bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.5%

Time: 12.5s

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: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot {\pi}^{2}\\ t_1 := \sin^{-1} \left(1 - x\right)\\ t_2 := \sqrt[3]{\mathsf{fma}\left(0.5 \cdot t_0, \pi, -{t_1}^{3}\right)}\\ \frac{t_2 \cdot \left(t_2 \cdot t_2\right)}{t_0 + t_1 \cdot \left(t_1 + 0.5 \cdot \pi\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.25 (pow PI 2.0)))
        (t_1 (asin (- 1.0 x)))
        (t_2 (cbrt (fma (* 0.5 t_0) PI (- (pow t_1 3.0))))))
   (/ (* t_2 (* t_2 t_2)) (+ t_0 (* t_1 (+ t_1 (* 0.5 PI)))))))

C

double code(double x) {
	double t_0 = 0.25 * pow(((double) M_PI), 2.0);
	double t_1 = asin((1.0 - x));
	double t_2 = cbrt(fma((0.5 * t_0), ((double) M_PI), -pow(t_1, 3.0)));
	return (t_2 * (t_2 * t_2)) / (t_0 + (t_1 * (t_1 + (0.5 * ((double) M_PI)))));
}

Julia

function code(x)
	t_0 = Float64(0.25 * (pi ^ 2.0))
	t_1 = asin(Float64(1.0 - x))
	t_2 = cbrt(fma(Float64(0.5 * t_0), pi, Float64(-(t_1 ^ 3.0))))
	return Float64(Float64(t_2 * Float64(t_2 * t_2)) / Float64(t_0 + Float64(t_1 * Float64(t_1 + Float64(0.5 * pi)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[(0.5 * t$95$0), $MachinePrecision] * Pi + (-N[Power[t$95$1, 3.0], $MachinePrecision])), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(t$95$2 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(t$95$1 * N[(t$95$1 + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 7.1%

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

   1. acos-asin 7.1%

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

   2. flip3-- 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

   5. div-inv 7.1%

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

   6. metadata-eval 7.1%

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

   7. div-inv 7.1%

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

   8. metadata-eval 7.1%

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

   9. div-inv 7.1%

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

   10. metadata-eval 7.1%

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

3. Applied egg-rr 7.1%

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

   1. swap-sqr 7.1%

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

   2. metadata-eval 7.1%

      \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot \color{blue}{0.25} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   3. distribute-rgt-out 7.1%

      \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \color{blue}{\sin^{-1} \left(1 - x\right) \cdot \left(\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\right)}} \]

   4. +-commutative 7.1%

      \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\left(\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\right)}} \]

   5. fma-def 7.1%

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

5. Simplified 7.1%

   \[\leadsto \color{blue}{\frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]
6. Step-by-step derivation

   1. unpow3 7.1%

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

   2. *-commutative 7.1%

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

   3. associate-*r* 7.1%

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

   4. fma-neg 10.5%

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

   5. *-commutative 10.5%

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

   6. *-commutative 10.5%

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

   7. swap-sqr 10.5%

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

   8. metadata-eval 10.5%

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

   9. pow2 10.5%

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

7. Applied egg-rr 10.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\pi}^{2}\right) \cdot 0.5, \pi, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
8. Step-by-step derivation

