bug323 (missed optimization)

Percentage Accurate: 6.7% → 10.4%

Time: 22.5s

Alternatives: 19

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.7% 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: 10.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot {\pi}^{2}\\ t_1 := \sin^{-1} \left(1 - x\right)\\ t_2 := {t_1}^{2}\\ t_3 := \mathsf{fma}\left(-t_1, t_1, t_2\right)\\ t_4 := t_0 - t_2\\ \frac{\frac{{t_4}^{3} + {t_3}^{3}}{{\left(t_0 - {\left({\left(\sqrt{t_1}\right)}^{2}\right)}^{2}\right)}^{2} + \left(t_3 \cdot t_3 - t_4 \cdot t_3\right)}}{t_1 + \pi \cdot 0.5} \end{array} \end{array} \]

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 2.0))
        (t_3 (fma (- t_1) t_1 t_2))
        (t_4 (- t_0 t_2)))
   (/
    (/
     (+ (pow t_4 3.0) (pow t_3 3.0))
     (+
      (pow (- t_0 (pow (pow (sqrt t_1) 2.0) 2.0)) 2.0)
      (- (* t_3 t_3) (* t_4 t_3))))
    (+ 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));
	double t_2 = pow(t_1, 2.0);
	double t_3 = fma(-t_1, t_1, t_2);
	double t_4 = t_0 - t_2;
	return ((pow(t_4, 3.0) + pow(t_3, 3.0)) / (pow((t_0 - pow(pow(sqrt(t_1), 2.0), 2.0)), 2.0) + ((t_3 * t_3) - (t_4 * t_3)))) / (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))
	t_2 = t_1 ^ 2.0
	t_3 = fma(Float64(-t_1), t_1, t_2)
	t_4 = Float64(t_0 - t_2)
	return Float64(Float64(Float64((t_4 ^ 3.0) + (t_3 ^ 3.0)) / Float64((Float64(t_0 - ((sqrt(t_1) ^ 2.0) ^ 2.0)) ^ 2.0) + Float64(Float64(t_3 * t_3) - Float64(t_4 * t_3)))) / 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]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, Block[{t$95$3 = N[((-t$95$1) * t$95$1 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 - t$95$2), $MachinePrecision]}, N[(N[(N[(N[Power[t$95$4, 3.0], $MachinePrecision] + N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(t$95$0 - N[Power[N[Power[N[Sqrt[t$95$1], $MachinePrecision], 2.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(t$95$4 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 6.3%

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

   1. acos-asin 6.3%

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

   2. flip-- 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

   5. div-inv 6.3%

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

   6. metadata-eval 6.3%

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

   7. div-inv 6.3%

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

   8. metadata-eval 6.3%

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

3. Applied egg-rr 6.3%

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

   1. add-cube-cbrt 9.7%

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

   2. pow3 9.7%

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

5. Applied egg-rr 9.7%

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

   1. prod-diff 9.7%

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

   2. rem-cube-cbrt 9.6%

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

   3. rem-cube-cbrt 9.7%

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

   4. fma-neg 9.7%

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

7. Applied egg-rr 9.7%

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

   1. add-sqr-sqrt 9.9%

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

   2. pow2 9.9%

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

9. Applied egg-rr 9.9%

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

10. Final simplification 9.9%

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

Alternative 2: 10.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot {\pi}^{2}\\ t_1 := \sin^{-1} \left(1 - x\right)\\ t_2 := {t_1}^{2}\\ t_3 := \mathsf{fma}\left(-t_1, t_1, t_2\right)\\ t_4 := t_0 - t_2\\ \frac{\frac{{t_4}^{3} + {t_3}^{3}}{{t_4}^{2} + \left(t_3 \cdot \mathsf{fma}\left(-{\left(\sqrt[3]{t_1}\right)}^{3}, t_1, t_2\right) + t_3 \cdot \left(t_2 - t_0\right)\right)}}{t_1 + \pi \cdot 0.5} \end{array} \end{array} \]

