Ian Simplification

Percentage Accurate: 6.9% → 8.4%

Time: 17.8s

Alternatives: 5

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 8.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\\ t_1 := \sin^{-1} t\_0\\ t_2 := \mathsf{fma}\left(t\_1, \mathsf{fma}\left(t\_1, 4, \pi\right), \pi \cdot \left(\pi \cdot 0.25\right)\right)\\ t_3 := \frac{0.125 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)}{t\_2}\\ t_4 := \cos^{-1} t\_0\\ t_5 := \mathsf{fma}\left(t\_4, 2, \pi\right)\\ t_6 := {\left(\mathsf{fma}\left(-4, {t\_4}^{2}, \pi \cdot \pi\right)\right)}^{3}\\ t_7 := \frac{t\_6}{t\_2 \cdot {t\_5}^{3}}\\ t_8 := \mathsf{fma}\left(t\_7, \frac{1}{t\_2} \cdot \mathsf{fma}\left(\pi, \pi \cdot \left(0.125 \cdot \pi\right), t\_6 \cdot {t\_5}^{-3}\right), {t\_3}^{2}\right)\\ \frac{{t\_3}^{3}}{t\_8} - \frac{{t\_7}^{3}}{t\_8} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (fma x -0.5 0.5)))
        (t_1 (asin t_0))
        (t_2 (fma t_1 (fma t_1 4.0 PI) (* PI (* PI 0.25))))
        (t_3 (/ (* 0.125 (* PI (* PI PI))) t_2))
        (t_4 (acos t_0))
        (t_5 (fma t_4 2.0 PI))
        (t_6 (pow (fma -4.0 (pow t_4 2.0) (* PI PI)) 3.0))
        (t_7 (/ t_6 (* t_2 (pow t_5 3.0))))
        (t_8
         (fma
          t_7
          (* (/ 1.0 t_2) (fma PI (* PI (* 0.125 PI)) (* t_6 (pow t_5 -3.0))))
          (pow t_3 2.0))))
   (- (/ (pow t_3 3.0) t_8) (/ (pow t_7 3.0) t_8))))

C

double code(double x) {
	double t_0 = sqrt(fma(x, -0.5, 0.5));
	double t_1 = asin(t_0);
	double t_2 = fma(t_1, fma(t_1, 4.0, ((double) M_PI)), (((double) M_PI) * (((double) M_PI) * 0.25)));
	double t_3 = (0.125 * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))) / t_2;
	double t_4 = acos(t_0);
	double t_5 = fma(t_4, 2.0, ((double) M_PI));
	double t_6 = pow(fma(-4.0, pow(t_4, 2.0), (((double) M_PI) * ((double) M_PI))), 3.0);
	double t_7 = t_6 / (t_2 * pow(t_5, 3.0));
	double t_8 = fma(t_7, ((1.0 / t_2) * fma(((double) M_PI), (((double) M_PI) * (0.125 * ((double) M_PI))), (t_6 * pow(t_5, -3.0)))), pow(t_3, 2.0));
	return (pow(t_3, 3.0) / t_8) - (pow(t_7, 3.0) / t_8);
}

Julia

function code(x)
	t_0 = sqrt(fma(x, -0.5, 0.5))
	t_1 = asin(t_0)
	t_2 = fma(t_1, fma(t_1, 4.0, pi), Float64(pi * Float64(pi * 0.25)))
	t_3 = Float64(Float64(0.125 * Float64(pi * Float64(pi * pi))) / t_2)
	t_4 = acos(t_0)
	t_5 = fma(t_4, 2.0, pi)
	t_6 = fma(-4.0, (t_4 ^ 2.0), Float64(pi * pi)) ^ 3.0
	t_7 = Float64(t_6 / Float64(t_2 * (t_5 ^ 3.0)))
	t_8 = fma(t_7, Float64(Float64(1.0 / t_2) * fma(pi, Float64(pi * Float64(0.125 * pi)), Float64(t_6 * (t_5 ^ -3.0)))), (t_3 ^ 2.0))
	return Float64(Float64((t_3 ^ 3.0) / t_8) - Float64((t_7 ^ 3.0) / t_8))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcSin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(t$95$1 * 4.0 + Pi), $MachinePrecision] + N[(Pi * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.125 * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[ArcCos[t$95$0], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * 2.0 + Pi), $MachinePrecision]}, Block[{t$95$6 = N[Power[N[(-4.0 * N[Power[t$95$4, 2.0], $MachinePrecision] + N[(Pi * Pi), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$7 = N[(t$95$6 / N[(t$95$2 * N[Power[t$95$5, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$7 * N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(Pi * N[(Pi * N[(0.125 * Pi), $MachinePrecision]), $MachinePrecision] + N[(t$95$6 * N[Power[t$95$5, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[t$95$3, 3.0], $MachinePrecision] / t$95$8), $MachinePrecision] - N[(N[Power[t$95$7, 3.0], $MachinePrecision] / t$95$8), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Initial program 7.7%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing

