bug323 (missed optimization)

Percentage Accurate: 6.8% → 10.3%

Time: 9.0s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 0.5}\\ t_1 := \sin^{-1} \left(1 - x\right)\\ t_2 := \sqrt[3]{\sqrt[3]{\sin^{-1} \left(\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\right) \cdot {t\_1}^{2}}}\\ t_3 := {t\_2}^{2}\\ t_4 := t\_2 \cdot t\_3\\ t_5 := \sqrt[3]{t\_1}\\ t_6 := {t\_5}^{2}\\ \left(\mathsf{fma}\left(t\_0, t\_0, -t\_4\right) + \mathsf{fma}\left(-t\_2, t\_3, t\_4\right)\right) + \mathsf{fma}\left(-t\_5, t\_6, t\_5 \cdot t\_6\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 0.5)))
        (t_1 (asin (- 1.0 x)))
        (t_2
         (cbrt (cbrt (* (asin (/ (fma x x -1.0) (- -1.0 x))) (pow t_1 2.0)))))
        (t_3 (pow t_2 2.0))
        (t_4 (* t_2 t_3))
        (t_5 (cbrt t_1))
        (t_6 (pow t_5 2.0)))
   (+
    (+ (fma t_0 t_0 (- t_4)) (fma (- t_2) t_3 t_4))
    (fma (- t_5) t_6 (* t_5 t_6)))))

C

double code(double x) {
	double t_0 = sqrt((((double) M_PI) * 0.5));
	double t_1 = asin((1.0 - x));
	double t_2 = cbrt(cbrt((asin((fma(x, x, -1.0) / (-1.0 - x))) * pow(t_1, 2.0))));
	double t_3 = pow(t_2, 2.0);
	double t_4 = t_2 * t_3;
	double t_5 = cbrt(t_1);
	double t_6 = pow(t_5, 2.0);
	return (fma(t_0, t_0, -t_4) + fma(-t_2, t_3, t_4)) + fma(-t_5, t_6, (t_5 * t_6));
}

Julia

function code(x)
	t_0 = sqrt(Float64(pi * 0.5))
	t_1 = asin(Float64(1.0 - x))
	t_2 = cbrt(cbrt(Float64(asin(Float64(fma(x, x, -1.0) / Float64(-1.0 - x))) * (t_1 ^ 2.0))))
	t_3 = t_2 ^ 2.0
	t_4 = Float64(t_2 * t_3)
	t_5 = cbrt(t_1)
	t_6 = t_5 ^ 2.0
	return Float64(Float64(fma(t_0, t_0, Float64(-t_4)) + fma(Float64(-t_2), t_3, t_4)) + fma(Float64(-t_5), t_6, Float64(t_5 * t_6)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Power[N[(N[ArcSin[N[(N[(x * x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Power[t$95$1, 1/3], $MachinePrecision]}, Block[{t$95$6 = N[Power[t$95$5, 2.0], $MachinePrecision]}, N[(N[(N[(t$95$0 * t$95$0 + (-t$95$4)), $MachinePrecision] + N[((-t$95$2) * t$95$3 + t$95$4), $MachinePrecision]), $MachinePrecision] + N[((-t$95$5) * t$95$6 + N[(t$95$5 * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 4.9%

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

   1. acos-asin 4.9%

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

   2. add-sqr-sqrt 3.0%

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

   3. add-cube-cbrt 8.7%

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

   4. prod-diff 8.7%

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

4. Applied egg-rr 8.7%

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

   1. flip-- 8.8%

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

   2. div-inv 8.8%

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

   3. metadata-eval 8.8%

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

   4. pow2 8.8%

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

6. Applied egg-rr 8.8%

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

   1. associate-*r/ 8.8%

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

   2. *-rgt-identity 8.8%

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

   3. remove-double-neg 8.8%

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

   4. distribute-frac-neg 8.8%

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

   5. distribute-frac-neg2 8.8%

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

   6. sub-neg 8.8%

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

   7. +-commutative 8.8%

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

   8. distribute-neg-in 8.8%

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

   9. unpow2 8.8%

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

   10. sqr-neg 8.8%

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

   11. unpow2 8.8%

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

   12. remove-double-neg 8.8%

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

   13. sub-neg 8.8%

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

   14. unpow2 8.8%

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

   15. sqr-neg 8.8%

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

   16. fmm-def 8.8%

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

   17. metadata-eval 8.8%

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

   18. distribute-neg-in 8.8%

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

   19. metadata-eval 8.8%

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

   20. unsub-neg 8.8%

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

8. Simplified 8.8%

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

9. Taylor expanded in x around 0 3.1%

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

11. Simplified 3.1%

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

    1. fmm-undef 3.1%

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

    2. add-cube-cbrt 8.8%

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

    3. prod-diff 8.8%

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

13. Applied egg-rr 8.8%

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

Alternative 2: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sin^{-1} \left(1 - x\right)}\\ t_1 := {t\_0}^{2}\\ t_2 := \sqrt{\pi \cdot 0.5}\\ \mathsf{fma}\left(-t\_0, t\_1, t\_0 \cdot t\_1\right) + \mathsf{fma}\left(t\_2, t\_2, t\_1 \cdot \left(-\sqrt[3]{\sin^{-1} \left(\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\right)}\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.9%