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 2.0))
        (t_3 (fma (- t_1) t_1 t_2))
        (t_4 (- t_0 t_2)))
   (/
    (/
     (+ (pow t_4 3.0) (pow t_3 3.0))
     (+
      (pow t_4 2.0)
      (+ (* t_3 (fma (- (pow (cbrt t_1) 3.0)) t_1 t_2)) (* t_3 (- t_2 t_0)))))
    (+ 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));
	double t_2 = pow(t_1, 2.0);
	double t_3 = fma(-t_1, t_1, t_2);
	double t_4 = t_0 - t_2;
	return ((pow(t_4, 3.0) + pow(t_3, 3.0)) / (pow(t_4, 2.0) + ((t_3 * fma(-pow(cbrt(t_1), 3.0), t_1, t_2)) + (t_3 * (t_2 - t_0))))) / (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))
	t_2 = t_1 ^ 2.0
	t_3 = fma(Float64(-t_1), t_1, t_2)
	t_4 = Float64(t_0 - t_2)
	return Float64(Float64(Float64((t_4 ^ 3.0) + (t_3 ^ 3.0)) / Float64((t_4 ^ 2.0) + Float64(Float64(t_3 * fma(Float64(-(cbrt(t_1) ^ 3.0)), t_1, t_2)) + Float64(t_3 * Float64(t_2 - t_0))))) / 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]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, Block[{t$95$3 = N[((-t$95$1) * t$95$1 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 - t$95$2), $MachinePrecision]}, N[(N[(N[(N[Power[t$95$4, 3.0], $MachinePrecision] + N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$4, 2.0], $MachinePrecision] + N[(N[(t$95$3 * N[((-N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]) * t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[(t$95$2 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 6.3%

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

   1. acos-asin 6.3%

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

   2. flip-- 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

   5. div-inv 6.3%

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

   6. metadata-eval 6.3%

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

   7. div-inv 6.3%

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

   8. metadata-eval 6.3%

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

3. Applied egg-rr 6.3%

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

   1. add-cube-cbrt 9.7%

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

   2. pow3 9.7%

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

5. Applied egg-rr 9.7%

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

   1. prod-diff 9.7%

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

   2. rem-cube-cbrt 9.6%

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

   3. rem-cube-cbrt 9.7%

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

   4. fma-neg 9.7%

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

7. Applied egg-rr 9.7%

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

   1. add-cube-cbrt 9.7%

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

   2. pow3 9.7%

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

9. Applied egg-rr 9.9%

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

10. Final simplification 9.9%

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

Alternative 3: 10.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot {\pi}^{2}\\ t_1 := \sin^{-1} \left(1 - x\right)\\ t_2 := {t_1}^{2}\\ t_3 := \mathsf{fma}\left(-t_1, t_1, t_2\right)\\ t_4 := t_0 - t_2\\ \frac{\frac{{t_4}^{3} + {t_3}^{3}}{\left(t_3 \cdot t_3 - t_4 \cdot t_3\right) + {\left(t_0 - {\left({\left(\sqrt[3]{t_1}\right)}^{3}\right)}^{2}\right)}^{2}}}{t_1 + \pi \cdot 0.5} \end{array} \end{array} \]

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 2.0))
        (t_3 (fma (- t_1) t_1 t_2))
        (t_4 (- t_0 t_2)))
   (/
    (/
     (+ (pow t_4 3.0) (pow t_3 3.0))
     (+
      (- (* t_3 t_3) (* t_4 t_3))
      (pow (- t_0 (pow (pow (cbrt t_1) 3.0) 2.0)) 2.0)))
    (+ 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));
	double t_2 = pow(t_1, 2.0);
	double t_3 = fma(-t_1, t_1, t_2);
	double t_4 = t_0 - t_2;
	return ((pow(t_4, 3.0) + pow(t_3, 3.0)) / (((t_3 * t_3) - (t_4 * t_3)) + pow((t_0 - pow(pow(cbrt(t_1), 3.0), 2.0)), 2.0))) / (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))
	t_2 = t_1 ^ 2.0
	t_3 = fma(Float64(-t_1), t_1, t_2)
	t_4 = Float64(t_0 - t_2)
	return Float64(Float64(Float64((t_4 ^ 3.0) + (t_3 ^ 3.0)) / Float64(Float64(Float64(t_3 * t_3) - Float64(t_4 * t_3)) + (Float64(t_0 - ((cbrt(t_1) ^ 3.0) ^ 2.0)) ^ 2.0))) / 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]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, Block[{t$95$3 = N[((-t$95$1) * t$95$1 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 - t$95$2), $MachinePrecision]}, N[(N[(N[(N[Power[t$95$4, 3.0], $MachinePrecision] + N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(t$95$4 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[Power[N[(t$95$0 - N[Power[N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 6.3%

