bug323 (missed optimization)

Percentage Accurate: 6.7% → 10.2%

Time: 43.4s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 10.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \mathsf{fma}\left(0.25, {\pi}^{2}, {t_0}^{2}\right)\\ t_2 := \mathsf{fma}\left(\pi, 0.5, t_0\right)\\ t_3 := {t_2}^{2}\\ \mathsf{fma}\left(\frac{0.0625 \cdot {\pi}^{4}}{\left(t_2 \cdot t_1\right) \cdot t_3}, t_3, \frac{{t_0}^{4}}{t_3} \cdot \frac{-t_2}{t_1}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x)))
        (t_1 (fma 0.25 (pow PI 2.0) (pow t_0 2.0)))
        (t_2 (fma PI 0.5 t_0))
        (t_3 (pow t_2 2.0)))
   (fma
    (/ (* 0.0625 (pow PI 4.0)) (* (* t_2 t_1) t_3))
    t_3
    (* (/ (pow t_0 4.0) t_3) (/ (- t_2) t_1)))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = fma(0.25, pow(((double) M_PI), 2.0), pow(t_0, 2.0));
	double t_2 = fma(((double) M_PI), 0.5, t_0);
	double t_3 = pow(t_2, 2.0);
	return fma(((0.0625 * pow(((double) M_PI), 4.0)) / ((t_2 * t_1) * t_3)), t_3, ((pow(t_0, 4.0) / t_3) * (-t_2 / t_1)));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = fma(0.25, (pi ^ 2.0), (t_0 ^ 2.0))
	t_2 = fma(pi, 0.5, t_0)
	t_3 = t_2 ^ 2.0
	return fma(Float64(Float64(0.0625 * (pi ^ 4.0)) / Float64(Float64(t_2 * t_1) * t_3)), t_3, Float64(Float64((t_0 ^ 4.0) / t_3) * Float64(Float64(-t_2) / t_1)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * 0.5 + t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 2.0], $MachinePrecision]}, N[(N[(N[(0.0625 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 * t$95$1), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * t$95$3 + N[(N[(N[Power[t$95$0, 4.0], $MachinePrecision] / t$95$3), $MachinePrecision] * N[((-t$95$2) / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 8.0%

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

   1. acos-asin 8.0%

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

   2. flip-- 8.0%

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

   3. div-sub 8.0%

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

   4. flip-- 8.0%

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

3. Applied egg-rr 8.0%

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

5. Applied egg-rr 11.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{{\pi}^{4} \cdot 0.0625}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}}}{\mathsf{fma}\left({\pi}^{2} \cdot 0.25, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot {\sin^{-1} \left(1 - x\right)}^{2}\right)}, {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}, -\frac{{\sin^{-1} \left(1 - x\right)}^{4}}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot \frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left({\pi}^{2}, 0.25, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right)} \]
6. Step-by-step derivation

   1. add-sqr-sqrt_binary64 11.4%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\frac{\frac{{\pi}^{4} \cdot 0.0625}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}}}{\mathsf{fma}\left({\pi}^{2} \cdot 0.25, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot {\sin^{-1} \left(1 - x\right)}^{2}\right)}, {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}, -\frac{{\sin^{-1} \left(1 - x\right)}^{4}}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot \frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left({\pi}^{2}, 0.25, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{{\pi}^{4} \cdot 0.0625}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}}}{\mathsf{fma}\left({\pi}^{2} \cdot 0.25, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot {\sin^{-1} \left(1 - x\right)}^{2}\right)}, {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}, -\frac{{\sin^{-1} \left(1 - x\right)}^{4}}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot \frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left({\pi}^{2}, 0.25, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right)}} \]

7. Applied rewrite-once 11.4%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\frac{\frac{{\pi}^{4} \cdot 0.0625}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}}}{\mathsf{fma}\left({\pi}^{2} \cdot 0.25, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot {\sin^{-1} \left(1 - x\right)}^{2}\right)}, {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}, -\frac{{\sin^{-1} \left(1 - x\right)}^{4}}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot \frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left({\pi}^{2}, 0.25, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{{\pi}^{4} \cdot 0.0625}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}}}{\mathsf{fma}\left({\pi}^{2} \cdot 0.25, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot {\sin^{-1} \left(1 - x\right)}^{2}\right)}, {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}, -\frac{{\sin^{-1} \left(1 - x\right)}^{4}}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot \frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left({\pi}^{2}, 0.25, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right)}} \]
8. Step-by-step derivation

   1. rem-square-sqrt 11.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{{\pi}^{4} \cdot 0.0625}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}}}{\mathsf{fma}\left({\pi}^{2} \cdot 0.25, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot {\sin^{-1} \left(1 - x\right)}^{2}\right)}, {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}, -\frac{{\sin^{-1} \left(1 - x\right)}^{4}}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot \frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left({\pi}^{2}, 0.25, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right)} \]