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

   1. acos-asin 4.9%

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

   2. add-sqr-sqrt 3.0%

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

   3. add-cube-cbrt 8.7%

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

   4. prod-diff 8.7%

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

4. Applied egg-rr 8.7%

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

   1. flip-- 8.8%

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

   2. div-inv 8.8%

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

   3. metadata-eval 8.8%

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

   4. pow2 8.8%

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

6. Applied egg-rr 8.8%

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

   1. associate-*r/ 8.8%

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

   2. *-rgt-identity 8.8%

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

   3. remove-double-neg 8.8%

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

   4. distribute-frac-neg 8.8%

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

   5. distribute-frac-neg2 8.8%

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

   6. sub-neg 8.8%

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

   7. +-commutative 8.8%

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

   8. distribute-neg-in 8.8%

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

   9. unpow2 8.8%

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

   10. sqr-neg 8.8%

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

   11. unpow2 8.8%

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

   12. remove-double-neg 8.8%

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

   13. sub-neg 8.8%

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

   14. unpow2 8.8%

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

   15. sqr-neg 8.8%

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

   16. fmm-def 8.8%

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

   17. metadata-eval 8.8%

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

   18. distribute-neg-in 8.8%

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

   19. metadata-eval 8.8%

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

   20. unsub-neg 8.8%

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

8. Simplified 8.8%

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

9. Final simplification 8.8%

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

Alternative 3: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 0.5}\\ t_1 := \sqrt[3]{\sin^{-1} \left(1 - x\right)}\\ t_2 := {t\_1}^{2}\\ \mathsf{fma}\left(t\_0, t\_0, t\_1 \cdot \left(-t\_2\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\right)}, t\_2, t\_1 \cdot t\_2\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.9%

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

   1. acos-asin 4.9%

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

   2. add-sqr-sqrt 3.0%

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

   3. add-cube-cbrt 8.7%

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

   4. prod-diff 8.7%

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

4. Applied egg-rr 8.7%

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

   1. flip-- 8.8%

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

   2. div-inv 8.8%

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

   3. metadata-eval 8.8%

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

   4. pow2 8.8%

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

6. Applied egg-rr 8.8%

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

   1. associate-*r/ 8.8%

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

   2. *-rgt-identity 8.8%

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

   3. remove-double-neg 8.8%

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

   4. distribute-frac-neg 8.8%

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

   5. distribute-frac-neg2 8.8%

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

   6. sub-neg 8.8%

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

   7. +-commutative 8.8%

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

   8. distribute-neg-in 8.8%

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

   9. unpow2 8.8%

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

   10. sqr-neg 8.8%

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

   11. unpow2 8.8%

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

   12. remove-double-neg 8.8%

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

   13. sub-neg 8.8%

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

   14. unpow2 8.8%

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

   15. sqr-neg 8.8%

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

   16. fmm-def 8.8%

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

   17. metadata-eval 8.8%

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

   18. distribute-neg-in 8.8%

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

   19. metadata-eval 8.8%

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

   20. unsub-neg 8.8%

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

8. Simplified 8.8%

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

9. Final simplification 8.8%

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

Alternative 4: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt{\pi \cdot 0.5}\\ t_2 := \sqrt[3]{t\_0}\\ \mathsf{fma}\left(t\_1, t\_1, -\sqrt[3]{\sin^{-1} \left(\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\right) \cdot {t\_0}^{2}}\right) + \mathsf{fma}\left(-t\_2, {t\_2}^{2}, t\_0\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.9%