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

   1. acos-asin 6.3%

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

   2. flip-- 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

   5. div-inv 6.3%

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

   6. metadata-eval 6.3%

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

   7. div-inv 6.3%

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

   8. metadata-eval 6.3%

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

3. Applied egg-rr 6.3%

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

   1. add-cube-cbrt 9.7%

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

   2. pow3 9.7%

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

5. Applied egg-rr 9.7%

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

   1. prod-diff 9.7%

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

   2. rem-cube-cbrt 9.6%

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

   3. rem-cube-cbrt 9.7%

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

   4. fma-neg 9.7%

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

7. Applied egg-rr 9.7%

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

   1. add-cube-cbrt 9.7%

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

   2. pow3 9.7%

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

9. Applied egg-rr 9.9%

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

10. Final simplification 9.9%

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

Alternative 4: 10.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := {t_0}^{2}\\ t_2 := \mathsf{fma}\left(-t_0, t_0, t_1\right)\\ t_3 := 0.25 \cdot {\pi}^{2} - t_1\\ \frac{\frac{{t_3}^{3} + {t_2}^{3}}{{t_3}^{2} + \left(t_2 \cdot t_2 + t_3 \cdot 0\right)}}{t_0 + \pi \cdot 0.5} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x)))
        (t_1 (pow t_0 2.0))
        (t_2 (fma (- t_0) t_0 t_1))
        (t_3 (- (* 0.25 (pow PI 2.0)) t_1)))
   (/
    (/
     (+ (pow t_3 3.0) (pow t_2 3.0))
     (+ (pow t_3 2.0) (+ (* t_2 t_2) (* t_3 0.0))))
    (+ t_0 (* PI 0.5)))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = pow(t_0, 2.0);
	double t_2 = fma(-t_0, t_0, t_1);
	double t_3 = (0.25 * pow(((double) M_PI), 2.0)) - t_1;
	return ((pow(t_3, 3.0) + pow(t_2, 3.0)) / (pow(t_3, 2.0) + ((t_2 * t_2) + (t_3 * 0.0)))) / (t_0 + (((double) M_PI) * 0.5));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = t_0 ^ 2.0
	t_2 = fma(Float64(-t_0), t_0, t_1)
	t_3 = Float64(Float64(0.25 * (pi ^ 2.0)) - t_1)
	return Float64(Float64(Float64((t_3 ^ 3.0) + (t_2 ^ 3.0)) / Float64((t_3 ^ 2.0) + Float64(Float64(t_2 * t_2) + Float64(t_3 * 0.0)))) / Float64(t_0 + Float64(pi * 0.5)))
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, 2.0], $MachinePrecision]}, Block[{t$95$2 = N[((-t$95$0) * t$95$0 + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]}, N[(N[(N[(N[Power[t$95$3, 3.0], $MachinePrecision] + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$3, 2.0], $MachinePrecision] + N[(N[(t$95$2 * t$95$2), $MachinePrecision] + N[(t$95$3 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 6.3%

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

   1. acos-asin 6.3%

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

   2. flip-- 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

   5. div-inv 6.3%

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

   6. metadata-eval 6.3%

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

   7. div-inv 6.3%

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

   8. metadata-eval 6.3%

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

3. Applied egg-rr 6.3%

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

   1. add-cube-cbrt 9.7%

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

   2. pow3 9.7%

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

5. Applied egg-rr 9.7%

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

   1. prod-diff 9.7%

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

   2. rem-cube-cbrt 9.6%

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

   3. rem-cube-cbrt 9.7%

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

   4. fma-neg 9.7%

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

7. Applied egg-rr 9.7%

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

8. Taylor expanded in x around 0 9.7%

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

   1. distribute-lft1-in 9.7%

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

   2. metadata-eval 9.7%

      \[\leadsto \frac{\frac{{\left(0.25 \cdot {\pi}^{2} - {\sin^{-1} \left(1 - x\right)}^{2}\right)}^{3} + {\left(\mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right), {\sin^{-1} \left(1 - x\right)}^{2}\right)\right)}^{3}}{{\left(0.25 \cdot {\pi}^{2} - {\sin^{-1} \left(1 - x\right)}^{2}\right)}^{2} + \left(\mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right), {\sin^{-1} \left(1 - x\right)}^{2}\right) \cdot \mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right), {\sin^{-1} \left(1 - x\right)}^{2}\right) - \left(0.25 \cdot {\pi}^{2} - {\sin^{-1} \left(1 - x\right)}^{2}\right) \cdot \left(\color{blue}{0} \cdot {\sin^{-1} \left(1 - x\right)}^{2}\right)\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   3. mul0-lft 9.7%

      \[\leadsto \frac{\frac{{\left(0.25 \cdot {\pi}^{2} - {\sin^{-1} \left(1 - x\right)}^{2}\right)}^{3} + {\left(\mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right), {\sin^{-1} \left(1 - x\right)}^{2}\right)\right)}^{3}}{{\left(0.25 \cdot {\pi}^{2} - {\sin^{-1} \left(1 - x\right)}^{2}\right)}^{2} + \left(\mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right), {\sin^{-1} \left(1 - x\right)}^{2}\right) \cdot \mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right), {\sin^{-1} \left(1 - x\right)}^{2}\right) - \left(0.25 \cdot {\pi}^{2} - {\sin^{-1} \left(1 - x\right)}^{2}\right) \cdot \color{blue}{0}\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

