bug323 (missed optimization)

Percentage Accurate: 7.1% → 10.6%

Time: 17.0s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 7.1% 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: 10.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := {t_0}^{3}\\ \frac{\frac{{\pi}^{6} \cdot 0.015625 - t_1 \cdot t_1}{{\left(\pi \cdot 0.5\right)}^{3} + {\left(\sqrt[3]{t_1}\right)}^{3}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (pow t_0 3.0)))
   (/
    (/
     (- (* (pow PI 6.0) 0.015625) (* t_1 t_1))
     (+ (pow (* PI 0.5) 3.0) (pow (cbrt t_1) 3.0)))
    (+ (* (* PI PI) 0.25) (* t_0 (fma PI 0.5 t_0))))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = pow(t_0, 3.0);
	return (((pow(((double) M_PI), 6.0) * 0.015625) - (t_1 * t_1)) / (pow((((double) M_PI) * 0.5), 3.0) + pow(cbrt(t_1), 3.0))) / (((((double) M_PI) * ((double) M_PI)) * 0.25) + (t_0 * fma(((double) M_PI), 0.5, t_0)));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = t_0 ^ 3.0
	return Float64(Float64(Float64(Float64((pi ^ 6.0) * 0.015625) - Float64(t_1 * t_1)) / Float64((Float64(pi * 0.5) ^ 3.0) + (cbrt(t_1) ^ 3.0))) / Float64(Float64(Float64(pi * pi) * 0.25) + Float64(t_0 * fma(pi, 0.5, t_0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 3.0], $MachinePrecision]}, N[(N[(N[(N[(N[Power[Pi, 6.0], $MachinePrecision] * 0.015625), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(Pi * 0.5), $MachinePrecision], 3.0], $MachinePrecision] + N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision] + N[(t$95$0 * N[(Pi * 0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 7.1%

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

   1. acos-asin 7.1%

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

   2. flip3-- 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

   5. div-inv 7.1%

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

   6. metadata-eval 7.1%

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

   7. div-inv 7.1%

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

   8. metadata-eval 7.1%

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

   9. div-inv 7.1%

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

   10. metadata-eval 7.1%

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

3. Applied egg-rr 7.1%

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

   1. swap-sqr 7.1%

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

   2. metadata-eval 7.1%

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

   3. distribute-rgt-out 7.1%

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

   4. +-commutative 7.1%

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

   5. fma-def 7.1%

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

5. Simplified 7.1%

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

   1. sub-neg 7.1%

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

   2. flip-+ 7.1%

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

   3. pow-prod-up 10.5%

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

   4. metadata-eval 10.5%

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

7. Applied egg-rr 10.5%

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

   1. rem-cbrt-cube 10.5%

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

9. Applied egg-rr 10.5%

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

    1. cancel-sign-sub 10.5%

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

    2. unpow-prod-down 10.5%

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

    3. metadata-eval 10.5%

       \[\leadsto \frac{\frac{{\pi}^{6} \cdot \color{blue}{0.015625} + {\sin^{-1} \left(1 - x\right)}^{3} \cdot \left(-{\sin^{-1} \left(1 - x\right)}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{3} - \left(-{\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{3}}\right)}^{3}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

11. Applied egg-rr 10.5%

    \[\leadsto \frac{\frac{\color{blue}{{\pi}^{6} \cdot 0.015625 + {\sin^{-1} \left(1 - x\right)}^{3} \cdot \left(-{\sin^{-1} \left(1 - x\right)}^{3}\right)}}{{\left(\pi \cdot 0.5\right)}^{3} - \left(-{\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{3}}\right)}^{3}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

12. Final simplification 10.5%

    \[\leadsto \frac{\frac{{\pi}^{6} \cdot 0.015625 - {\sin^{-1} \left(1 - x\right)}^{3} \cdot {\sin^{-1} \left(1 - x\right)}^{3}}{{\left(\pi \cdot 0.5\right)}^{3} + {\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{3}}\right)}^{3}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