4. Applied rewrites 7.8%

   \[\leadsto \color{blue}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot 0.125\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right) \cdot \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right)\right)} - \frac{{\left(2 \cdot \sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3}}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right) \cdot \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right)\right)}} \]

6. Applied rewrites 8.7%

   \[\leadsto \frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot 0.125\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right) \cdot \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right)\right)} - \frac{\color{blue}{\frac{{\left(\pi \cdot \pi - {\left(-2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{2}\right)}^{3}}{{\left(\pi - -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3}}}}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right) \cdot \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right)\right)} \]

8. Applied rewrites 8.7%

   \[\leadsto \color{blue}{\frac{{\left(\frac{0.125 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right), \pi \cdot \left(\pi \cdot 0.25\right)\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(-4, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{2}, \pi \cdot \pi\right)\right)}^{3}}{{\left(\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 2, \pi\right)\right)}^{3} \cdot \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right), \pi \cdot \left(\pi \cdot 0.25\right)\right)}, \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right), \pi \cdot \left(\pi \cdot 0.25\right)\right)} \cdot \mathsf{fma}\left(\pi, \pi \cdot \left(\pi \cdot 0.125\right), {\left(\mathsf{fma}\left(-4, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{2}, \pi \cdot \pi\right)\right)}^{3} \cdot {\left(\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 2, \pi\right)\right)}^{-3}\right), {\left(\frac{0.125 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right), \pi \cdot \left(\pi \cdot 0.25\right)\right)}\right)}^{2}\right)} - \frac{{\left(\frac{{\left(\mathsf{fma}\left(-4, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{2}, \pi \cdot \pi\right)\right)}^{3}}{{\left(\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 2, \pi\right)\right)}^{3} \cdot \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right), \pi \cdot \left(\pi \cdot 0.25\right)\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(-4, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{2}, \pi \cdot \pi\right)\right)}^{3}}{{\left(\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 2, \pi\right)\right)}^{3} \cdot \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right), \pi \cdot \left(\pi \cdot 0.25\right)\right)}, \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right), \pi \cdot \left(\pi \cdot 0.25\right)\right)} \cdot \mathsf{fma}\left(\pi, \pi \cdot \left(\pi \cdot 0.125\right), {\left(\mathsf{fma}\left(-4, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{2}, \pi \cdot \pi\right)\right)}^{3} \cdot {\left(\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 2, \pi\right)\right)}^{-3}\right), {\left(\frac{0.125 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right), \pi \cdot \left(\pi \cdot 0.25\right)\right)}\right)}^{2}\right)}} \]

9. Final simplification 8.7%

   \[\leadsto \frac{{\left(\frac{0.125 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right), \pi \cdot \left(\pi \cdot 0.25\right)\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(-4, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{2}, \pi \cdot \pi\right)\right)}^{3}}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right), \pi \cdot \left(\pi \cdot 0.25\right)\right) \cdot {\left(\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 2, \pi\right)\right)}^{3}}, \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right), \pi \cdot \left(\pi \cdot 0.25\right)\right)} \cdot \mathsf{fma}\left(\pi, \pi \cdot \left(0.125 \cdot \pi\right), {\left(\mathsf{fma}\left(-4, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{2}, \pi \cdot \pi\right)\right)}^{3} \cdot {\left(\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 2, \pi\right)\right)}^{-3}\right), {\left(\frac{0.125 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right), \pi \cdot \left(\pi \cdot 0.25\right)\right)}\right)}^{2}\right)} - \frac{{\left(\frac{{\left(\mathsf{fma}\left(-4, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{2}, \pi \cdot \pi\right)\right)}^{3}}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right), \pi \cdot \left(\pi \cdot 0.25\right)\right) \cdot {\left(\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 2, \pi\right)\right)}^{3}}\right)}^{3}}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(-4, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{2}, \pi \cdot \pi\right)\right)}^{3}}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right), \pi \cdot \left(\pi \cdot 0.25\right)\right) \cdot {\left(\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 2, \pi\right)\right)}^{3}}, \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right), \pi \cdot \left(\pi \cdot 0.25\right)\right)} \cdot \mathsf{fma}\left(\pi, \pi \cdot \left(0.125 \cdot \pi\right), {\left(\mathsf{fma}\left(-4, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{2}, \pi \cdot \pi\right)\right)}^{3} \cdot {\left(\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 2, \pi\right)\right)}^{-3}\right), {\left(\frac{0.125 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right), \pi \cdot \left(\pi \cdot 0.25\right)\right)}\right)}^{2}\right)} \]
10. Add Preprocessing

