bug323 (missed optimization)

Percentage Accurate: 7.0% → 10.6%

Time: 10.1s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\pi \cdot 0.5\right)}^{1.5}\\ t_1 := \sin^{-1} \left(1 - x\right)\\ t_2 := \sqrt[3]{\mathsf{fma}\left({\pi}^{2}, 0.25, t_1 \cdot \mathsf{fma}\left(\pi, 0.5, t_1\right)\right)}\\ \frac{\mathsf{fma}\left(t_0, t_0, -{t_1}^{3}\right)}{t_2 \cdot \left(t_2 \cdot t_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (* PI 0.5) 1.5))
        (t_1 (asin (- 1.0 x)))
        (t_2 (cbrt (fma (pow PI 2.0) 0.25 (* t_1 (fma PI 0.5 t_1))))))
   (/ (fma t_0 t_0 (- (pow t_1 3.0))) (* t_2 (* t_2 t_2)))))

C

double code(double x) {
	double t_0 = pow((((double) M_PI) * 0.5), 1.5);
	double t_1 = asin((1.0 - x));
	double t_2 = cbrt(fma(pow(((double) M_PI), 2.0), 0.25, (t_1 * fma(((double) M_PI), 0.5, t_1))));
	return fma(t_0, t_0, -pow(t_1, 3.0)) / (t_2 * (t_2 * t_2));
}

Julia

function code(x)
	t_0 = Float64(pi * 0.5) ^ 1.5
	t_1 = asin(Float64(1.0 - x))
	t_2 = cbrt(fma((pi ^ 2.0), 0.25, Float64(t_1 * fma(pi, 0.5, t_1))))
	return Float64(fma(t_0, t_0, Float64(-(t_1 ^ 3.0))) / Float64(t_2 * Float64(t_2 * t_2)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(Pi * 0.5), $MachinePrecision], 1.5], $MachinePrecision]}, Block[{t$95$1 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.25 + N[(t$95$1 * N[(Pi * 0.5 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(t$95$0 * t$95$0 + (-N[Power[t$95$1, 3.0], $MachinePrecision])), $MachinePrecision] / N[(t$95$2 * N[(t$95$2 * t$95$2), $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. sqr-pow 7.1%

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

   2. fma-neg 10.8%

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

   3. metadata-eval 10.8%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{\color{blue}{1.5}}, {\left(\pi \cdot 0.5\right)}^{\left(\frac{3}{2}\right)}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   4. metadata-eval 10.8%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{\color{blue}{1.5}}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

7. Applied egg-rr 10.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{1.5}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
8. Step-by-step derivation

   1. add-cube-cbrt 10.8%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{1.5}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\color{blue}{\left(\sqrt[3]{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \cdot \sqrt[3]{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}\right) \cdot \sqrt[3]{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}}} \]

9. Applied egg-rr 10.8%

   \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{1.5}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left({\pi}^{2}, 0.25, \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left({\pi}^{2}, 0.25, \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left({\pi}^{2}, 0.25, \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}}} \]

10. Final simplification 10.8%

    \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{1.5}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\sqrt[3]{\mathsf{fma}\left({\pi}^{2}, 0.25, \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left({\pi}^{2}, 0.25, \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left({\pi}^{2}, 0.25, \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}\right)} \]

Alternative 2: 10.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\pi \cdot 0.5\right)}^{1.5}\\ t_1 := \sin^{-1} \left(1 - x\right)\\ t_2 := \sqrt[3]{\mathsf{fma}\left({\pi}^{2}, 0.25, t_1 \cdot \mathsf{fma}\left(\pi, 0.5, t_1\right)\right)}\\ \frac{\mathsf{fma}\left(t_0, t_0, -{t_1}^{3}\right)}{t_2 \cdot \left(t_2 \cdot {\left(t_1 \cdot \left(\pi \cdot 0.5 + t_1\right) + {\pi}^{2} \cdot 0.25\right)}^{0.3333333333333333}\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (* PI 0.5) 1.5))
        (t_1 (asin (- 1.0 x)))
        (t_2 (cbrt (fma (pow PI 2.0) 0.25 (* t_1 (fma PI 0.5 t_1))))))
   (/
    (fma t_0 t_0 (- (pow t_1 3.0)))
    (*
     t_2
     (*
      t_2
      (pow
       (+ (* t_1 (+ (* PI 0.5) t_1)) (* (pow PI 2.0) 0.25))
       0.3333333333333333))))))

