bug323 (missed optimization)

Percentage Accurate: 7.0% → 10.4%

Time: 6.9s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. acos-asin 6.5%

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

   2. *-un-lft-identity 6.5%

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

   3. add-sqr-sqrt 10.0%

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

   4. prod-diff 10.0%

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

   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. flip-- 10.1%

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

   2. div-inv 10.1%

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

   3. metadata-eval 10.1%

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

   4. pow2 10.1%

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

6. Applied egg-rr 10.1%

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

   1. associate-*r/ 10.1%

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

   2. *-rgt-identity 10.1%

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

   3. remove-double-neg 10.1%

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

   4. distribute-frac-neg 10.1%

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

   5. distribute-frac-neg2 10.1%

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

   6. sub-neg 10.1%

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

   7. +-commutative 10.1%

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

   8. distribute-neg-in 10.1%

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

   9. unpow2 10.1%

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

   10. sqr-neg 10.1%

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

   11. unpow2 10.1%

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

   12. remove-double-neg 10.1%

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

   13. sub-neg 10.1%

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

   14. unpow2 10.1%

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

   15. sqr-neg 10.1%

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

   16. fmm-def 10.1%

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

   17. metadata-eval 10.1%

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

   18. distribute-neg-in 10.1%

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

   19. metadata-eval 10.1%

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

   20. unsub-neg 10.1%

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

8. Simplified 10.1%

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

   1. asin-acos 10.1%

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

   2. div-inv 10.1%

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

   3. metadata-eval 10.1%

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

10. Applied egg-rr 10.1%

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

Alternative 2: 10.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(acos(Float64(1.0 - x)) + fma(Float64(-sqrt(asin(Float64(fma(x, x, -1.0) / Float64(-1.0 - x))))), sqrt(t_0), t_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[((-N[Sqrt[N[ArcSin[N[(N[(x * x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]) * N[Sqrt[t$95$0], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.5%

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

   1. acos-asin 6.5%

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

   2. *-un-lft-identity 6.5%

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

   3. add-sqr-sqrt 10.0%

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

   4. prod-diff 10.0%

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

   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. flip-- 10.1%

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

   2. div-inv 10.1%

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

   3. metadata-eval 10.1%

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

   4. pow2 10.1%

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

6. Applied egg-rr 10.1%

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

   1. associate-*r/ 10.1%

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

   2. *-rgt-identity 10.1%

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

   3. remove-double-neg 10.1%

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

   4. distribute-frac-neg 10.1%

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

   5. distribute-frac-neg2 10.1%

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

   6. sub-neg 10.1%

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

   7. +-commutative 10.1%

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

   8. distribute-neg-in 10.1%

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

   9. unpow2 10.1%

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

   10. sqr-neg 10.1%

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

   11. unpow2 10.1%

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

   12. remove-double-neg 10.1%

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

   13. sub-neg 10.1%

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

   14. unpow2 10.1%

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

   15. sqr-neg 10.1%

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

   16. fmm-def 10.1%

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

   17. metadata-eval 10.1%

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

   18. distribute-neg-in 10.1%

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

   19. metadata-eval 10.1%

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

   20. unsub-neg 10.1%

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

8. Simplified 10.1%

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

Alternative 3: 10.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = sqrt(t_0)
	return Float64(acos(Float64(1.0 - x)) + fma(Float64(-t_1), t_1, t_0))
end

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. acos-asin 6.5%

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

   2. *-un-lft-identity 6.5%

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

   3. add-sqr-sqrt 10.0%

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

   4. prod-diff 10.0%

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

   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. Add Preprocessing

Alternative 4: 9.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.6e-17) (acos x) (pow (cbrt (acos (- 1.0 x))) 3.0)))

C

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

Java

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 5.6e-17], N[ArcCos[x], $MachinePrecision], N[Power[N[Power[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.8%

