bug323 (missed optimization)

Percentage Accurate: 7.1% → 10.7%

Time: 13.7s

Alternatives: 8

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: 10.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

   1. acos-asin 6.7%

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

   2. flip3-- 6.7%

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

   3. div-inv 6.7%

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

   4. metadata-eval 6.7%

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

   5. div-inv 6.7%

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

   6. metadata-eval 6.7%

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

   7. div-inv 6.7%

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

   8. metadata-eval 6.7%

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

   9. div-inv 6.7%

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

   10. metadata-eval 6.7%

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

3. Applied egg-rr 6.7%

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

   1. swap-sqr 6.7%

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

   2. metadata-eval 6.7%

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

   3. distribute-rgt-out 6.7%

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

   4. +-commutative 6.7%

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

   5. fma-def 6.7%

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

5. Simplified 6.7%

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

   1. unpow3 6.7%

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

   2. *-commutative 6.7%

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

   3. associate-*r* 6.7%

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

   4. fma-neg 10.2%

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

   5. *-commutative 10.2%

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

   6. *-commutative 10.2%

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

   7. swap-sqr 10.2%

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

   8. metadata-eval 10.2%

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

   9. pow2 10.2%

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

7. Applied egg-rr 10.2%

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

8. Taylor expanded in x around 0 10.2%

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

   1. rem-cbrt-cube 10.2%

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

10. Applied egg-rr 10.2%

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

11. Final simplification 10.2%

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

Alternative 2: 10.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

   1. acos-asin 6.7%

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

   2. flip3-- 6.7%

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

   3. div-inv 6.7%

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

   4. metadata-eval 6.7%

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

   5. div-inv 6.7%

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

   6. metadata-eval 6.7%

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

   7. div-inv 6.7%

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

   8. metadata-eval 6.7%

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

   9. div-inv 6.7%

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

   10. metadata-eval 6.7%

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

3. Applied egg-rr 6.7%

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

   1. swap-sqr 6.7%

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

   2. metadata-eval 6.7%

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

   3. distribute-rgt-out 6.7%

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

   4. +-commutative 6.7%

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

   5. fma-def 6.7%

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

5. Simplified 6.7%

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

   1. unpow3 6.7%

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

   2. *-commutative 6.7%

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

   3. associate-*r* 6.7%

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

   4. fma-neg 10.2%

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

   5. *-commutative 10.2%

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

   6. *-commutative 10.2%

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

   7. swap-sqr 10.2%

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

   8. metadata-eval 10.2%

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

   9. pow2 10.2%

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

7. Applied egg-rr 10.2%

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

8. Taylor expanded in x around 0 10.2%

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

9. Final simplification 10.2%

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

Alternative 3: 10.6% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (asin (- 1.0 x)))))
   (+ (acos (- 1.0 x)) (fma (- t_0) (pow t_0 2.0) (pow t_0 3.0)))))

C

double code(double x) {
	double t_0 = cbrt(asin((1.0 - x)));
	return acos((1.0 - x)) + fma(-t_0, pow(t_0, 2.0), pow(t_0, 3.0));
}

Julia

function code(x)
	t_0 = cbrt(asin(Float64(1.0 - x)))
	return Float64(acos(Float64(1.0 - x)) + fma(Float64(-t_0), (t_0 ^ 2.0), (t_0 ^ 3.0)))
end

Wolfram

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

Derivation

1. Initial program 6.7%

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

   1. add-cbrt-cube 6.7%

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

   2. pow1/3 6.7%

      \[\leadsto \color{blue}{{\left(\left(\cos^{-1} \left(1 - x\right) \cdot \cos^{-1} \left(1 - x\right)\right) \cdot \cos^{-1} \left(1 - x\right)\right)}^{0.3333333333333333}} \]

   3. pow3 6.7%

      \[\leadsto {\color{blue}{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}}^{0.3333333333333333} \]

3. Applied egg-rr 6.7%

   \[\leadsto \color{blue}{{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}^{0.3333333333333333}} \]
4. Step-by-step derivation

   1. unpow1/3 6.7%

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

   2. rem-cbrt-cube 6.7%

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

   3. acos-asin 6.7%

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

   4. div-inv 6.7%

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

   5. metadata-eval 6.7%

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

   6. add-cube-cbrt 10.1%

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

   7. prod-diff 10.1%

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

5. Applied egg-rr 10.1%

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

   1. add-cube-cbrt 10.1%

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

   2. pow3 10.1%

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

7. Applied egg-rr 10.2%

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

8. Final simplification 10.2%

   \[\leadsto \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(1 - x\right)}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}\right) \]