C

double code(double x) {
	double t_0 = pow((((double) M_PI) * 0.5), 1.5);
	double t_1 = asin((1.0 - x));
	double t_2 = cbrt(fma(pow(((double) M_PI), 2.0), 0.25, (t_1 * fma(((double) M_PI), 0.5, t_1))));
	return fma(t_0, t_0, -pow(t_1, 3.0)) / (t_2 * (t_2 * pow(((t_1 * ((((double) M_PI) * 0.5) + t_1)) + (pow(((double) M_PI), 2.0) * 0.25)), 0.3333333333333333)));
}

Julia

function code(x)
	t_0 = Float64(pi * 0.5) ^ 1.5
	t_1 = asin(Float64(1.0 - x))
	t_2 = cbrt(fma((pi ^ 2.0), 0.25, Float64(t_1 * fma(pi, 0.5, t_1))))
	return Float64(fma(t_0, t_0, Float64(-(t_1 ^ 3.0))) / Float64(t_2 * Float64(t_2 * (Float64(Float64(t_1 * Float64(Float64(pi * 0.5) + t_1)) + Float64((pi ^ 2.0) * 0.25)) ^ 0.3333333333333333))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(Pi * 0.5), $MachinePrecision], 1.5], $MachinePrecision]}, Block[{t$95$1 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.25 + N[(t$95$1 * N[(Pi * 0.5 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(t$95$0 * t$95$0 + (-N[Power[t$95$1, 3.0], $MachinePrecision])), $MachinePrecision] / N[(t$95$2 * N[(t$95$2 * N[Power[N[(N[(t$95$1 * N[(N[(Pi * 0.5), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $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. sqr-pow 7.1%

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

   2. fma-neg 10.8%

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

   3. metadata-eval 10.8%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{\color{blue}{1.5}}, {\left(\pi \cdot 0.5\right)}^{\left(\frac{3}{2}\right)}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   4. metadata-eval 10.8%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{\color{blue}{1.5}}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

7. Applied egg-rr 10.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{1.5}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
8. Step-by-step derivation

   1. add-cube-cbrt 10.8%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{1.5}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\color{blue}{\left(\sqrt[3]{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \cdot \sqrt[3]{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}\right) \cdot \sqrt[3]{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}}} \]

9. Applied egg-rr 10.8%

   \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{1.5}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left({\pi}^{2}, 0.25, \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left({\pi}^{2}, 0.25, \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left({\pi}^{2}, 0.25, \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}}} \]

10. Taylor expanded in x around 0 10.8%

    \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{1.5}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\left(\color{blue}{{\left(\left(\sin^{-1} \left(1 - x\right) + 0.5 \cdot \pi\right) \cdot \sin^{-1} \left(1 - x\right) + 0.25 \cdot {\pi}^{2}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{\mathsf{fma}\left({\pi}^{2}, 0.25, \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left({\pi}^{2}, 0.25, \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}} \]

11. Final simplification 10.8%

    \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{1.5}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\sqrt[3]{\mathsf{fma}\left({\pi}^{2}, 0.25, \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left({\pi}^{2}, 0.25, \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)} \cdot {\left(\sin^{-1} \left(1 - x\right) \cdot \left(\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\right) + {\pi}^{2} \cdot 0.25\right)}^{0.3333333333333333}\right)} \]

Alternative 3: 10.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\pi \cdot 0.5\right)}^{1.5}\\ t_1 := \sin^{-1} \left(1 - x\right)\\ \frac{\mathsf{fma}\left(t_0, t_0, -{t_1}^{3}\right)}{0.25 \cdot \left(\pi \cdot \pi\right) + \mathsf{fma}\left(\pi, 0.5, t_1\right) \cdot {\left(\sqrt[3]{t_1}\right)}^{3}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (* PI 0.5) 1.5)) (t_1 (asin (- 1.0 x))))
   (/
    (fma t_0 t_0 (- (pow t_1 3.0)))
    (+ (* 0.25 (* PI PI)) (* (fma PI 0.5 t_1) (pow (cbrt t_1) 3.0))))))

C

double code(double x) {
	double t_0 = pow((((double) M_PI) * 0.5), 1.5);
	double t_1 = asin((1.0 - x));
	return fma(t_0, t_0, -pow(t_1, 3.0)) / ((0.25 * (((double) M_PI) * ((double) M_PI))) + (fma(((double) M_PI), 0.5, t_1) * pow(cbrt(t_1), 3.0)));
}