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

      1. add-cube-cbrt 3.8%

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

      2. pow3 3.8%

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

   4. Applied egg-rr 3.8%

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

   5. Taylor expanded in x around inf 6.6%

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

      1. neg-mul-1 6.6%

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

   7. Simplified 6.6%

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

      1. rem-cube-cbrt 6.6%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 6.6%

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

      4. sqr-neg 6.6%

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

      5. sqrt-unprod 6.6%

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

      6. add-sqr-sqrt 6.6%

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

   9. Applied egg-rr 6.6%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 56.5%

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

      1. add-cube-cbrt 56.6%

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

      2. pow3 56.6%

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

   4. Applied egg-rr 56.6%

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

Alternative 5: 9.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.8%

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

      1. add-cube-cbrt 3.8%

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

      2. pow3 3.8%

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

   4. Applied egg-rr 3.8%

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

   5. Taylor expanded in x around inf 6.6%

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

      1. neg-mul-1 6.6%

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

   7. Simplified 6.6%

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

      1. rem-cube-cbrt 6.6%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 6.6%

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

      4. sqr-neg 6.6%

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

      5. sqrt-unprod 6.6%

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

      6. add-sqr-sqrt 6.6%

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

   9. Applied egg-rr 6.6%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 56.5%

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

      1. add-log-exp 56.5%

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

   4. Applied egg-rr 56.5%

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

Alternative 6: 10.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. acos-asin 6.5%

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

   2. add-cube-cbrt 4.7%

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

   3. *-un-lft-identity 4.7%

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

   4. prod-diff 4.7%

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

   5. cbrt-unprod 4.7%

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

   6. pow2 4.7%

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

   7. div-inv 4.7%

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

   8. metadata-eval 4.7%

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

   9. div-inv 4.7%

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

   10. metadata-eval 4.7%

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

4. Applied egg-rr 4.7%

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

   1. fma-undefine 4.7%

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

   2. *-rgt-identity 4.7%

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

   3. *-rgt-identity 4.7%

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

   4. +-commutative 4.7%

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

   5. sub-neg 4.7%

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

   6. +-inverses 4.7%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{2}}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right) \cdot 1\right) + \color{blue}{0} \]

   7. +-rgt-identity 4.7%

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

   8. fmm-undef 4.7%

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

   9. *-rgt-identity 4.7%

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

6. Simplified 4.7%

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

   1. cbrt-unprod 10.0%

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

   2. unpow2 10.0%

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

   3. pow3 10.0%

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

8. Applied egg-rr 10.0%

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

Alternative 7: 9.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} x\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(-1 + \left(\cos^{-1} \left(1 - x\right) + 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.6e-17) (acos x) (+ -1.0 (+ -1.0 (+ (acos (- 1.0 x)) 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = acos(x);
	} else {
		tmp = -1.0 + (-1.0 + (acos((1.0 - x)) + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.6d-17) then
        tmp = acos(x)
    else
        tmp = (-1.0d0) + ((-1.0d0) + (acos((1.0d0 - x)) + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = Math.acos(x);
	} else {
		tmp = -1.0 + (-1.0 + (Math.acos((1.0 - x)) + 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.6e-17:
		tmp = math.acos(x)
	else:
		tmp = -1.0 + (-1.0 + (math.acos((1.0 - x)) + 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = acos(x);
	else
		tmp = Float64(-1.0 + Float64(-1.0 + Float64(acos(Float64(1.0 - x)) + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.6e-17)
		tmp = acos(x);
	else
		tmp = -1.0 + (-1.0 + (acos((1.0 - x)) + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.6e-17], N[ArcCos[x], $MachinePrecision], N[(-1.0 + N[(-1.0 + N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.8%