Alternative 2: 8.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\\ t_1 := \cos^{-1} t\_0 \cdot -2\\ t_2 := \sin^{-1} t\_0\\ t_3 := \mathsf{fma}\left(\pi, \pi \cdot 0.25, t\_2 \cdot \mathsf{fma}\left(t\_2, 4, \pi\right)\right)\\ \frac{\left(\pi \cdot \pi\right) \cdot \left(0.125 \cdot \pi\right)}{t\_3} - \frac{\frac{{\left(\pi \cdot \pi - {t\_1}^{2}\right)}^{3}}{{\left(\pi - t\_1\right)}^{3}}}{t\_3} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (fma x -0.5 0.5)))
        (t_1 (* (acos t_0) -2.0))
        (t_2 (asin t_0))
        (t_3 (fma PI (* PI 0.25) (* t_2 (fma t_2 4.0 PI)))))
   (-
    (/ (* (* PI PI) (* 0.125 PI)) t_3)
    (/ (/ (pow (- (* PI PI) (pow t_1 2.0)) 3.0) (pow (- PI t_1) 3.0)) t_3))))

C

double code(double x) {
	double t_0 = sqrt(fma(x, -0.5, 0.5));
	double t_1 = acos(t_0) * -2.0;
	double t_2 = asin(t_0);
	double t_3 = fma(((double) M_PI), (((double) M_PI) * 0.25), (t_2 * fma(t_2, 4.0, ((double) M_PI))));
	return (((((double) M_PI) * ((double) M_PI)) * (0.125 * ((double) M_PI))) / t_3) - ((pow(((((double) M_PI) * ((double) M_PI)) - pow(t_1, 2.0)), 3.0) / pow((((double) M_PI) - t_1), 3.0)) / t_3);
}

Julia

function code(x)
	t_0 = sqrt(fma(x, -0.5, 0.5))
	t_1 = Float64(acos(t_0) * -2.0)
	t_2 = asin(t_0)
	t_3 = fma(pi, Float64(pi * 0.25), Float64(t_2 * fma(t_2, 4.0, pi)))
	return Float64(Float64(Float64(Float64(pi * pi) * Float64(0.125 * pi)) / t_3) - Float64(Float64((Float64(Float64(pi * pi) - (t_1 ^ 2.0)) ^ 3.0) / (Float64(pi - t_1) ^ 3.0)) / t_3))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcCos[t$95$0], $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[ArcSin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(Pi * N[(Pi * 0.25), $MachinePrecision] + N[(t$95$2 * N[(t$95$2 * 4.0 + Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(0.125 * Pi), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] - N[(N[(N[Power[N[(N[(Pi * Pi), $MachinePrecision] - N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[(Pi - t$95$1), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 7.7%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing

4. Applied rewrites 7.8%

   \[\leadsto \color{blue}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot 0.125\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right) \cdot \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right)\right)} - \frac{{\left(2 \cdot \sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3}}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right) \cdot \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right)\right)}} \]

6. Applied rewrites 8.7%

   \[\leadsto \frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot 0.125\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right) \cdot \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right)\right)} - \frac{\color{blue}{\frac{{\left(\pi \cdot \pi - {\left(-2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{2}\right)}^{3}}{{\left(\pi - -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3}}}}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right) \cdot \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right)\right)} \]

7. Final simplification 8.7%

   \[\leadsto \frac{\left(\pi \cdot \pi\right) \cdot \left(0.125 \cdot \pi\right)}{\mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right) \cdot \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right)\right)} - \frac{\frac{{\left(\pi \cdot \pi - {\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right) \cdot -2\right)}^{2}\right)}^{3}}{{\left(\pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right) \cdot -2\right)}^{3}}}{\mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right) \cdot \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \pi\right)\right)} \]
8. Add Preprocessing

Alternative 3: 8.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\\ t_1 := \mathsf{fma}\left(2, t\_0, \pi \cdot 0.5\right)\\ \frac{{\left(t\_0 \cdot 2\right)}^{2}}{t\_1} - \frac{\pi \cdot \left(\pi \cdot 0.25\right)}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (sqrt (fma x -0.5 0.5)))) (t_1 (fma 2.0 t_0 (* PI 0.5))))
   (- (/ (pow (* t_0 2.0) 2.0) t_1) (/ (* PI (* PI 0.25)) t_1))))

C

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

Julia

function code(x)
	t_0 = acos(sqrt(fma(x, -0.5, 0.5)))
	t_1 = fma(2.0, t_0, Float64(pi * 0.5))
	return Float64(Float64((Float64(t_0 * 2.0) ^ 2.0) / t_1) - Float64(Float64(pi * Float64(pi * 0.25)) / t_1))
end

Wolfram

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

Derivation

1. Initial program 7.7%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. sub-neg N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right)} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right) \]

