bug323 (missed optimization)

Percentage Accurate: 6.7% → 10.2%

Time: 10.1s

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: 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: 10.2% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (+
    (acos (- 1.0 x))
    (-
     t_0
     (pow (pow (* t_0 (- (asin (/ (fma x x -1.0) (+ 1.0 x))))) 0.25) 2.0)))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return acos((1.0 - x)) + (t_0 - pow(pow((t_0 * -asin((fma(x, x, -1.0) / (1.0 + x)))), 0.25), 2.0));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(acos(Float64(1.0 - x)) + Float64(t_0 - ((Float64(t_0 * Float64(-asin(Float64(fma(x, x, -1.0) / Float64(1.0 + x))))) ^ 0.25) ^ 2.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[(t$95$0 - N[Power[N[Power[N[(t$95$0 * (-N[ArcSin[N[(N[(x * x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], 0.25], $MachinePrecision], 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 9.7%

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

   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. flip-- 9.7%

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

   2. div-inv 9.7%

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

   3. metadata-eval 9.7%

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

   4. pow2 9.7%

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

6. Applied egg-rr 9.7%

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

   1. associate-*r/ 9.7%

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

   2. *-rgt-identity 9.7%

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

   3. remove-double-neg 9.7%

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

   4. distribute-frac-neg 9.7%

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

   5. distribute-frac-neg2 9.7%

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

   6. sub-neg 9.7%

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

   7. +-commutative 9.7%

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

   8. distribute-neg-in 9.7%

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

   9. unpow2 9.7%

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

   10. sqr-neg 9.7%

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

   11. unpow2 9.7%

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

   12. remove-double-neg 9.7%

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

   13. sub-neg 9.7%

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

   14. unpow2 9.7%

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

   15. sqr-neg 9.7%

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

   16. fma-neg 9.7%

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

   17. metadata-eval 9.7%

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

   18. distribute-neg-in 9.7%

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

   19. metadata-eval 9.7%

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

   20. unsub-neg 9.7%

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

8. Simplified 9.7%

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

9. Taylor expanded in x around 0 5.9%

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

11. Simplified 5.9%

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

    1. add-sqr-sqrt 9.8%

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

    2. pow2 9.8%

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

    3. pow1/2 9.8%

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

    4. sqrt-pow1 9.8%

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

    5. *-commutative 9.8%

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

    6. asin-neg 9.8%

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

    7. +-commutative 9.8%

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

    8. metadata-eval 9.8%

       \[\leadsto \cos^{-1} \left(1 - x\right) + \left(\left(-{\left({\left(\sin^{-1} \left(1 - x\right) \cdot \left(-\sin^{-1} \left(\frac{\mathsf{fma}\left(x, x, -1\right)}{x + 1}\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}\right) + \sin^{-1} \left(1 - x\right)\right) \]

13. Applied egg-rr 9.8%

    \[\leadsto \cos^{-1} \left(1 - x\right) + \left(\left(-\color{blue}{{\left({\left(\sin^{-1} \left(1 - x\right) \cdot \left(-\sin^{-1} \left(\frac{\mathsf{fma}\left(x, x, -1\right)}{x + 1}\right)\right)\right)}^{0.25}\right)}^{2}}\right) + \sin^{-1} \left(1 - x\right)\right) \]

14. Final simplification 9.8%

    \[\leadsto \cos^{-1} \left(1 - x\right) + \left(\sin^{-1} \left(1 - x\right) - {\left({\left(\sin^{-1} \left(1 - x\right) \cdot \left(-\sin^{-1} \left(\frac{\mathsf{fma}\left(x, x, -1\right)}{1 + x}\right)\right)\right)}^{0.25}\right)}^{2}\right) \]
15. Add Preprocessing

Alternative 2: 6.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\pi \cdot 0.5 - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - {\left(\sqrt[3]{\sin^{-1} 1}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 1.0)
   (- (* PI 0.5) (expm1 (log1p (asin (- 1.0 x)))))
   (- (* PI 0.5) (pow (cbrt (asin 1.0)) 3.0))))

C

double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = (((double) M_PI) * 0.5) - expm1(log1p(asin((1.0 - x))));
	} else {
		tmp = (((double) M_PI) * 0.5) - pow(cbrt(asin(1.0)), 3.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = (Math.PI * 0.5) - Math.expm1(Math.log1p(Math.asin((1.0 - x))));
	} else {
		tmp = (Math.PI * 0.5) - Math.pow(Math.cbrt(Math.asin(1.0)), 3.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = Float64(Float64(pi * 0.5) - expm1(log1p(asin(Float64(1.0 - x)))));
	else
		tmp = Float64(Float64(pi * 0.5) - (cbrt(asin(1.0)) ^ 3.0));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 1