10. Simplified 9.7%

    \[\leadsto \frac{\frac{{\left(0.25 \cdot {\pi}^{2} - {\sin^{-1} \left(1 - x\right)}^{2}\right)}^{3} + {\left(\mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right), {\sin^{-1} \left(1 - x\right)}^{2}\right)\right)}^{3}}{{\left(0.25 \cdot {\pi}^{2} - {\sin^{-1} \left(1 - x\right)}^{2}\right)}^{2} + \left(\mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right), {\sin^{-1} \left(1 - x\right)}^{2}\right) \cdot \mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right), {\sin^{-1} \left(1 - x\right)}^{2}\right) - \left(0.25 \cdot {\pi}^{2} - {\sin^{-1} \left(1 - x\right)}^{2}\right) \cdot \color{blue}{0}\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

11. Final simplification 9.7%

    \[\leadsto \frac{\frac{{\left(0.25 \cdot {\pi}^{2} - {\sin^{-1} \left(1 - x\right)}^{2}\right)}^{3} + {\left(\mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right), {\sin^{-1} \left(1 - x\right)}^{2}\right)\right)}^{3}}{{\left(0.25 \cdot {\pi}^{2} - {\sin^{-1} \left(1 - x\right)}^{2}\right)}^{2} + \left(\mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right), {\sin^{-1} \left(1 - x\right)}^{2}\right) \cdot \mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right), {\sin^{-1} \left(1 - x\right)}^{2}\right) + \left(0.25 \cdot {\pi}^{2} - {\sin^{-1} \left(1 - x\right)}^{2}\right) \cdot 0\right)}}{\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5} \]

Alternative 5: 10.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = t_0 ^ 2.0
	return Float64(Float64(Float64(Float64(0.25 * (pi ^ 2.0)) - t_1) + fma(Float64(-t_0), t_0, t_1)) / Float64(t_0 + Float64(pi * 0.5)))
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, 2.0], $MachinePrecision]}, N[(N[(N[(N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] + N[((-t$95$0) * t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 6.3%

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

   1. acos-asin 6.3%

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

   2. flip-- 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

   5. div-inv 6.3%

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

   6. metadata-eval 6.3%

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

   7. div-inv 6.3%

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

   8. metadata-eval 6.3%

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

3. Applied egg-rr 6.3%

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

   1. add-cube-cbrt 9.7%

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

   2. pow3 9.7%

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

5. Applied egg-rr 9.7%

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

   1. add-sqr-sqrt 9.7%

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

   2. prod-diff 9.7%

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

7. Applied egg-rr 9.7%

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

8. Final simplification 9.7%

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

Alternative 6: 10.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

   1. acos-asin 6.3%

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

   2. flip-- 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

   5. div-inv 6.3%

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

   6. metadata-eval 6.3%

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

   7. div-inv 6.3%

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

   8. metadata-eval 6.3%

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

3. Applied egg-rr 6.3%

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

   1. flip-- 6.3%

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

   2. add-sqr-sqrt 9.7%

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

   3. prod-diff 9.7%

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

   4. add-sqr-sqrt 9.7%

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

   5. fma-neg 9.7%

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

   6. metadata-eval 9.7%

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

   7. div-inv 9.7%

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

   8. acos-asin 9.7%

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

   9. add-sqr-sqrt 9.7%

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

5. Applied egg-rr 9.7%

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

   1. add-sqr-sqrt 9.9%

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

   2. pow2 9.9%

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

7. Applied egg-rr 9.7%

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

8. Final simplification 9.7%

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

Alternative 7: 10.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	return (((Math.PI * 0.5) * (Math.PI * 0.5)) - (t_0 * Math.pow(Math.cbrt(t_0), 3.0))) / (t_0 + (Math.PI * 0.5));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