Alternative 4: 10.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

   1. acos-asin 6.7%

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

   2. sub-neg 6.7%

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

   3. div-inv 6.7%

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

   4. metadata-eval 6.7%

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

3. Applied egg-rr 6.7%

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

   1. sub-neg 6.7%

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

5. Simplified 6.7%

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

   1. add-cube-cbrt 10.1%

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

   2. pow3 10.1%

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

7. Applied egg-rr 10.1%

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

8. Final simplification 10.1%

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

Alternative 5: 10.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

   1. acos-asin 6.7%

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

   2. sub-neg 6.7%

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

   3. div-inv 6.7%

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

   4. metadata-eval 6.7%

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

3. Applied egg-rr 6.7%

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

   1. sub-neg 6.7%

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

5. Simplified 6.7%

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

   1. add-sqr-sqrt 10.1%

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

   2. pow2 10.1%

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

7. Applied egg-rr 10.1%

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

8. Final simplification 10.1%

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

Alternative 6: 7.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left({\cos^{-1} \left(1 - x\right)}^{0.16666666666666666}\right)}^{6} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (pow (acos (- 1.0 x)) 0.16666666666666666) 6.0))

C

double code(double x) {
	return pow(pow(acos((1.0 - x)), 0.16666666666666666), 6.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (acos((1.0d0 - x)) ** 0.16666666666666666d0) ** 6.0d0
end function

Java

public static double code(double x) {
	return Math.pow(Math.pow(Math.acos((1.0 - x)), 0.16666666666666666), 6.0);
}

Python

def code(x):
	return math.pow(math.pow(math.acos((1.0 - x)), 0.16666666666666666), 6.0)

Julia

function code(x)
	return (acos(Float64(1.0 - x)) ^ 0.16666666666666666) ^ 6.0
end

MATLAB

function tmp = code(x)
	tmp = (acos((1.0 - x)) ^ 0.16666666666666666) ^ 6.0;
end

Wolfram

code[x_] := N[Power[N[Power[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 0.16666666666666666], $MachinePrecision], 6.0], $MachinePrecision]

Derivation

1. Initial program 6.7%

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

   1. add-log-exp 6.7%

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

3. Applied egg-rr 6.7%

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

   1. add-log-exp 6.7%

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

   2. pow1 6.7%

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

   3. metadata-eval 6.7%

      \[\leadsto {\cos^{-1} \left(1 - x\right)}^{\color{blue}{\left(0.16666666666666666 \cdot 6\right)}} \]

   4. metadata-eval 6.7%

      \[\leadsto {\cos^{-1} \left(1 - x\right)}^{\left(0.16666666666666666 \cdot \color{blue}{\left(2 \cdot 3\right)}\right)} \]

   5. pow-pow 6.7%

      \[\leadsto \color{blue}{{\left({\cos^{-1} \left(1 - x\right)}^{0.16666666666666666}\right)}^{\left(2 \cdot 3\right)}} \]

   6. metadata-eval 6.7%

      \[\leadsto {\left({\cos^{-1} \left(1 - x\right)}^{0.16666666666666666}\right)}^{\color{blue}{6}} \]

5. Applied egg-rr 6.7%

   \[\leadsto \color{blue}{{\left({\cos^{-1} \left(1 - x\right)}^{0.16666666666666666}\right)}^{6}} \]

6. Final simplification 6.7%

   \[\leadsto {\left({\cos^{-1} \left(1 - x\right)}^{0.16666666666666666}\right)}^{6} \]

Alternative 7: 7.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}^{0.3333333333333333} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (pow (acos (- 1.0 x)) 3.0) 0.3333333333333333))

C

double code(double x) {
	return pow(pow(acos((1.0 - x)), 3.0), 0.3333333333333333);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (acos((1.0d0 - x)) ** 3.0d0) ** 0.3333333333333333d0
end function

Java

public static double code(double x) {
	return Math.pow(Math.pow(Math.acos((1.0 - x)), 3.0), 0.3333333333333333);
}

Python

def code(x):
	return math.pow(math.pow(math.acos((1.0 - x)), 3.0), 0.3333333333333333)

Julia

function code(x)
	return (acos(Float64(1.0 - x)) ^ 3.0) ^ 0.3333333333333333
end

MATLAB

function tmp = code(x)
	tmp = (acos((1.0 - x)) ^ 3.0) ^ 0.3333333333333333;
end

Wolfram

code[x_] := N[Power[N[Power[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]

Derivation

1. Initial program 6.7%

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

   1. add-cbrt-cube 6.7%

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

   2. pow1/3 6.7%

      \[\leadsto \color{blue}{{\left(\left(\cos^{-1} \left(1 - x\right) \cdot \cos^{-1} \left(1 - x\right)\right) \cdot \cos^{-1} \left(1 - x\right)\right)}^{0.3333333333333333}} \]

   3. pow3 6.7%

      \[\leadsto {\color{blue}{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}}^{0.3333333333333333} \]

3. Applied egg-rr 6.7%

   \[\leadsto \color{blue}{{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}^{0.3333333333333333}} \]

4. Final simplification 6.7%

   \[\leadsto {\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}^{0.3333333333333333} \]

Alternative 8: 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 6.7%

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

2. Final simplification 6.7%

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

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 2023230 
(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)))