   1. add-cube-cbrt 10.5%

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

   2. *-commutative 10.5%

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

   3. *-commutative 10.5%

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

   4. *-commutative 10.5%

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

9. Applied egg-rr 10.5%

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

10. Taylor expanded in x around 0 10.5%

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

11. Final simplification 10.5%

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

Alternative 2: 10.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := {t_0}^{3}\\ \frac{\mathsf{fma}\left(0.5 \cdot \left(0.25 \cdot {\pi}^{2}\right), \pi, -t_1\right)}{0.25 \cdot \left(\pi \cdot \pi\right) + \sqrt[3]{t_1} \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (pow t_0 3.0)))
   (/
    (fma (* 0.5 (* 0.25 (pow PI 2.0))) PI (- t_1))
    (+ (* 0.25 (* PI PI)) (* (cbrt t_1) (fma PI 0.5 t_0))))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = pow(t_0, 3.0);
	return fma((0.5 * (0.25 * pow(((double) M_PI), 2.0))), ((double) M_PI), -t_1) / ((0.25 * (((double) M_PI) * ((double) M_PI))) + (cbrt(t_1) * fma(((double) M_PI), 0.5, t_0)));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = t_0 ^ 3.0
	return Float64(fma(Float64(0.5 * Float64(0.25 * (pi ^ 2.0))), pi, Float64(-t_1)) / Float64(Float64(0.25 * Float64(pi * pi)) + Float64(cbrt(t_1) * fma(pi, 0.5, t_0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 3.0], $MachinePrecision]}, N[(N[(N[(0.5 * N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi + (-t$95$1)), $MachinePrecision] / N[(N[(0.25 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + N[(N[Power[t$95$1, 1/3], $MachinePrecision] * N[(Pi * 0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 7.1%

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

   1. acos-asin 7.1%

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

   2. flip3-- 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

   5. div-inv 7.1%

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

   6. metadata-eval 7.1%

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

   7. div-inv 7.1%

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

   8. metadata-eval 7.1%

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

   9. div-inv 7.1%

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

   10. metadata-eval 7.1%

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

3. Applied egg-rr 7.1%

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

   1. swap-sqr 7.1%

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

   2. metadata-eval 7.1%

      \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot \color{blue}{0.25} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   3. distribute-rgt-out 7.1%

      \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \color{blue}{\sin^{-1} \left(1 - x\right) \cdot \left(\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\right)}} \]

   4. +-commutative 7.1%

      \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\left(\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\right)}} \]

   5. fma-def 7.1%

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

5. Simplified 7.1%

   \[\leadsto \color{blue}{\frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]
6. Step-by-step derivation

   1. unpow3 7.1%

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

   2. *-commutative 7.1%

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

   3. associate-*r* 7.1%

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

   4. fma-neg 10.5%

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

   5. *-commutative 10.5%

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

   6. *-commutative 10.5%

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

   7. swap-sqr 10.5%

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

   8. metadata-eval 10.5%

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

   9. pow2 10.5%

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

7. Applied egg-rr 10.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\pi}^{2}\right) \cdot 0.5, \pi, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
8. Step-by-step derivation

   1. rem-cbrt-cube 10.5%

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

9. Applied egg-rr 10.5%

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

10. Final simplification 10.5%

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

Alternative 3: 10.5% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/
    (exp (log (fma (* 0.5 (* 0.25 (pow PI 2.0))) PI (- (pow t_0 3.0)))))
    (+ (* 0.25 (* PI PI)) (* t_0 (fma PI 0.5 t_0))))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return exp(log(fma((0.5 * (0.25 * pow(((double) M_PI), 2.0))), ((double) M_PI), -pow(t_0, 3.0)))) / ((0.25 * (((double) M_PI) * ((double) M_PI))) + (t_0 * fma(((double) M_PI), 0.5, t_0)));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(exp(log(fma(Float64(0.5 * Float64(0.25 * (pi ^ 2.0))), pi, Float64(-(t_0 ^ 3.0))))) / Float64(Float64(0.25 * Float64(pi * pi)) + Float64(t_0 * fma(pi, 0.5, t_0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[Exp[N[Log[N[(N[(0.5 * N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi + (-N[Power[t$95$0, 3.0], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(N[(0.25 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(Pi * 0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.1%

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

   1. acos-asin 7.1%

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

   2. flip3-- 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

   5. div-inv 7.1%

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

   6. metadata-eval 7.1%

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

   7. div-inv 7.1%

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

   8. metadata-eval 7.1%

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

   9. div-inv 7.1%

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

   10. metadata-eval 7.1%

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

3. Applied egg-rr 7.1%

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

   1. swap-sqr 7.1%

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

   2. metadata-eval 7.1%

      \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot \color{blue}{0.25} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   3. distribute-rgt-out 7.1%

      \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \color{blue}{\sin^{-1} \left(1 - x\right) \cdot \left(\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\right)}} \]

   4. +-commutative 7.1%

      \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\left(\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\right)}} \]

   5. fma-def 7.1%

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

5. Simplified 7.1%

   \[\leadsto \color{blue}{\frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]
6. Step-by-step derivation