   7. div-inv N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right) \]

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-acos.f64 8.7

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

   12. lift-/.f64 N/A

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

   13. lift--.f64 N/A

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

   14. div-sub N/A

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

   15. metadata-eval N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. div-inv N/A

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

   19. metadata-eval N/A

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

   20. distribute-rgt-neg-in N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. lower-fma.f64 N/A

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

4. Applied rewrites 8.7%

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

5. Taylor expanded in x around 0

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

   1. sub-neg N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)\right)\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot x}\right)\right)\right)\right) \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot x}}\right)\right)\right)\right) \]

   4. distribute-lft-in N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{PI}\left(\right) - \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) + 2 \cdot \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)\right)\right)} \]

   5. associate--r+ N/A

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

   6. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)\right)} \]

   7. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)} + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \color{blue}{-2} \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + -2 \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{-2} \cdot \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)\right) \]

7. Applied rewrites 8.7%

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

9. Applied rewrites 8.7%

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

10. Final simplification 8.7%

    \[\leadsto \frac{{\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right) \cdot 2\right)}^{2}}{\mathsf{fma}\left(2, \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), \pi \cdot 0.5\right)} - \frac{\pi \cdot \left(\pi \cdot 0.25\right)}{\mathsf{fma}\left(2, \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), \pi \cdot 0.5\right)} \]
11. Add Preprocessing

Alternative 4: 8.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fma 2.0 (acos (sqrt (fma -0.5 x 0.5))) (* PI -0.5)))

C

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

Julia

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

Wolfram

code[x_] := N[(2.0 * N[ArcCos[N[Sqrt[N[(-0.5 * x + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(Pi * -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. sub-neg N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right)} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right) \]

   7. div-inv N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right) \]

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-acos.f64 8.7

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

   12. lift-/.f64 N/A

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

   13. lift--.f64 N/A

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

   14. div-sub N/A

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

   15. metadata-eval N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. div-inv N/A

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

   19. metadata-eval N/A

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

   20. distribute-rgt-neg-in N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. lower-fma.f64 N/A

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

4. Applied rewrites 8.7%

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

5. Taylor expanded in x around 0

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

   1. sub-neg N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)\right)\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot x}\right)\right)\right)\right) \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot x}}\right)\right)\right)\right) \]

   4. distribute-lft-in N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{PI}\left(\right) - \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) + 2 \cdot \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)\right)\right)} \]

   5. associate--r+ N/A

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

   6. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)\right)} \]

   7. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)} + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \color{blue}{-2} \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + -2 \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{-2} \cdot \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)\right) \]

7. Applied rewrites 8.7%

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

Alternative 5: 5.4% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (fma 2.0 (acos (sqrt 0.5)) (* PI -0.5)))

C

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

Julia

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

Wolfram

code[x_] := N[(2.0 * N[ArcCos[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision] + N[(Pi * -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. sub-neg N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right)} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right) \]

   7. div-inv N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right) \]

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-acos.f64 8.7

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

   12. lift-/.f64 N/A

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

   13. lift--.f64 N/A

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

   14. div-sub N/A

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

   15. metadata-eval N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. div-inv N/A

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

   19. metadata-eval N/A

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

   20. distribute-rgt-neg-in N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. lower-fma.f64 N/A

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

4. Applied rewrites 8.7%

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

5. Taylor expanded in x around 0

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

   1. sub-neg N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)\right)\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot x}\right)\right)\right)\right) \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot x}}\right)\right)\right)\right) \]

   4. distribute-lft-in N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{PI}\left(\right) - \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) + 2 \cdot \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)\right)\right)} \]

   5. associate--r+ N/A

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

   6. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 2 \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)\right)} \]

   7. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)} + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \color{blue}{-2} \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + -2 \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{-2} \cdot \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)\right) \]

7. Applied rewrites 8.7%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(2, \cos^{-1} \left(\sqrt{\frac{1}{2}}\right), \mathsf{PI}\left(\right) \cdot \frac{-1}{2}\right) \]
9. Step-by-step derivation

10. Applied rewrites 5.1%

    \[\leadsto \mathsf{fma}\left(2, \cos^{-1} \left(\sqrt{0.5}\right), \pi \cdot -0.5\right) \]
11. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (asin x))

C

double code(double x) {
	return asin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = asin(x)
end function

Java

public static double code(double x) {
	return Math.asin(x);
}

Python

def code(x):
	return math.asin(x)

Julia

function code(x)
	return asin(x)
end

MATLAB

function tmp = code(x)
	tmp = asin(x);
end

Wolfram

code[x_] := N[ArcSin[x], $MachinePrecision]

Reproduce

herbie shell --seed 2024234 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :alt
  (! :herbie-platform default (asin x))

  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))