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

      1. add-cube-cbrt 3.8%

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

      2. pow3 3.8%

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

   4. Applied egg-rr 3.8%

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

   5. Taylor expanded in x around inf 6.6%

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

      1. neg-mul-1 6.6%

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

   7. Simplified 6.6%

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

      1. rem-cube-cbrt 6.6%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 6.6%

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

      4. sqr-neg 6.6%

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

      5. sqrt-unprod 6.6%

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

      6. add-sqr-sqrt 6.6%

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

   9. Applied egg-rr 6.6%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 56.5%

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

      1. expm1-log1p-u 56.5%

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

      2. expm1-undefine 56.5%

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

      3. log1p-undefine 56.5%

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

      4. rem-exp-log 56.5%

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

   4. Applied egg-rr 56.5%

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

      1. expm1-log1p-u 56.5%

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

      2. expm1-undefine 56.5%

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

      3. log1p-undefine 56.5%

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

      4. +-commutative 56.5%

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

      5. add-exp-log 56.5%

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

      6. +-commutative 56.5%

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

      7. associate-+l+ 56.5%

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

      8. metadata-eval 56.5%

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

   6. Applied egg-rr 56.5%

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

4. Final simplification 9.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} x\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(-1 + \left(\cos^{-1} \left(1 - x\right) + 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 9.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.8%

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

      1. add-cube-cbrt 3.8%

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

      2. pow3 3.8%

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

   4. Applied egg-rr 3.8%

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

   5. Taylor expanded in x around inf 6.6%

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

      1. neg-mul-1 6.6%

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

   7. Simplified 6.6%

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

      1. rem-cube-cbrt 6.6%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 6.6%

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

      4. sqr-neg 6.6%

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

      5. sqrt-unprod 6.6%

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

      6. add-sqr-sqrt 6.6%

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

   9. Applied egg-rr 6.6%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 56.5%

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

      1. acos-asin 56.5%

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

      2. sub-neg 56.5%

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

      3. div-inv 56.5%

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

      4. metadata-eval 56.5%

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

   4. Applied egg-rr 56.5%

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

      1. sub-neg 56.5%

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

   6. Simplified 56.5%

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

Alternative 9: 9.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.8%

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

      1. add-cube-cbrt 3.8%

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

      2. pow3 3.8%

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

   4. Applied egg-rr 3.8%

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

   5. Taylor expanded in x around inf 6.6%

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

      1. neg-mul-1 6.6%

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

   7. Simplified 6.6%

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

      1. rem-cube-cbrt 6.6%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 6.6%

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

      4. sqr-neg 6.6%

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

      5. sqrt-unprod 6.6%

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

      6. add-sqr-sqrt 6.6%

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

   9. Applied egg-rr 6.6%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 56.5%

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

      1. expm1-log1p-u 56.5%

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

      2. expm1-undefine 56.5%

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

      3. log1p-undefine 56.5%

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

      4. rem-exp-log 56.5%

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

   4. Applied egg-rr 56.5%

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

      1. associate--l+ 56.5%

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

      2. +-commutative 56.5%

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

      3. sub-neg 56.5%

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

      4. metadata-eval 56.5%

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

   6. Applied egg-rr 56.5%

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

4. Final simplification 9.1%

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

Alternative 10: 9.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.8%

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

      1. add-cube-cbrt 3.8%

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

      2. pow3 3.8%

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

   4. Applied egg-rr 3.8%

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

   5. Taylor expanded in x around inf 6.6%

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

      1. neg-mul-1 6.6%

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

   7. Simplified 6.6%

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

      1. rem-cube-cbrt 6.6%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 6.6%

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

      4. sqr-neg 6.6%

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

      5. sqrt-unprod 6.6%

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

      6. add-sqr-sqrt 6.6%

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

   9. Applied egg-rr 6.6%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 56.5%

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

Alternative 11: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return acos(x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. add-cube-cbrt 6.5%

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

   2. pow3 6.5%

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

4. Applied egg-rr 6.5%

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

5. Taylor expanded in x around inf 6.9%

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

   1. neg-mul-1 6.9%

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

7. Simplified 6.9%

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

   1. rem-cube-cbrt 6.9%

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

   2. add-sqr-sqrt 0.0%

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

   3. sqrt-unprod 6.9%

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

   4. sqr-neg 6.9%

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

   5. sqrt-unprod 6.9%

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

   6. add-sqr-sqrt 6.9%

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

9. Applied egg-rr 6.9%

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

Alternative 12: 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 6.5%

   \[\cos^{-1} \left(1 - x\right) \]
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 2024185 
(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)))