   1. unpow3 7.1%

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

   2. *-commutative 7.1%

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

   3. associate-*r* 7.1%

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

   4. fma-neg 10.5%

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

   5. *-commutative 10.5%

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

   6. *-commutative 10.5%

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

   7. swap-sqr 10.5%

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

   8. metadata-eval 10.5%

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

   9. pow2 10.5%

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

7. Applied egg-rr 10.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(0.25 \cdot {\pi}^{2}\right) \cdot 0.5, \pi, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
8. Step-by-step derivation

   1. add-exp-log 10.5%

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

   2. *-commutative 10.5%

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

9. Applied egg-rr 10.5%

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

10. Final simplification 10.5%

    \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(0.5 \cdot \left(0.25 \cdot {\pi}^{2}\right), \pi, -{\sin^{-1} \left(1 - x\right)}^{3}\right)\right)}}{0.25 \cdot \left(\pi \cdot \pi\right) + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

Alternative 4: 10.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot {\pi}^{2}\\ t_1 := \sin^{-1} \left(1 - x\right)\\ \frac{\mathsf{fma}\left(0.5 \cdot t_0, \pi, -{t_1}^{3}\right)}{t_0 + t_1 \cdot \left(t_1 + 0.5 \cdot \pi\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.25 (pow PI 2.0))) (t_1 (asin (- 1.0 x))))
   (/
    (fma (* 0.5 t_0) PI (- (pow t_1 3.0)))
    (+ t_0 (* t_1 (+ t_1 (* 0.5 PI)))))))

C

double code(double x) {
	double t_0 = 0.25 * pow(((double) M_PI), 2.0);
	double t_1 = asin((1.0 - x));
	return fma((0.5 * t_0), ((double) M_PI), -pow(t_1, 3.0)) / (t_0 + (t_1 * (t_1 + (0.5 * ((double) M_PI)))));
}

Julia

function code(x)
	t_0 = Float64(0.25 * (pi ^ 2.0))
	t_1 = asin(Float64(1.0 - x))
	return Float64(fma(Float64(0.5 * t_0), pi, Float64(-(t_1 ^ 3.0))) / Float64(t_0 + Float64(t_1 * Float64(t_1 + Float64(0.5 * pi)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(0.5 * t$95$0), $MachinePrecision] * Pi + (-N[Power[t$95$1, 3.0], $MachinePrecision])), $MachinePrecision] / N[(t$95$0 + N[(t$95$1 * N[(t$95$1 + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 7.1%

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

   1. acos-asin 7.1%

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

   2. flip3-- 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

   5. div-inv 7.1%

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

   6. metadata-eval 7.1%

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

   7. div-inv 7.1%

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

   8. metadata-eval 7.1%

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

   9. div-inv 7.1%

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

   10. metadata-eval 7.1%

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

3. Applied egg-rr 7.1%

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

   1. swap-sqr 7.1%

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

   2. metadata-eval 7.1%

      \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot \color{blue}{0.25} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

   3. distribute-rgt-out 7.1%

      \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \color{blue}{\sin^{-1} \left(1 - x\right) \cdot \left(\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\right)}} \]

   4. +-commutative 7.1%

      \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\left(\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\right)}} \]

   5. fma-def 7.1%

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

5. Simplified 7.1%

   \[\leadsto \color{blue}{\frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]
6. Step-by-step derivation

   1. unpow3 7.1%

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

   2. *-commutative 7.1%

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

   3. associate-*r* 7.1%

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

   4. fma-neg 10.5%

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

   5. *-commutative 10.5%

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

   6. *-commutative 10.5%

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

   7. swap-sqr 10.5%

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

   8. metadata-eval 10.5%

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

   9. pow2 10.5%

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

7. Applied egg-rr 10.5%

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

8. Taylor expanded in x around 0 10.5%

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

9. Final simplification 10.5%

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

Alternative 5: 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 7.1%

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

   1. add-sqr-sqrt 7.1%

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

   2. pow2 7.1%

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

3. Applied egg-rr 7.1%

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

   1. unpow2 7.1%

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

   2. add-sqr-sqrt 7.1%

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

   3. acos-asin 7.1%

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

   4. div-inv 7.1%

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

   5. metadata-eval 7.1%

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

   6. add-sqr-sqrt 10.4%

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

   7. prod-diff 10.4%

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

   8. add-sqr-sqrt 10.4%

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

   9. fma-neg 10.4%

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

   10. metadata-eval 10.4%

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

   11. div-inv 10.4%

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

   12. acos-asin 10.5%

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

   13. add-sqr-sqrt 10.4%

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

5. Applied egg-rr 10.4%

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

6. Final simplification 10.4%

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

Alternative 6: 10.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. acos-asin 7.1%