Julia

function code(x)
	t_0 = Float64(pi * 0.5) ^ 1.5
	t_1 = asin(Float64(1.0 - x))
	return Float64(fma(t_0, t_0, Float64(-(t_1 ^ 3.0))) / Float64(Float64(0.25 * Float64(pi * pi)) + Float64(fma(pi, 0.5, t_1) * (cbrt(t_1) ^ 3.0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(Pi * 0.5), $MachinePrecision], 1.5], $MachinePrecision]}, Block[{t$95$1 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 * t$95$0 + (-N[Power[t$95$1, 3.0], $MachinePrecision])), $MachinePrecision] / N[(N[(0.25 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + N[(N[(Pi * 0.5 + t$95$1), $MachinePrecision] * N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $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. sqr-pow 7.1%

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

   2. fma-neg 10.8%

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

   3. metadata-eval 10.8%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{\color{blue}{1.5}}, {\left(\pi \cdot 0.5\right)}^{\left(\frac{3}{2}\right)}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   4. metadata-eval 10.8%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{\color{blue}{1.5}}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

7. Applied egg-rr 10.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{1.5}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
8. Step-by-step derivation

   1. add-cube-cbrt 10.7%

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

   2. pow3 10.7%

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

9. Applied egg-rr 10.8%

   \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{1.5}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

10. Final simplification 10.8%

    \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{1.5}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{0.25 \cdot \left(\pi \cdot \pi\right) + \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

Alternative 4: 10.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\pi \cdot 0.5\right)}^{1.5}\\ t_1 := \sin^{-1} \left(1 - x\right)\\ \frac{\mathsf{fma}\left(t_0, t_0, -{t_1}^{3}\right)}{t_1 \cdot \left(\pi \cdot 0.5 + t_1\right) + {\pi}^{2} \cdot 0.25} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (* PI 0.5) 1.5)) (t_1 (asin (- 1.0 x))))
   (/
    (fma t_0 t_0 (- (pow t_1 3.0)))
    (+ (* t_1 (+ (* PI 0.5) t_1)) (* (pow PI 2.0) 0.25)))))

C

double code(double x) {
	double t_0 = pow((((double) M_PI) * 0.5), 1.5);
	double t_1 = asin((1.0 - x));
	return fma(t_0, t_0, -pow(t_1, 3.0)) / ((t_1 * ((((double) M_PI) * 0.5) + t_1)) + (pow(((double) M_PI), 2.0) * 0.25));
}

Julia

function code(x)
	t_0 = Float64(pi * 0.5) ^ 1.5
	t_1 = asin(Float64(1.0 - x))
	return Float64(fma(t_0, t_0, Float64(-(t_1 ^ 3.0))) / Float64(Float64(t_1 * Float64(Float64(pi * 0.5) + t_1)) + Float64((pi ^ 2.0) * 0.25)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(Pi * 0.5), $MachinePrecision], 1.5], $MachinePrecision]}, Block[{t$95$1 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 * t$95$0 + (-N[Power[t$95$1, 3.0], $MachinePrecision])), $MachinePrecision] / N[(N[(t$95$1 * N[(N[(Pi * 0.5), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.25), $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. sqr-pow 7.1%

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

   2. fma-neg 10.8%

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

   3. metadata-eval 10.8%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{\color{blue}{1.5}}, {\left(\pi \cdot 0.5\right)}^{\left(\frac{3}{2}\right)}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

   4. metadata-eval 10.8%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{\color{blue}{1.5}}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

7. Applied egg-rr 10.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{1.5}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

8. Taylor expanded in x around 0 10.8%

   \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{1.5}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\color{blue}{\left(\sin^{-1} \left(1 - x\right) + 0.5 \cdot \pi\right) \cdot \sin^{-1} \left(1 - x\right) + 0.25 \cdot {\pi}^{2}}} \]

9. Final simplification 10.8%

   \[\leadsto \frac{\mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{1.5}, {\left(\pi \cdot 0.5\right)}^{1.5}, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\sin^{-1} \left(1 - x\right) \cdot \left(\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\right) + {\pi}^{2} \cdot 0.25} \]

Alternative 5: 10.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. acos-asin 7.1%

