbrt-rft 7.4%

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

      6. unpow2 7.4%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 6.5%

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

   8. Applied egg-rr 6.5%

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

      1. +-commutative 6.5%

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

      2. asin-acos 6.5%

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

      3. div-inv 6.5%

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

      4. metadata-eval 6.5%

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

      5. associate-+l- 6.5%

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

   10. Applied egg-rr 6.5%

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

       1. sub-neg 6.5%

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

       2. neg-sub0 6.5%

          \[\leadsto \pi \cdot 0.5 + \color{blue}{\left(0 - \left(\cos^{-1} \left(1 - x\right) - \pi \cdot 0.5\right)\right)} \]

       3. associate-+l- 6.5%

          \[\leadsto \pi \cdot 0.5 + \color{blue}{\left(\left(0 - \cos^{-1} \left(1 - x\right)\right) + \pi \cdot 0.5\right)} \]

       4. neg-sub0 6.5%

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

       5. +-commutative 6.5%

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

       6. sub-neg 6.5%

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

       7. associate--l+ 6.5%

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

       8. distribute-lft-out 6.5%

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

       9. metadata-eval 6.5%

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

       10. *-rgt-identity 6.5%

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

   12. Simplified 6.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 52.4%

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

Alternative 8: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

Developer target: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

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

  (acos (- 1.0 x)))