   1. acos-asin 6.3%

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

   2. flip-- 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

   5. div-inv 6.3%

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

   6. metadata-eval 6.3%

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

   7. div-inv 6.3%

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

   8. metadata-eval 6.3%

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

3. Applied egg-rr 6.3%

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

   1. add-cube-cbrt 9.7%

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

   2. pow3 9.7%

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

5. Applied egg-rr 9.7%

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

6. Final simplification 9.7%

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

Alternative 8: 10.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(1 - e^{\mathsf{log1p}\left({t_0}^{2}\right)}\right)}{t_0 + \pi \cdot 0.5} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/
    (+ (* (* PI 0.5) (* PI 0.5)) (- 1.0 (exp (log1p (pow t_0 2.0)))))
    (+ t_0 (* PI 0.5)))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return (((((double) M_PI) * 0.5) * (((double) M_PI) * 0.5)) + (1.0 - exp(log1p(pow(t_0, 2.0))))) / (t_0 + (((double) M_PI) * 0.5));
}

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	return (((Math.PI * 0.5) * (Math.PI * 0.5)) + (1.0 - Math.exp(Math.log1p(Math.pow(t_0, 2.0))))) / (t_0 + (Math.PI * 0.5));
}

Python

def code(x):
	t_0 = math.asin((1.0 - x))
	return (((math.pi * 0.5) * (math.pi * 0.5)) + (1.0 - math.exp(math.log1p(math.pow(t_0, 2.0))))) / (t_0 + (math.pi * 0.5))

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(Float64(Float64(Float64(pi * 0.5) * Float64(pi * 0.5)) + Float64(1.0 - exp(log1p((t_0 ^ 2.0))))) / Float64(t_0 + Float64(pi * 0.5)))
end

Wolfram

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

Derivation

1. Initial program 6.3%

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

   1. acos-asin 6.3%

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

   2. flip-- 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

   5. div-inv 6.3%

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

   6. metadata-eval 6.3%

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

   7. div-inv 6.3%

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

   8. metadata-eval 6.3%

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

3. Applied egg-rr 6.3%

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

   1. expm1-log1p-u 6.3%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   2. expm1-udef 9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{\left(e^{\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)\right)} - 1\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   3. pow2 9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \left(e^{\mathsf{log1p}\left(\color{blue}{{\sin^{-1} \left(1 - x\right)}^{2}}\right)} - 1\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

5. Applied egg-rr 9.7%

   \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{\left(e^{\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{2}\right)} - 1\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

6. Final simplification 9.7%

   \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(1 - e^{\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{2}\right)}\right)}{\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5} \]

Alternative 9: 10.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

   1. acos-asin 6.3%

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

   2. flip-- 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

   5. div-inv 6.3%

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

   6. metadata-eval 6.3%

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

   7. div-inv 6.3%

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

   8. metadata-eval 6.3%

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

3. Applied egg-rr 6.3%

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

   1. flip-- 6.3%

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

   2. add-sqr-sqrt 9.7%

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

   3. prod-diff 9.7%

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

   4. add-sqr-sqrt 9.7%

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

   5. fma-neg 9.7%

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

   6. metadata-eval 9.7%

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

   7. div-inv 9.7%

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

   8. acos-asin 9.7%

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

   9. add-sqr-sqrt 9.7%

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

5. Applied egg-rr 9.7%

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

6. Final simplification 9.7%

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

Alternative 10: 9.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;t_0 + 2 \cdot \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \log \left(\sqrt[3]{e^{t_0}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= x 5.6e-17)
     (+ t_0 (* 2.0 (asin (- 1.0 x))))
     (* 3.0 (log (cbrt (exp t_0)))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (x <= 5.6e-17) {
		tmp = t_0 + (2.0 * asin((1.0 - x)));
	} else {
		tmp = 3.0 * log(cbrt(exp(t_0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (x <= 5.6e-17) {
		tmp = t_0 + (2.0 * Math.asin((1.0 - x)));
	} else {
		tmp = 3.0 * Math.log(Math.cbrt(Math.exp(t_0)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = Float64(t_0 + Float64(2.0 * asin(Float64(1.0 - x))));
	else
		tmp = Float64(3.0 * log(cbrt(exp(t_0))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.6e-17], N[(t$95$0 + N[(2.0 * N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[Log[N[Power[N[Exp[t$95$0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   5. Simplified 3.9%