9. Simplified 11.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.0625 \cdot {\pi}^{4}}{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot \mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)\right) \cdot {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}}, {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}, \frac{{\sin^{-1} \left(1 - x\right)}^{4}}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot \left(-\frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right)\right)} \]

10. Final simplification 11.4%

    \[\leadsto \mathsf{fma}\left(\frac{0.0625 \cdot {\pi}^{4}}{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot \mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)\right) \cdot {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}}, {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}, \frac{{\sin^{-1} \left(1 - x\right)}^{4}}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot \frac{-\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right) \]

Alternative 2: 10.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \mathsf{fma}\left(\pi, 0.5, t_0\right)\\ t_2 := \mathsf{fma}\left(0.25, {\pi}^{2}, {t_0}^{2}\right)\\ \mathsf{fma}\left({\pi}^{4}, \frac{\frac{0.0625}{t_1}}{t_2}, \left({t_0}^{4} \cdot \frac{-t_1}{t_2}\right) \cdot {t_1}^{-2}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x)))
        (t_1 (fma PI 0.5 t_0))
        (t_2 (fma 0.25 (pow PI 2.0) (pow t_0 2.0))))
   (fma
    (pow PI 4.0)
    (/ (/ 0.0625 t_1) t_2)
    (* (* (pow t_0 4.0) (/ (- t_1) t_2)) (pow t_1 -2.0)))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = fma(((double) M_PI), 0.5, t_0);
	double t_2 = fma(0.25, pow(((double) M_PI), 2.0), pow(t_0, 2.0));
	return fma(pow(((double) M_PI), 4.0), ((0.0625 / t_1) / t_2), ((pow(t_0, 4.0) * (-t_1 / t_2)) * pow(t_1, -2.0)));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = fma(pi, 0.5, t_0)
	t_2 = fma(0.25, (pi ^ 2.0), (t_0 ^ 2.0))
	return fma((pi ^ 4.0), Float64(Float64(0.0625 / t_1) / t_2), Float64(Float64((t_0 ^ 4.0) * Float64(Float64(-t_1) / t_2)) * (t_1 ^ -2.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(Pi * 0.5 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[Power[Pi, 4.0], $MachinePrecision] * N[(N[(0.0625 / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(N[(N[Power[t$95$0, 4.0], $MachinePrecision] * N[((-t$95$1) / t$95$2), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$1, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 8.0%

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

   1. acos-asin 8.0%

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

   2. flip-- 8.0%

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

   3. div-sub 8.0%

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

   4. flip-- 8.0%

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

3. Applied egg-rr 8.0%

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

5. Applied egg-rr 11.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{{\pi}^{4} \cdot 0.0625}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}}}{\mathsf{fma}\left({\pi}^{2} \cdot 0.25, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot {\sin^{-1} \left(1 - x\right)}^{2}\right)}, {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}, -\frac{{\sin^{-1} \left(1 - x\right)}^{4}}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot \frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left({\pi}^{2}, 0.25, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right)} \]
6. Step-by-step derivation

   1. add-sqr-sqrt_binary64 11.4%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\frac{\frac{{\pi}^{4} \cdot 0.0625}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}}}{\mathsf{fma}\left({\pi}^{2} \cdot 0.25, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot {\sin^{-1} \left(1 - x\right)}^{2}\right)}, {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}, -\frac{{\sin^{-1} \left(1 - x\right)}^{4}}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot \frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left({\pi}^{2}, 0.25, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{{\pi}^{4} \cdot 0.0625}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}}}{\mathsf{fma}\left({\pi}^{2} \cdot 0.25, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot {\sin^{-1} \left(1 - x\right)}^{2}\right)}, {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}, -\frac{{\sin^{-1} \left(1 - x\right)}^{4}}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot \frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left({\pi}^{2}, 0.25, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right)}} \]

7. Applied rewrite-once 11.4%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\frac{\frac{{\pi}^{4} \cdot 0.0625}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}}}{\mathsf{fma}\left({\pi}^{2} \cdot 0.25, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot {\sin^{-1} \left(1 - x\right)}^{2}\right)}, {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}, -\frac{{\sin^{-1} \left(1 - x\right)}^{4}}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot \frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left({\pi}^{2}, 0.25, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{{\pi}^{4} \cdot 0.0625}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}}}{\mathsf{fma}\left({\pi}^{2} \cdot 0.25, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot {\sin^{-1} \left(1 - x\right)}^{2}\right)}, {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}, -\frac{{\sin^{-1} \left(1 - x\right)}^{4}}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot \frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left({\pi}^{2}, 0.25, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right)}} \]
8. Step-by-step derivation