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

   1. acos-asin 4.9%

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

   2. add-sqr-sqrt 3.0%

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

   3. add-cube-cbrt 8.7%

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

   4. prod-diff 8.7%

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

4. Applied egg-rr 8.7%

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

   1. flip-- 8.8%

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

   2. div-inv 8.8%

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

   3. metadata-eval 8.8%

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

   4. pow2 8.8%

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

6. Applied egg-rr 8.8%

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

   1. associate-*r/ 8.8%

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

   2. *-rgt-identity 8.8%

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

   3. remove-double-neg 8.8%

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

   4. distribute-frac-neg 8.8%

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

   5. distribute-frac-neg2 8.8%

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

   6. sub-neg 8.8%

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

   7. +-commutative 8.8%

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

   8. distribute-neg-in 8.8%

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

   9. unpow2 8.8%

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

   10. sqr-neg 8.8%

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

   11. unpow2 8.8%

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

   12. remove-double-neg 8.8%

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

   13. sub-neg 8.8%

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

   14. unpow2 8.8%

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

   15. sqr-neg 8.8%

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

   16. fmm-def 8.8%

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

   17. metadata-eval 8.8%

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

   18. distribute-neg-in 8.8%

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

   19. metadata-eval 8.8%

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

   20. unsub-neg 8.8%

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

8. Simplified 8.8%

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

9. Taylor expanded in x around 0 3.1%

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

11. Simplified 3.1%

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

12. Taylor expanded in x around 0 8.8%

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

Alternative 5: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 0.5}\\ t_1 := \sin^{-1} \left(1 - x\right)\\ \mathsf{fma}\left(t\_0, t\_0, {\left(\sqrt[3]{t\_1}\right)}^{2} \cdot \left(-\sqrt[3]{\sin^{-1} \left(\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\right)}\right)\right) + t\_1 \cdot 0 \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 0.5))) (t_1 (asin (- 1.0 x))))
   (+
    (fma
     t_0
     t_0
     (* (pow (cbrt t_1) 2.0) (- (cbrt (asin (/ (fma x x -1.0) (- -1.0 x)))))))
    (* t_1 0.0))))

C

double code(double x) {
	double t_0 = sqrt((((double) M_PI) * 0.5));
	double t_1 = asin((1.0 - x));
	return fma(t_0, t_0, (pow(cbrt(t_1), 2.0) * -cbrt(asin((fma(x, x, -1.0) / (-1.0 - x)))))) + (t_1 * 0.0);
}

Julia

function code(x)
	t_0 = sqrt(Float64(pi * 0.5))
	t_1 = asin(Float64(1.0 - x))
	return Float64(fma(t_0, t_0, Float64((cbrt(t_1) ^ 2.0) * Float64(-cbrt(asin(Float64(fma(x, x, -1.0) / Float64(-1.0 - x))))))) + Float64(t_1 * 0.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 * t$95$0 + N[(N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * (-N[Power[N[ArcSin[N[(N[(x * x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * 0.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 4.9%

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

   1. acos-asin 4.9%

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

   2. add-sqr-sqrt 3.0%

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

   3. add-cube-cbrt 8.7%

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

   4. prod-diff 8.7%

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

4. Applied egg-rr 8.7%

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

   1. flip-- 8.8%

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

   2. div-inv 8.8%

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

   3. metadata-eval 8.8%

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

   4. pow2 8.8%

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

6. Applied egg-rr 8.8%

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

   1. associate-*r/ 8.8%

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

   2. *-rgt-identity 8.8%

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

   3. remove-double-neg 8.8%

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

   4. distribute-frac-neg 8.8%

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

   5. distribute-frac-neg2 8.8%

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

   6. sub-neg 8.8%

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

   7. +-commutative 8.8%

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

   8. distribute-neg-in 8.8%

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

   9. unpow2 8.8%

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

   10. sqr-neg 8.8%

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

   11. unpow2 8.8%

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

   12. remove-double-neg 8.8%

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

   13. sub-neg 8.8%

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

   14. unpow2 8.8%

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

   15. sqr-neg 8.8%

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

   16. fmm-def 8.8%

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

   17. metadata-eval 8.8%

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

   18. distribute-neg-in 8.8%

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

   19. metadata-eval 8.8%

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

   20. unsub-neg 8.8%

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

8. Simplified 8.8%

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

   1. fma-undefine 8.7%

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

   2. distribute-lft-neg-in 8.7%

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

   3. neg-mul-1 8.7%

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

   4. unpow2 8.7%

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

   5. rem-3cbrt-rft 3.1%

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

   6. unpow2 3.1%

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

   7. rem-3cbrt-rft 8.7%

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

   8. *-un-lft-identity 8.7%

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

   9. distribute-rgt-out 8.7%

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

   10. metadata-eval 8.7%

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

10. Applied egg-rr 8.7%

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

11. Final simplification 8.7%

    \[\leadsto \mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2} \cdot \left(-\sqrt[3]{\sin^{-1} \left(\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\right)}\right)\right) + \sin^{-1} \left(1 - x\right) \cdot 0 \]
12. Add Preprocessing