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

      1. add-sqr-sqrt 7.6%

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

      2. pow2 7.6%

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

   7. Applied egg-rr 7.4%

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

      1. unpow2 7.4%

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

      2. prod-diff 7.4%

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

      3. add-sqr-sqrt 7.5%

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

      4. fma-neg 7.5%

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

      5. metadata-eval 7.5%

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

      6. div-inv 7.5%

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

      7. acos-asin 7.5%

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

      8. add-sqr-sqrt 7.4%

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

      9. fma-udef 7.4%

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

   9. Applied egg-rr 6.5%

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

       1. count-2 6.5%

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

   11. Simplified 6.5%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 55.1%

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

      1. acos-asin 55.1%

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

      2. flip-- 55.1%

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

      3. div-inv 55.1%

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

      4. metadata-eval 55.1%

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

      5. div-inv 55.1%

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

      6. metadata-eval 55.1%

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

      7. div-inv 55.1%

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

      8. metadata-eval 55.1%

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

   3. Applied egg-rr 55.1%

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

      1. flip-- 55.1%

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

      2. metadata-eval 55.1%

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

      3. div-inv 55.1%

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

      4. acos-asin 55.1%

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

      5. add-log-exp 55.1%

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

      6. add-cube-cbrt 55.2%

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

      7. log-prod 55.2%

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

      8. pow2 55.2%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right) \]

   5. Applied egg-rr 55.2%

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

      1. log-pow 55.2%

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

      2. distribute-lft1-in 55.2%

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

      3. metadata-eval 55.2%

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

   7. Simplified 55.2%

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

4. Final simplification 8.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right) + 2 \cdot \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)\\ \end{array} \]

Alternative 11: 10.2% 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 6.3%

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

   1. acos-asin 6.3%

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

   2. sub-neg 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

3. Applied egg-rr 6.3%

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

   1. sub-neg 6.3%

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

5. Simplified 6.3%

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

   1. add-cube-cbrt 9.7%

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

   2. pow3 9.7%

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

7. Applied egg-rr 9.6%

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

8. Final simplification 9.6%

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

Alternative 12: 10.2% 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 6.3%

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

   1. acos-asin 6.3%

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

   2. sub-neg 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

3. Applied egg-rr 6.3%

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

   1. sub-neg 6.3%

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

5. Simplified 6.3%

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

   1. add-sqr-sqrt 9.9%

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

   2. pow2 9.9%

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

7. Applied egg-rr 9.7%

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

8. Final simplification 9.7%

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

Alternative 13: 9.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;t_0 + 2 \cdot \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log \left(e^{t_0}\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= x 5.6e-17)
     (+ t_0 (* 2.0 (asin (- 1.0 x))))
     (+ 1.0 (+ (log (exp t_0)) -1.0)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (x <= 5.6e-17) {
		tmp = t_0 + (2.0 * asin((1.0 - x)));
	} else {
		tmp = 1.0 + (log(exp(t_0)) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = acos((1.0d0 - x))
    if (x <= 5.6d-17) then
        tmp = t_0 + (2.0d0 * asin((1.0d0 - x)))
    else
        tmp = 1.0d0 + (log(exp(t_0)) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (x <= 5.6e-17) {
		tmp = t_0 + (2.0 * Math.asin((1.0 - x)));
	} else {
		tmp = 1.0 + (Math.log(Math.exp(t_0)) + -1.0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if x <= 5.6e-17:
		tmp = t_0 + (2.0 * math.asin((1.0 - x)))
	else:
		tmp = 1.0 + (math.log(math.exp(t_0)) + -1.0)
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = Float64(t_0 + Float64(2.0 * asin(Float64(1.0 - x))));
	else
		tmp = Float64(1.0 + Float64(log(exp(t_0)) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (x <= 5.6e-17)
		tmp = t_0 + (2.0 * asin((1.0 - x)));
	else
		tmp = 1.0 + (log(exp(t_0)) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.6e-17], N[(t$95$0 + N[(2.0 * N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   5. Simplified 3.9%