   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. sub-neg 5.9%

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

      3. div-inv 5.9%

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

      4. metadata-eval 5.9%

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

   4. Applied egg-rr 5.9%

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

      1. sub-neg 5.9%

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

   6. Simplified 5.9%

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

      1. expm1-log1p-u 5.9%

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

   8. Applied egg-rr 5.9%

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

   if 1 < (-.f64 #s(literal 1 binary64) x)

   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. sub-neg 5.9%

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

      3. div-inv 5.9%

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

      4. metadata-eval 5.9%

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

   4. Applied egg-rr 5.9%

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

      1. sub-neg 5.9%

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

   6. Simplified 5.9%

      \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]
   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. Taylor expanded in x around 0 7.8%

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

Alternative 3: 10.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi \cdot 0.5 - {\left(\sqrt[3]{\sin^{-1} 1}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \cos^{-1} \left(1 - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17)
   (- (* PI 0.5) (pow (cbrt (asin 1.0)) 3.0))
   (+ -1.0 (+ 1.0 (acos (- 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = (((double) M_PI) * 0.5) - pow(cbrt(asin(1.0)), 3.0);
	} else {
		tmp = -1.0 + (1.0 + acos((1.0 - x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = (Math.PI * 0.5) - Math.pow(Math.cbrt(Math.asin(1.0)), 3.0);
	} else {
		tmp = -1.0 + (1.0 + Math.acos((1.0 - x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(Float64(pi * 0.5) - (cbrt(asin(1.0)) ^ 3.0));
	else
		tmp = Float64(-1.0 + Float64(1.0 + acos(Float64(1.0 - x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Power[N[ArcSin[1.0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(1.0 + N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.8%

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

      1. acos-asin 3.8%

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

      2. sub-neg 3.8%

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

      3. div-inv 3.8%

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

      4. metadata-eval 3.8%

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

   4. Applied egg-rr 3.8%

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

      1. sub-neg 3.8%

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

   6. Simplified 3.8%

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

      1. add-cube-cbrt 7.7%

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

      2. pow3 7.7%

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

   8. Applied egg-rr 7.7%

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

   9. Taylor expanded in x around 0 7.7%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 56.9%

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

      1. expm1-log1p-u 56.9%

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

      2. expm1-undefine 56.9%

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

      3. log1p-undefine 56.9%

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

      4. rem-exp-log 56.9%

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

   4. Applied egg-rr 56.9%

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

4. Final simplification 9.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi \cdot 0.5 - {\left(\sqrt[3]{\sin^{-1} 1}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \cos^{-1} \left(1 - x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 10.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := N[(N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.5), $MachinePrecision] - N[ArcSin[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. sub-neg 5.9%

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

   3. div-inv 5.9%

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

   4. metadata-eval 5.9%

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

4. Applied egg-rr 5.9%

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

   1. sub-neg 5.9%

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

6. Simplified 5.9%

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

   1. add-cbrt-cube 9.7%

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

   2. pow3 9.7%

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

8. Applied egg-rr 9.7%

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

Alternative 5: 10.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := N[(N[(Pi * N[Power[N[Power[0.5, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] - N[ArcSin[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. sub-neg 5.9%

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

   3. div-inv 5.9%

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

   4. metadata-eval 5.9%

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

4. Applied egg-rr 5.9%

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

   1. sub-neg 5.9%

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

6. Simplified 5.9%

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

   1. add-cube-cbrt 4.1%

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

   2. pow3 4.1%

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

8. Applied egg-rr 4.1%

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

9. Taylor expanded in x around 0 9.7%

   \[\leadsto \color{blue}{\pi \cdot {\left(\sqrt[3]{0.5}\right)}^{3} - \sin^{-1} \left(1 - x\right)} \]
10. 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 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. sub-neg 5.9%

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

   3. div-inv 5.9%

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

   4. metadata-eval 5.9%

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

4. Applied egg-rr 5.9%

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

   1. sub-neg 5.9%

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

6. Simplified 5.9%

   \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]
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: 6.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 1

   1. Initial program 5.9%

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

      1. expm1-log1p-u 5.9%

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

      2. expm1-undefine 5.9%

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

      3. log1p-undefine 5.9%

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

      4. rem-exp-log 5.9%

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

   4. Applied egg-rr 5.9%

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

   if 1 < (-.f64 #s(literal 1 binary64) x)

   1. Initial program 5.9%

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

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

4. Final simplification 5.9%

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

Alternative 8: 6.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 1

   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. sub-neg 5.9%

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

      3. div-inv 5.9%

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

      4. metadata-eval 5.9%

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

   4. Applied egg-rr 5.9%

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

      1. sub-neg 5.9%

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

   6. Simplified 5.9%

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

   if 1 < (-.f64 #s(literal 1 binary64) x)

   1. Initial program 5.9%

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

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

Alternative 9: 6.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 1

   1. Initial program 5.9%

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

   if 1 < (-.f64 #s(literal 1 binary64) x)

   1. Initial program 5.9%

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

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

Alternative 10: 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 5.9%

   \[\cos^{-1} \left(1 - x\right) \]
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 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: 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 2024110 
(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)))