Alternative 2: 10.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := {t_0}^{3}\\ \frac{\frac{{\pi}^{6} \cdot 0.015625 - t_1 \cdot t_1}{t_1 + {\left(\pi \cdot 0.5\right)}^{3}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (pow t_0 3.0)))
   (/
    (/ (- (* (pow PI 6.0) 0.015625) (* t_1 t_1)) (+ t_1 (pow (* PI 0.5) 3.0)))
    (+ (* (* PI PI) 0.25) (* t_0 (fma PI 0.5 t_0))))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = pow(t_0, 3.0);
	return (((pow(((double) M_PI), 6.0) * 0.015625) - (t_1 * t_1)) / (t_1 + pow((((double) M_PI) * 0.5), 3.0))) / (((((double) M_PI) * ((double) M_PI)) * 0.25) + (t_0 * fma(((double) M_PI), 0.5, t_0)));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = t_0 ^ 3.0
	return Float64(Float64(Float64(Float64((pi ^ 6.0) * 0.015625) - Float64(t_1 * t_1)) / Float64(t_1 + (Float64(pi * 0.5) ^ 3.0))) / Float64(Float64(Float64(pi * pi) * 0.25) + Float64(t_0 * fma(pi, 0.5, t_0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 3.0], $MachinePrecision]}, N[(N[(N[(N[(N[Power[Pi, 6.0], $MachinePrecision] * 0.015625), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[Power[N[(Pi * 0.5), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision] + N[(t$95$0 * N[(Pi * 0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 7.1%

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

   1. acos-asin 7.1%

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

   2. flip3-- 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

   5. div-inv 7.1%

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

   6. metadata-eval 7.1%

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

   7. div-inv 7.1%

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

   8. metadata-eval 7.1%

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

   9. div-inv 7.1%

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

   10. metadata-eval 7.1%

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

3. Applied egg-rr 7.1%

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

   1. swap-sqr 7.1%

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

   2. metadata-eval 7.1%

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

   3. distribute-rgt-out 7.1%

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

   4. +-commutative 7.1%

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

   5. fma-def 7.1%

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

5. Simplified 7.1%

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

   1. sub-neg 7.1%

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

   2. flip-+ 7.1%

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

   3. pow-prod-up 10.5%

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

   4. metadata-eval 10.5%

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

7. Applied egg-rr 10.5%

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

   1. cancel-sign-sub 10.5%

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

   2. unpow-prod-down 10.5%

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

   3. metadata-eval 10.5%

      \[\leadsto \frac{\frac{{\pi}^{6} \cdot \color{blue}{0.015625} + {\sin^{-1} \left(1 - x\right)}^{3} \cdot \left(-{\sin^{-1} \left(1 - x\right)}^{3}\right)}{{\left(\pi \cdot 0.5\right)}^{3} - \left(-{\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{3}}\right)}^{3}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

9. Applied egg-rr 10.5%

   \[\leadsto \frac{\frac{\color{blue}{{\pi}^{6} \cdot 0.015625 + {\sin^{-1} \left(1 - x\right)}^{3} \cdot \left(-{\sin^{-1} \left(1 - x\right)}^{3}\right)}}{{\left(\pi \cdot 0.5\right)}^{3} - \left(-{\sin^{-1} \left(1 - x\right)}^{3}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

10. Final simplification 10.5%

    \[\leadsto \frac{\frac{{\pi}^{6} \cdot 0.015625 - {\sin^{-1} \left(1 - x\right)}^{3} \cdot {\sin^{-1} \left(1 - x\right)}^{3}}{{\sin^{-1} \left(1 - x\right)}^{3} + {\left(\pi \cdot 0.5\right)}^{3}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

Alternative 3: 10.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[((-t$95$0) * t$95$0 + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.1%

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

   1. expm1-log1p-u 7.1%

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

   2. expm1-udef 7.1%

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

   3. log1p-udef 7.1%

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

   4. add-exp-log 7.1%

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

3. Applied egg-rr 7.1%

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

   1. *-rgt-identity 7.1%

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

   2. add-exp-log 7.1%

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

   3. *-rgt-identity 7.1%

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

   4. log1p-udef 7.1%

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

   5. expm1-udef 7.1%

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

   6. expm1-log1p-u 7.1%

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

   7. acos-asin 7.1%

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

   8. div-inv 7.1%

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

   9. metadata-eval 7.1%

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

   10. add-sqr-sqrt 10.4%

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

   11. prod-diff 10.4%

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

5. Applied egg-rr 10.4%

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

   1. add-sqr-sqrt 10.4%

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

   2. pow2 10.4%

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

7. Applied egg-rr 10.4%

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

8. Final simplification 10.4%

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

Alternative 4: 10.5% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 0.5))))
   (- (/ PI 2.0) (fma t_0 t_0 (- (acos (- 1.0 x)))))))