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

   2. flip3-- 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

   5. div-inv 7.1%

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

   6. metadata-eval 7.1%

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

   7. div-inv 7.1%

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

   8. metadata-eval 7.1%

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

   9. div-inv 7.1%

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

   10. metadata-eval 7.1%

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

3. Applied egg-rr 7.1%

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

   1. swap-sqr 7.1%

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

   2. metadata-eval 7.1%

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

   3. distribute-rgt-out 7.1%

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

   4. +-commutative 7.1%

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

   5. fma-def 7.1%

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

5. Simplified 7.1%

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

   1. unpow3 7.1%

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

   2. fma-neg 10.8%

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

   3. pow2 10.8%

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

7. Applied egg-rr 10.8%

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

8. Final simplification 10.8%

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

Alternative 6: 10.5% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (cbrt t_0)))
   (+ (acos (- 1.0 x)) (fma (- t_1) (pow t_1 2.0) t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = cbrt(t_0);
	return acos((1.0 - x)) + fma(-t_1, pow(t_1, 2.0), t_0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = cbrt(t_0)
	return Float64(acos(Float64(1.0 - x)) + fma(Float64(-t_1), (t_1 ^ 2.0), t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 1/3], $MachinePrecision]}, N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[((-t$95$1) * N[Power[t$95$1, 2.0], $MachinePrecision] + t$95$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. add-exp-log 7.1%

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

   2. log1p-udef 7.1%

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

   3. expm1-udef 7.1%

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

   4. expm1-log1p-u 7.1%

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

   5. acos-asin 7.1%

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

   6. div-inv 7.1%

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

   7. metadata-eval 7.1%

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

   8. add-cube-cbrt 10.7%

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

   9. prod-diff 10.7%

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

5. Applied egg-rr 10.8%

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

6. Final simplification 10.8%

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

Alternative 7: 10.5% 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 7.1%

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

   1. acos-asin 7.1%

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

   2. sub-neg 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

3. Applied egg-rr 7.1%

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

   1. sub-neg 7.1%

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

5. Simplified 7.1%

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

   1. add-cube-cbrt 10.7%

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

   2. pow3 10.7%

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

7. Applied egg-rr 10.7%

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

8. Final simplification 10.7%

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

Alternative 8: 7.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (+ (+ (+ 2.0 (+ (+ 1.0 (acos (- 1.0 x))) -1.0)) -1.0) -1.0))

C

double code(double x) {
	return ((2.0 + ((1.0 + acos((1.0 - x))) + -1.0)) + -1.0) + -1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return ((2.0 + ((1.0 + math.acos((1.0 - x))) + -1.0)) + -1.0) + -1.0

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(2.0 + N[(N[(1.0 + N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $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. expm1-log1p-u 7.1%

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

   2. expm1-udef 7.1%

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

   3. log1p-udef 7.1%

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

   4. +-commutative 7.1%

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

   5. add-exp-log 7.1%

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

   6. +-commutative 7.1%

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

   7. associate-+l+ 7.1%

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

   8. metadata-eval 7.1%

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

5. Applied egg-rr 7.1%

   \[\leadsto \color{blue}{\left(\left(\cos^{-1} \left(1 - x\right) + 2\right) - 1\right)} - 1 \]
6. 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%

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

7. Applied egg-rr 7.1%

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

8. Final simplification 7.1%

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

Alternative 9: 7.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (+ (+ 2.0 (acos (- 1.0 x))) -2.0))

C

double code(double x) {
	return (2.0 + acos((1.0 - x))) + -2.0;
}

Fortran

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

Java

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

Python

def code(x):
	return (2.0 + math.acos((1.0 - x))) + -2.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. 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. expm1-log1p-u 7.1%

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

   2. expm1-udef 7.1%

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

   3. log1p-udef 7.1%

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

   4. +-commutative 7.1%

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

   5. add-exp-log 7.1%

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

   6. +-commutative 7.1%

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

   7. associate-+l+ 7.1%

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

   8. metadata-eval 7.1%

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

5. Applied egg-rr 7.1%

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

   1. associate--l- 7.1%

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

   2. metadata-eval 7.1%

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

   3. sub-neg 7.1%

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

   4. +-commutative 7.1%

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

   5. metadata-eval 7.1%

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

7. Applied egg-rr 7.1%

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

8. Final simplification 7.1%

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

Alternative 10: 7.0% accurate, 1.0× speedup

Math

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

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(Float64(1.0 + 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[(N[(1.0 + N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $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. Final simplification 7.1%

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

Alternative 11: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 2023216 
(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)))