   1. rem-square-sqrt 11.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{{\pi}^{4} \cdot 0.0625}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}}}{\mathsf{fma}\left({\pi}^{2} \cdot 0.25, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot {\sin^{-1} \left(1 - x\right)}^{2}\right)}, {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}, -\frac{{\sin^{-1} \left(1 - x\right)}^{4}}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot \frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left({\pi}^{2}, 0.25, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right)} \]

9. Simplified 11.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.0625 \cdot {\pi}^{4}}{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot \mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)\right) \cdot {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}}, {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}, \frac{{\sin^{-1} \left(1 - x\right)}^{4}}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot \left(-\frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right)\right)} \]
10. Step-by-step derivation

    1. fma-udef 8.0%

       \[\leadsto \color{blue}{\frac{0.0625 \cdot {\pi}^{4}}{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot \mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)\right) \cdot {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2} + \frac{{\sin^{-1} \left(1 - x\right)}^{4}}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot \left(-\frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right)} \]

    2. +-commutative 8.0%

       \[\leadsto \color{blue}{\frac{{\sin^{-1} \left(1 - x\right)}^{4}}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot \left(-\frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right) + \frac{0.0625 \cdot {\pi}^{4}}{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right) \cdot \mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)\right) \cdot {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \cdot {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}} \]

11. Applied egg-rr 8.0%

    \[\leadsto \color{blue}{{\sin^{-1} \left(1 - x\right)}^{4} \cdot \left(\frac{-\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)} \cdot {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{-2}\right) + {\pi}^{4} \cdot \frac{\frac{0.0625}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}}{\mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)}} \]
12. Step-by-step derivation

    1. +-commutative 8.0%

       \[\leadsto \color{blue}{{\pi}^{4} \cdot \frac{\frac{0.0625}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}}{\mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)} + {\sin^{-1} \left(1 - x\right)}^{4} \cdot \left(\frac{-\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)} \cdot {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{-2}\right)} \]

    2. fma-def 11.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left({\pi}^{4}, \frac{\frac{0.0625}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}}{\mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)}, {\sin^{-1} \left(1 - x\right)}^{4} \cdot \left(\frac{-\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)} \cdot {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{-2}\right)\right)} \]

    3. associate-*r* 11.4%

       \[\leadsto \mathsf{fma}\left({\pi}^{4}, \frac{\frac{0.0625}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}}{\mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)}, \color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{4} \cdot \frac{-\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right) \cdot {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{-2}}\right) \]

13. Simplified 11.4%

    \[\leadsto \color{blue}{\mathsf{fma}\left({\pi}^{4}, \frac{\frac{0.0625}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}}{\mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)}, \left({\sin^{-1} \left(1 - x\right)}^{4} \cdot \frac{-\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right) \cdot {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{-2}\right)} \]

14. Final simplification 11.4%

    \[\leadsto \mathsf{fma}\left({\pi}^{4}, \frac{\frac{0.0625}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}}{\mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)}, \left({\sin^{-1} \left(1 - x\right)}^{4} \cdot \frac{-\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}{\mathsf{fma}\left(0.25, {\pi}^{2}, {\sin^{-1} \left(1 - x\right)}^{2}\right)}\right) \cdot {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{-2}\right) \]

Alternative 3: 10.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

   1. acos-asin 8.0%

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

   2. sub-neg 8.0%

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

   3. +-commutative 8.0%

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

   4. flip-+ 8.0%

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

   5. sqr-neg 8.0%

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

   6. pow2 8.0%

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

   7. pow2 8.0%

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

   8. div-inv 8.0%

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

   9. metadata-eval 8.0%

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

   10. div-inv 8.0%

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

   11. metadata-eval 8.0%

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

3. Applied egg-rr 8.0%

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

   1. sub-neg 8.0%

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

   2. distribute-rgt-neg-in 8.0%

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

   3. metadata-eval 8.0%

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

5. Applied egg-rr 8.0%

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

   1. +-commutative 8.0%

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

   2. sub-neg 8.0%

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

7. Simplified 8.0%

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

   1. unpow2 8.0%

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

   2. fma-neg 11.4%

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

   3. unpow-prod-down 11.4%

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

   4. distribute-rgt-neg-in 11.4%

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

   5. metadata-eval 11.4%

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

   6. metadata-eval 11.4%

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

9. Applied egg-rr 11.4%

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

10. Final simplification 11.4%

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

Alternative 4: 10.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return 1.0 / ((2.0 / -t_0) / fma(((double) M_PI), (-1.0 / t_0), 2.0));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(1.0 / Float64(Float64(2.0 / Float64(-t_0)) / fma(pi, Float64(-1.0 / t_0), 2.0)))
end