C

double code(double x) {
	double t_0 = sqrt((((double) M_PI) * 0.5));
	return (((double) M_PI) / 2.0) - fma(t_0, t_0, -acos((1.0 - x)));
}

Julia

function code(x)
	t_0 = sqrt(Float64(pi * 0.5))
	return Float64(Float64(pi / 2.0) - fma(t_0, t_0, Float64(-acos(Float64(1.0 - x)))))
end

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. add-cube-cbrt 7.1%

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

   2. pow3 7.1%

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

3. Applied egg-rr 7.1%

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

   1. add-log-exp 7.1%

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

5. Applied egg-rr 7.1%

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

   1. unpow3 7.1%

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

   2. add-log-exp 7.1%

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

   3. add-log-exp 7.1%

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

   4. add-log-exp 7.1%

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

   5. add-cube-cbrt 7.1%

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

   6. acos-asin 7.1%

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

7. Applied egg-rr 7.1%

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

   1. asin-acos 7.1%

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

   2. div-inv 7.1%

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

   3. metadata-eval 7.1%

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

   4. add-sqr-sqrt 10.4%

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

   5. fma-neg 10.4%

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

9. Applied egg-rr 10.4%

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

10. Final simplification 10.4%

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

Alternative 5: 10.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. add-exp-log 7.1%

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

3. Applied egg-rr 7.1%

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

   1. acos-asin 7.1%

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

   2. sub-neg 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

5. Applied egg-rr 7.1%

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

   1. sub-neg 7.1%

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

7. Simplified 7.1%

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

   1. add-sqr-sqrt 10.4%

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

   2. pow2 10.4%

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

9. Applied egg-rr 10.4%

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

10. Final simplification 10.4%

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

Alternative 6: 10.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (pow (cbrt (asin (- 1.0 x))) 3.0)))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - pow(cbrt(asin((1.0 - x))), 3.0);
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - Math.pow(Math.cbrt(Math.asin((1.0 - x))), 3.0);
}

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - (cbrt(asin(Float64(1.0 - x))) ^ 3.0))
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[Power[N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

   1. add-cube-cbrt 7.1%

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

   2. pow3 7.1%

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

3. Applied egg-rr 7.1%

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

   1. add-log-exp 7.1%

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

5. Applied egg-rr 7.1%

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

   1. unpow3 7.1%

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

   2. add-log-exp 7.1%

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

   3. add-log-exp 7.1%

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

   4. add-log-exp 7.1%

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

   5. add-cube-cbrt 7.1%

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

   6. acos-asin 7.1%

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

7. Applied egg-rr 7.1%

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

   1. add-cube-cbrt 10.3%

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

   2. pow3 10.3%

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

9. Applied egg-rr 10.3%

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

10. Final simplification 10.3%

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

Alternative 7: 10.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. add-cube-cbrt 7.1%

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

   2. pow3 7.1%

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

3. Applied egg-rr 7.1%

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

   1. add-log-exp 7.1%

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

5. Applied egg-rr 7.1%

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

   1. unpow3 7.1%

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

   2. add-log-exp 7.1%

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

   3. add-log-exp 7.1%

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

   4. add-log-exp 7.1%

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

   5. add-cube-cbrt 7.1%

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

   6. acos-asin 7.1%

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

7. Applied egg-rr 7.1%

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

   1. add-sqr-sqrt 10.4%

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

   2. pow2 10.4%

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

9. Applied egg-rr 10.4%

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

10. Final simplification 10.4%

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

Alternative 8: 7.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[(1.0 + N[Power[N[Power[N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(Pi * 0.5), $MachinePrecision] ^ 2 + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 x) < 1