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

   2. sub-neg 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

3. Applied egg-rr 7.1%

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

   1. sub-neg 7.1%

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

5. Simplified 7.1%

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

   1. asin-acos 7.1%

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

   2. div-inv 7.1%

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

   3. metadata-eval 7.1%

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

   4. *-commutative 7.1%

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

   5. add-sqr-sqrt 10.4%

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

   6. associate-*r* 10.4%

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

   7. fma-neg 10.4%

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

7. Applied egg-rr 10.4%

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

8. Final simplification 10.4%

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

Alternative 7: 10.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. acos-asin 7.1%

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

   2. sub-neg 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

3. Applied egg-rr 7.1%

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

   1. sub-neg 7.1%

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

5. Simplified 7.1%

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

   1. add-cube-cbrt 10.3%

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

   2. pow3 10.3%

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

7. Applied egg-rr 10.3%

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

8. Final simplification 10.3%

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

Alternative 8: 10.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. acos-asin 7.1%

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

   2. sub-neg 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

3. Applied egg-rr 7.1%

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

   1. sub-neg 7.1%

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

5. Simplified 7.1%

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

   1. add-sqr-sqrt 10.4%

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

   2. pow2 10.4%

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

7. Applied egg-rr 10.4%

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

8. Final simplification 10.4%

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

Alternative 9: 6.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \log \left(1 + \mathsf{expm1}\left(\cos^{-1} \left(1 - x\right)\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (+ 1.0 (expm1 (acos (- 1.0 x))))))

C

double code(double x) {
	return log((1.0 + expm1(acos((1.0 - x)))));
}

Java

public static double code(double x) {
	return Math.log((1.0 + Math.expm1(Math.acos((1.0 - x)))));
}

Python

def code(x):
	return math.log((1.0 + math.expm1(math.acos((1.0 - x)))))

Julia

function code(x)
	return log(Float64(1.0 + expm1(acos(Float64(1.0 - x)))))
end

Wolfram

code[x_] := N[Log[N[(1.0 + N[(Exp[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 7.1%

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

   1. add-sqr-sqrt 7.1%

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

   2. pow2 7.1%

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

3. Applied egg-rr 7.1%

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

   1. unpow2 7.1%

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

   2. add-sqr-sqrt 7.1%

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

   3. log1p-expm1-u 7.1%

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

   4. log1p-udef 7.1%

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

5. Applied egg-rr 7.1%

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

6. Final simplification 7.1%

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

Alternative 10: 6.9% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (pow (sqrt (acos (- 1.0 x))) 2.0))

C

double code(double x) {
	return pow(sqrt(acos((1.0 - x))), 2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(acos((1.0d0 - x))) ** 2.0d0
end function

Java

public static double code(double x) {
	return Math.pow(Math.sqrt(Math.acos((1.0 - x))), 2.0);
}

Python

def code(x):
	return math.pow(math.sqrt(math.acos((1.0 - x))), 2.0)

Julia

function code(x)
	return sqrt(acos(Float64(1.0 - x))) ^ 2.0
end

MATLAB

function tmp = code(x)
	tmp = sqrt(acos((1.0 - x))) ^ 2.0;
end

Wolfram

code[x_] := N[Power[N[Sqrt[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 7.1%

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

   1. add-sqr-sqrt 7.1%

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

   2. pow2 7.1%

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

3. Applied egg-rr 7.1%

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

4. Final simplification 7.1%

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

Alternative 11: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Final simplification 7.1%

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

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 2023257 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :herbie-target
  (* 2.0 (asin (sqrt (/ x 2.0))))

  (acos (- 1.0 x)))