Alternative 6: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \left(t\_0 \cdot 0 + \pi \cdot {\left(\sqrt{0.5}\right)}^{2}\right) - \sqrt[3]{\sin^{-1} \left(\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\right) \cdot {t\_0}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (-
    (+ (* t_0 0.0) (* PI (pow (sqrt 0.5) 2.0)))
    (cbrt (* (asin (/ (fma x x -1.0) (- -1.0 x))) (pow t_0 2.0))))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return ((t_0 * 0.0) + (((double) M_PI) * pow(sqrt(0.5), 2.0))) - cbrt((asin((fma(x, x, -1.0) / (-1.0 - x))) * pow(t_0, 2.0)));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(Float64(Float64(t_0 * 0.0) + Float64(pi * (sqrt(0.5) ^ 2.0))) - cbrt(Float64(asin(Float64(fma(x, x, -1.0) / Float64(-1.0 - x))) * (t_0 ^ 2.0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(t$95$0 * 0.0), $MachinePrecision] + N[(Pi * N[Power[N[Sqrt[0.5], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[N[(N[ArcSin[N[(N[(x * x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 4.9%

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

   1. acos-asin 4.9%

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

   2. add-sqr-sqrt 3.0%

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

   3. add-cube-cbrt 8.7%

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

   4. prod-diff 8.7%

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

4. Applied egg-rr 8.7%

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

   1. flip-- 8.8%

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

   2. div-inv 8.8%

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

   3. metadata-eval 8.8%

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

   4. pow2 8.8%

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

6. Applied egg-rr 8.8%

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

   1. associate-*r/ 8.8%

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

   2. *-rgt-identity 8.8%

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

   3. remove-double-neg 8.8%

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

   4. distribute-frac-neg 8.8%

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

   5. distribute-frac-neg2 8.8%

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

   6. sub-neg 8.8%

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

   7. +-commutative 8.8%

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

   8. distribute-neg-in 8.8%

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

   9. unpow2 8.8%

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

   10. sqr-neg 8.8%

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

   11. unpow2 8.8%

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

   12. remove-double-neg 8.8%

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

   13. sub-neg 8.8%

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

   14. unpow2 8.8%

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

   15. sqr-neg 8.8%

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

   16. fmm-def 8.8%

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

   17. metadata-eval 8.8%

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

   18. distribute-neg-in 8.8%

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

   19. metadata-eval 8.8%

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

   20. unsub-neg 8.8%

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

8. Simplified 8.8%

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

9. Taylor expanded in x around 0 3.1%

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

11. Simplified 3.1%

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

12. Taylor expanded in x around 0 8.7%

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

    1. fma-define 8.7%

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

    2. sub-neg 8.7%

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

    3. mul-1-neg 8.7%

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

    4. fma-define 8.7%

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

    5. sub-neg 8.7%

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

    6. mul-1-neg 8.7%

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

14. Simplified 8.7%

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

15. Final simplification 8.7%

    \[\leadsto \left(\sin^{-1} \left(1 - x\right) \cdot 0 + \pi \cdot {\left(\sqrt{0.5}\right)}^{2}\right) - \sqrt[3]{\sin^{-1} \left(\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\right) \cdot {\sin^{-1} \left(1 - x\right)}^{2}} \]
16. Add Preprocessing

Alternative 7: 10.3% 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 4.9%

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

   1. acos-asin 4.9%

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

   2. *-un-lft-identity 4.9%

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

   3. add-sqr-sqrt 8.7%

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

   4. prod-diff 8.7%

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

   5. add-sqr-sqrt 8.7%

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

   6. fmm-def 8.7%

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

   7. *-un-lft-identity 8.7%

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

   8. acos-asin 8.7%

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

   9. add-sqr-sqrt 8.7%

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

4. Applied egg-rr 8.7%

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

Alternative 8: 10.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 4.9%

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

   1. acos-asin 4.9%

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

   2. add-sqr-sqrt 3.0%

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

   3. add-cube-cbrt 8.7%

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

   4. prod-diff 8.7%

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

4. Applied egg-rr 8.7%

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

5. Taylor expanded in x around 0 8.7%

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

7. Simplified 8.7%

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

Alternative 9: 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 4.9%

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

3. Taylor expanded in x around inf 6.8%

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

   1. neg-mul-1 6.8%

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

5. Simplified 6.8%

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

   1. *-un-lft-identity 6.8%

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

   2. add-sqr-sqrt 0.0%

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

   3. sqrt-unprod 6.8%

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

   4. sqr-neg 6.8%

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

   5. sqrt-unprod 6.8%

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

   6. add-sqr-sqrt 6.8%

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

7. Applied egg-rr 6.8%

   \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]
8. Step-by-step derivation

   1. *-lft-identity 6.8%

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

9. Simplified 6.8%

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

Alternative 10: 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 4.9%

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

3. Taylor expanded in x around 0 3.8%

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

Developer Target 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 2024181 
(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)))