   1. Initial program 7.1%

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

      1. expm1-log1p-u 7.1%

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

      2. expm1-udef 7.1%

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

      3. log1p-udef 7.1%

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

      4. add-exp-log 7.1%

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

   3. Applied egg-rr 7.1%

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

      1. associate--l+ 7.1%

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

      2. +-commutative 7.1%

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

      3. sub-neg 7.1%

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

      4. metadata-eval 7.1%

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

   5. Applied egg-rr 7.1%

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

      1. add-cbrt-cube 7.1%

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

      2. pow3 7.1%

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

   7. Applied egg-rr 7.1%

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

   if 1 < (-.f64 1 x)

   1. Initial program 7.1%

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

      1. add-cube-cbrt 7.1%

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

      2. pow3 7.1%

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

   3. Applied egg-rr 7.1%

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

      1. add-log-exp 7.1%

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

   5. Applied egg-rr 7.1%

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

      1. unpow3 7.1%

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

      2. add-log-exp 7.1%

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

      3. add-log-exp 7.1%

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

      4. add-log-exp 7.1%

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

      5. add-cube-cbrt 7.1%

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

      6. acos-asin 7.1%

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

   7. Applied egg-rr 7.1%

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

      1. div-inv 7.1%

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

      2. metadata-eval 7.1%

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

      3. add-sqr-sqrt 7.1%

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

      4. sqrt-unprod 7.1%

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

      5. flip3-- 7.1%

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

   9. Applied egg-rr 6.8%

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

4. Final simplification 7.1%

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

Alternative 9: 7.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\pi \cdot 0.5 - t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\pi \cdot 0.5, t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (if (<= (- 1.0 x) 1.0) (- (* PI 0.5) t_0) (hypot (* PI 0.5) t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = (((double) M_PI) * 0.5) - t_0;
	} else {
		tmp = hypot((((double) M_PI) * 0.5), t_0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = (Math.PI * 0.5) - t_0;
	} else {
		tmp = Math.hypot((Math.PI * 0.5), t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.asin((1.0 - x))
	tmp = 0
	if (1.0 - x) <= 1.0:
		tmp = (math.pi * 0.5) - t_0
	else:
		tmp = math.hypot((math.pi * 0.5), t_0)
	return tmp

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = Float64(Float64(pi * 0.5) - t_0);
	else
		tmp = hypot(Float64(pi * 0.5), t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = asin((1.0 - x));
	tmp = 0.0;
	if ((1.0 - x) <= 1.0)
		tmp = (pi * 0.5) - t_0;
	else
		tmp = hypot((pi * 0.5), t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[(N[(Pi * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision], N[Sqrt[N[(Pi * 0.5), $MachinePrecision] ^ 2 + t$95$0 ^ 2], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 x) < 1

   1. Initial program 7.1%

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

      1. acos-asin 7.1%

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

      2. sub-neg 7.1%

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

      3. div-inv 7.1%

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

      4. metadata-eval 7.1%

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

   3. Applied egg-rr 7.1%

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

      1. sub-neg 7.1%

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

   5. Simplified 7.1%

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

   if 1 < (-.f64 1 x)

   1. Initial program 7.1%

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

      1. add-cube-cbrt 7.1%

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

      2. pow3 7.1%

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

   3. Applied egg-rr 7.1%

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

      1. add-log-exp 7.1%

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

   5. Applied egg-rr 7.1%

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

      1. unpow3 7.1%

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

      2. add-log-exp 7.1%

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

      3. add-log-exp 7.1%

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

      4. add-log-exp 7.1%

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

      5. add-cube-cbrt 7.1%

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

      6. acos-asin 7.1%

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

   7. Applied egg-rr 7.1%

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

      1. div-inv 7.1%

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

      2. metadata-eval 7.1%

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

      3. add-sqr-sqrt 7.1%

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

      4. sqrt-unprod 7.1%

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

      5. flip3-- 7.1%

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

   9. Applied egg-rr 6.8%

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

4. Final simplification 7.1%

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

Alternative 10: 9.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.asin((1.0 - x)) + (math.pi * 0.5)
	else:
		tmp = 1.0 + (math.acos((1.0 - x)) + -1.0)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = asin((1.0 - x)) + (pi * 0.5);
	else
		tmp = 1.0 + (acos((1.0 - x)) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. add-cube-cbrt 3.9%