Wolfram

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

Derivation

1. Initial program 8.0%

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

   1. acos-asin 8.0%

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

   2. sub-neg 8.0%

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

   3. div-inv 8.0%

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

   4. metadata-eval 8.0%

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

3. Applied egg-rr 8.0%

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

   1. sub-neg 8.0%

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

5. Simplified 8.0%

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

   1. /-rgt-identity 8.0%

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

   2. clear-num 8.0%

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

7. Applied egg-rr 8.0%

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

   1. metadata-eval 8.0%

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

   2. div-inv 8.0%

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

   3. frac-2neg 8.0%

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

   4. metadata-eval 8.0%

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

   5. frac-sub 7.9%

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

   6. clear-num 8.0%

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

   7. frac-2neg 8.0%

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

   8. metadata-eval 8.0%

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

   9. distribute-neg-frac 8.0%

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

   10. metadata-eval 8.0%

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

   11. un-div-inv 8.0%

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

   12. metadata-eval 8.0%

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

   13. fma-neg 11.3%

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

9. Applied egg-rr 11.3%

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

10. Final simplification 11.3%

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

Alternative 5: 10.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

   1. acos-asin 8.0%

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

   2. sub-neg 8.0%

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

   3. div-inv 8.0%

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

   4. metadata-eval 8.0%

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

3. Applied egg-rr 8.0%

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

   1. sub-neg 8.0%

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

5. Simplified 8.0%

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

   1. /-rgt-identity 8.0%

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

   2. clear-num 8.0%

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

7. Applied egg-rr 8.0%

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

   1. metadata-eval 8.0%

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

   2. div-inv 8.0%

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

   3. frac-2neg 8.0%

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

   4. metadata-eval 8.0%

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

   5. frac-sub 7.9%

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

   6. div-inv 7.9%

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

   7. metadata-eval 7.9%

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

   8. fma-neg 11.3%

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

   9. distribute-neg-frac 11.3%

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

   10. metadata-eval 11.3%

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

   11. metadata-eval 11.3%

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

   12. frac-2neg 11.3%

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

   13. metadata-eval 11.3%

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

   14. distribute-neg-frac 11.3%

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

   15. metadata-eval 11.3%

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

9. Applied egg-rr 11.3%

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

    1. associate-/r/ 11.3%

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

    2. metadata-eval 11.3%

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

    3. mul-1-neg 11.3%

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

    4. associate-*r* 11.3%

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

    5. metadata-eval 11.3%

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

11. Simplified 11.3%

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

12. Final simplification 11.3%

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

Alternative 6: 6.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ 1.0 (log (exp (acos (- 1.0 x)))))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 / log(exp(acos(Float64(1.0 - x))))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

   1. acos-asin 8.0%

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

   2. flip3-- 8.0%

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

   3. clear-num 8.0%

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

   4. clear-num 8.0%

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

   5. flip3-- 8.0%

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

   6. acos-asin 8.0%

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

3. Applied egg-rr 8.0%

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

   1. add-log-exp_binary64 8.0%

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

5. Applied rewrite-once 8.0%

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

6. Final simplification 8.0%

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

Alternative 7: 6.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

   1. add-log-exp_binary64 8.0%

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

3. Applied rewrite-once 8.0%

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

4. Final simplification 8.0%

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

Alternative 8: 6.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

   1. acos-asin 8.0%

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

   2. flip3-- 8.0%

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

   3. clear-num 8.0%

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

   4. clear-num 8.0%

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

   5. flip3-- 8.0%

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

   6. acos-asin 8.0%

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

3. Applied egg-rr 8.0%

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

   1. acos-asin 8.0%

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

   2. sub-neg 8.0%

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

   3. div-inv 8.0%

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

   4. metadata-eval 8.0%

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

5. Applied egg-rr 8.0%

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

   1. sub-neg 8.0%

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

7. Simplified 8.0%

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

8. Final simplification 8.0%

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

Alternative 9: 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 8.0%

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

   1. acos-asin 8.0%

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

   2. sub-neg 8.0%

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

   3. div-inv 8.0%

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

   4. metadata-eval 8.0%

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

3. Applied egg-rr 8.0%

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

   1. sub-neg 8.0%

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

5. Simplified 8.0%

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

6. Final simplification 8.0%

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

Alternative 10: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

   1. acos-asin 8.0%

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

   2. flip3-- 8.0%

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

   3. clear-num 8.0%

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

   4. clear-num 8.0%

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

   5. flip3-- 8.0%

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

   6. acos-asin 8.0%

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

3. Applied egg-rr 8.0%

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

4. Final simplification 8.0%

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

Alternative 11: 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 8.0%

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

2. Final simplification 8.0%

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

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 2023297 
(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)))