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

      1. add-sqr-sqrt 7.6%

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

      2. pow2 7.6%

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

   7. Applied egg-rr 7.4%

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

      1. unpow2 7.4%

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

      2. prod-diff 7.4%

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

      3. add-sqr-sqrt 7.5%

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

      4. fma-neg 7.5%

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

      5. metadata-eval 7.5%

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

      6. div-inv 7.5%

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

      7. acos-asin 7.5%

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

      8. add-sqr-sqrt 7.4%

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

      9. fma-udef 7.4%

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

   9. Applied egg-rr 6.5%

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

       1. count-2 6.5%

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

   11. Simplified 6.5%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 55.1%

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

      1. acos-asin 55.1%

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

      2. flip-- 55.1%

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

      3. div-inv 55.1%

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

      4. metadata-eval 55.1%

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

      5. div-inv 55.1%

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

      6. metadata-eval 55.1%

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

      7. div-inv 55.1%

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

      8. metadata-eval 55.1%

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

   3. Applied egg-rr 55.1%

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

      1. flip-- 55.1%

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

      2. metadata-eval 55.1%

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

      3. div-inv 55.1%

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

      4. acos-asin 55.1%

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

      5. expm1-log1p-u 55.1%

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

      6. expm1-udef 55.1%

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

      7. log1p-udef 55.1%

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

      8. add-exp-log 55.1%

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

      9. associate--l+ 55.1%

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

      10. +-commutative 55.1%

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

      11. sub-neg 55.1%

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

      12. metadata-eval 55.1%

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

   5. Applied egg-rr 55.1%

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

      1. add-log-exp 55.1%

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

   7. Applied egg-rr 55.1%

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

4. Final simplification 8.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right) + 2 \cdot \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log \left(e^{\cos^{-1} \left(1 - x\right)}\right) + -1\right)\\ \end{array} \]

Alternative 14: 9.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;t_0 + 2 \cdot \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \log \left(e^{t_0}\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= x 5.6e-17)
     (+ t_0 (* 2.0 (asin (- 1.0 x))))
     (+ (+ 1.0 (log (exp t_0))) -1.0))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (x <= 5.6e-17) {
		tmp = t_0 + (2.0 * asin((1.0 - x)));
	} else {
		tmp = (1.0 + log(exp(t_0))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = acos((1.0d0 - x))
    if (x <= 5.6d-17) then
        tmp = t_0 + (2.0d0 * asin((1.0d0 - x)))
    else
        tmp = (1.0d0 + log(exp(t_0))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (x <= 5.6e-17) {
		tmp = t_0 + (2.0 * Math.asin((1.0 - x)));
	} else {
		tmp = (1.0 + Math.log(Math.exp(t_0))) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if x <= 5.6e-17:
		tmp = t_0 + (2.0 * math.asin((1.0 - x)))
	else:
		tmp = (1.0 + math.log(math.exp(t_0))) + -1.0
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = Float64(t_0 + Float64(2.0 * asin(Float64(1.0 - x))));
	else
		tmp = Float64(Float64(1.0 + log(exp(t_0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (x <= 5.6e-17)
		tmp = t_0 + (2.0 * asin((1.0 - x)));
	else
		tmp = (1.0 + log(exp(t_0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.6e-17], N[(t$95$0 + N[(2.0 * N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   5. Simplified 3.9%

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

      1. add-sqr-sqrt 7.6%

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

      2. pow2 7.6%

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

   7. Applied egg-rr 7.4%

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

      1. unpow2 7.4%

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

      2. prod-diff 7.4%

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

      3. add-sqr-sqrt 7.5%

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

      4. fma-neg 7.5%

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

      5. metadata-eval 7.5%

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

      6. div-inv 7.5%

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

      7. acos-asin 7.5%

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

      8. add-sqr-sqrt 7.4%

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

      9. fma-udef 7.4%

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

   9. Applied egg-rr 6.5%

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

       1. count-2 6.5%

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

   11. Simplified 6.5%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 55.1%

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

      1. expm1-log1p-u 55.1%

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

      2. expm1-udef 55.1%

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

      3. log1p-udef 55.1%

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

      4. add-exp-log 55.1%

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

   3. Applied egg-rr 55.1%

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

      1. add-log-exp 55.1%

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

   5. Applied egg-rr 55.1%

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

4. Final simplification 8.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right) + 2 \cdot \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \log \left(e^{\cos^{-1} \left(1 - x\right)}\right)\right) + -1\\ \end{array} \]

Alternative 15: 9.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;t_0 + 2 \cdot \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{t_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= x 5.6e-17) (+ t_0 (* 2.0 (asin (- 1.0 x)))) (log (exp t_0)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (x <= 5.6e-17) {
		tmp = t_0 + (2.0 * asin((1.0 - x)));
	} else {
		tmp = log(exp(t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = acos((1.0d0 - x))
    if (x <= 5.6d-17) then
        tmp = t_0 + (2.0d0 * asin((1.0d0 - x)))
    else
        tmp = log(exp(t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (x <= 5.6e-17) {
		tmp = t_0 + (2.0 * Math.asin((1.0 - x)));
	} else {
		tmp = Math.log(Math.exp(t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if x <= 5.6e-17:
		tmp = t_0 + (2.0 * math.asin((1.0 - x)))
	else:
		tmp = math.log(math.exp(t_0))
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = Float64(t_0 + Float64(2.0 * asin(Float64(1.0 - x))));
	else
		tmp = log(exp(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (x <= 5.6e-17)
		tmp = t_0 + (2.0 * asin((1.0 - x)));
	else
		tmp = log(exp(t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.6e-17], N[(t$95$0 + N[(2.0 * N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   5. Simplified 3.9%