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

      2. pow3 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. add-log-exp 3.9%

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

   5. Applied egg-rr 3.9%

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

      1. unpow3 3.9%

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

      2. add-log-exp 3.9%

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

      3. add-log-exp 3.9%

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

      4. add-log-exp 3.9%

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

      5. add-cube-cbrt 3.9%

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

      6. acos-asin 3.9%

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

   7. Applied egg-rr 3.9%

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

      1. div-inv 3.9%

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

      2. metadata-eval 3.9%

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

      3. add-cube-cbrt 7.3%

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

      4. unpow2 7.3%

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

      5. cancel-sign-sub-inv 7.3%

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

      6. +-commutative 7.3%

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

      7. fma-def 7.3%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 6.5%

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

      10. sqr-neg 6.5%

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

      11. sqrt-unprod 6.5%

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

      12. add-sqr-sqrt 6.5%

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

   9. Applied egg-rr 6.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 58.5%

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

      1. expm1-log1p-u 58.5%

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

      2. expm1-udef 58.5%

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

      3. log1p-udef 58.5%

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

      4. add-exp-log 58.5%

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

   3. Applied egg-rr 58.5%

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

      1. associate--l+ 58.5%

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

      2. +-commutative 58.5%

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

      3. sub-neg 58.5%

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

      4. metadata-eval 58.5%

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

   5. Applied egg-rr 58.5%

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

4. Final simplification 9.5%

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

Alternative 11: 9.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;t_0 + \pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (if (<= x 5.5e-17) (+ t_0 (* PI 0.5)) (- (* PI 0.5) t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = t_0 + (((double) M_PI) * 0.5);
	} else {
		tmp = (((double) M_PI) * 0.5) - t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = t_0 + (Math.PI * 0.5);
	} else {
		tmp = (Math.PI * 0.5) - t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.asin((1.0 - x))
	tmp = 0
	if x <= 5.5e-17:
		tmp = t_0 + (math.pi * 0.5)
	else:
		tmp = (math.pi * 0.5) - t_0
	return tmp

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(t_0 + Float64(pi * 0.5));
	else
		tmp = Float64(Float64(pi * 0.5) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = asin((1.0 - x));
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = t_0 + (pi * 0.5);
	else
		tmp = (pi * 0.5) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(t$95$0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. add-cube-cbrt 3.9%

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

      2. pow3 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. add-log-exp 3.9%

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

   5. Applied egg-rr 3.9%

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

      1. unpow3 3.9%

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

      2. add-log-exp 3.9%

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

      3. add-log-exp 3.9%

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

      4. add-log-exp 3.9%

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

      5. add-cube-cbrt 3.9%

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

      6. acos-asin 3.9%

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

   7. Applied egg-rr 3.9%

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

      1. div-inv 3.9%

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

      2. metadata-eval 3.9%

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

      3. add-cube-cbrt 7.3%

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

      4. unpow2 7.3%

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

      5. cancel-sign-sub-inv 7.3%

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

      6. +-commutative 7.3%

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

      7. fma-def 7.3%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 6.5%

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

      10. sqr-neg 6.5%

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

      11. sqrt-unprod 6.5%

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

      12. add-sqr-sqrt 6.5%

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

   9. Applied egg-rr 6.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 58.5%

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

      1. acos-asin 58.5%

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

      2. sub-neg 58.5%

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

      3. div-inv 58.5%

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

      4. metadata-eval 58.5%

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

   3. Applied egg-rr 58.5%

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

      1. sub-neg 58.5%

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

   5. Simplified 58.5%

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

4. Final simplification 9.5%

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

Alternative 12: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. expm1-log1p-u 7.1%

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

   2. expm1-udef 7.1%

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

   3. log1p-udef 7.1%

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

   4. add-exp-log 7.1%

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

3. Applied egg-rr 7.1%

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

   1. associate--l+ 7.1%

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

   2. +-commutative 7.1%

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

   3. sub-neg 7.1%

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

   4. metadata-eval 7.1%

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

5. Applied egg-rr 7.1%

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

6. Final simplification 7.1%

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

Alternative 13: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Final simplification 7.1%

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

Developer target: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023271 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

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

  (acos (- 1.0 x)))