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

      1. add-sqr-sqrt 7.6%

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

      2. pow2 7.6%

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

   7. Applied egg-rr 7.4%

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

      1. unpow2 7.4%

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

      2. prod-diff 7.4%

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

      3. add-sqr-sqrt 7.5%

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

      4. fma-neg 7.5%

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

      5. metadata-eval 7.5%

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

      6. div-inv 7.5%

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

      7. acos-asin 7.5%

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

      8. add-sqr-sqrt 7.4%

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

      9. fma-udef 7.4%

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

   9. Applied egg-rr 6.5%

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

       1. count-2 6.5%

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

   11. Simplified 6.5%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 55.1%

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

      1. add-log-exp 55.1%

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

   3. Applied egg-rr 55.1%

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

4. Final simplification 8.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right) + 2 \cdot \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)\\ \end{array} \]

Alternative 16: 9.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right) + 2 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (if (<= x 5.6e-17) (+ (acos (- 1.0 x)) (* 2.0 t_0)) (- (* PI 0.5) t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double tmp;
	if (x <= 5.6e-17) {
		tmp = acos((1.0 - x)) + (2.0 * t_0);
	} else {
		tmp = (((double) M_PI) * 0.5) - t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	double tmp;
	if (x <= 5.6e-17) {
		tmp = Math.acos((1.0 - x)) + (2.0 * t_0);
	} else {
		tmp = (Math.PI * 0.5) - t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.asin((1.0 - x))
	tmp = 0
	if x <= 5.6e-17:
		tmp = math.acos((1.0 - x)) + (2.0 * t_0)
	else:
		tmp = (math.pi * 0.5) - t_0
	return tmp

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = Float64(acos(Float64(1.0 - x)) + Float64(2.0 * t_0));
	else
		tmp = Float64(Float64(pi * 0.5) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = asin((1.0 - x));
	tmp = 0.0;
	if (x <= 5.6e-17)
		tmp = acos((1.0 - x)) + (2.0 * t_0);
	else
		tmp = (pi * 0.5) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.6e-17], N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   5. Simplified 3.9%

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

      1. add-sqr-sqrt 7.6%

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

      2. pow2 7.6%

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

   7. Applied egg-rr 7.4%

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

      1. unpow2 7.4%

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

      2. prod-diff 7.4%

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

      3. add-sqr-sqrt 7.5%

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

      4. fma-neg 7.5%

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

      5. metadata-eval 7.5%

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

      6. div-inv 7.5%

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

      7. acos-asin 7.5%

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

      8. add-sqr-sqrt 7.4%

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

      9. fma-udef 7.4%

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

   9. Applied egg-rr 6.5%

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

       1. count-2 6.5%

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

   11. Simplified 6.5%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 55.1%

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

      1. acos-asin 55.1%

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

      2. sub-neg 55.1%

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

      3. div-inv 55.1%

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

      4. metadata-eval 55.1%

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

   3. Applied egg-rr 55.1%

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

      1. sub-neg 55.1%

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

   5. Simplified 55.1%

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

4. Final simplification 8.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right) + 2 \cdot \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)\\ \end{array} \]

Alternative 17: 6.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (((double) M_PI) * 0.5) - asin((1.0 - x));
}

Java

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

Python

def code(x):
	return (math.pi * 0.5) - math.asin((1.0 - x))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

   1. acos-asin 6.3%

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

   2. sub-neg 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

3. Applied egg-rr 6.3%

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

   1. sub-neg 6.3%

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

5. Simplified 6.3%

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

6. Final simplification 6.3%

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

Alternative 18: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

   1. acos-asin 6.3%

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

   2. flip-- 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

   5. div-inv 6.3%

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

   6. metadata-eval 6.3%

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

   7. div-inv 6.3%

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

   8. metadata-eval 6.3%

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

3. Applied egg-rr 6.3%

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

   1. flip-- 6.3%

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

   2. metadata-eval 6.3%

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

   3. div-inv 6.3%

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

   4. acos-asin 6.3%

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

   5. expm1-log1p-u 6.3%

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

   6. expm1-udef 6.3%

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

   7. log1p-udef 6.3%

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

   8. add-exp-log 6.3%

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

   9. associate--l+ 6.3%

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

   10. +-commutative 6.3%

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

   11. sub-neg 6.3%

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

   12. metadata-eval 6.3%

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

5. Applied egg-rr 6.3%

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

6. Final simplification 6.3%

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

Alternative 19: 6.7% 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 6.3%

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

2. Final simplification 6.3%

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

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 2023200 
(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)))