Ian Simplification

Percentage Accurate: 6.9% → 8.4%

Time: 2.3min

Alternatives: 7

Speedup: 1.0×

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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot \pi\right) \cdot 0.25\\ t_1 := \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\\ t_2 := {t\_1}^{2}\\ \frac{t\_0 + t\_1 \cdot \left(t\_1 \cdot 4 - \pi\right)}{{\left(t\_0 + 4 \cdot t\_2\right)}^{2} - \left(\pi \cdot \pi\right) \cdot t\_2} \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.125 + {t\_1}^{3} \cdot 8\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* PI PI) 0.25))
        (t_1 (acos (sqrt (+ 0.5 (* -0.5 x)))))
        (t_2 (pow t_1 2.0)))
   (*
    (/
     (+ t_0 (* t_1 (- (* t_1 4.0) PI)))
     (- (pow (+ t_0 (* 4.0 t_2)) 2.0) (* (* PI PI) t_2)))
    (+ (* (* PI (* PI PI)) -0.125) (* (pow t_1 3.0) 8.0)))))

C

double code(double x) {
	double t_0 = (((double) M_PI) * ((double) M_PI)) * 0.25;
	double t_1 = acos(sqrt((0.5 + (-0.5 * x))));
	double t_2 = pow(t_1, 2.0);
	return ((t_0 + (t_1 * ((t_1 * 4.0) - ((double) M_PI)))) / (pow((t_0 + (4.0 * t_2)), 2.0) - ((((double) M_PI) * ((double) M_PI)) * t_2))) * (((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * -0.125) + (pow(t_1, 3.0) * 8.0));
}

Java

public static double code(double x) {
	double t_0 = (Math.PI * Math.PI) * 0.25;
	double t_1 = Math.acos(Math.sqrt((0.5 + (-0.5 * x))));
	double t_2 = Math.pow(t_1, 2.0);
	return ((t_0 + (t_1 * ((t_1 * 4.0) - Math.PI))) / (Math.pow((t_0 + (4.0 * t_2)), 2.0) - ((Math.PI * Math.PI) * t_2))) * (((Math.PI * (Math.PI * Math.PI)) * -0.125) + (Math.pow(t_1, 3.0) * 8.0));
}

Python

def code(x):
	t_0 = (math.pi * math.pi) * 0.25
	t_1 = math.acos(math.sqrt((0.5 + (-0.5 * x))))
	t_2 = math.pow(t_1, 2.0)
	return ((t_0 + (t_1 * ((t_1 * 4.0) - math.pi))) / (math.pow((t_0 + (4.0 * t_2)), 2.0) - ((math.pi * math.pi) * t_2))) * (((math.pi * (math.pi * math.pi)) * -0.125) + (math.pow(t_1, 3.0) * 8.0))

Julia

function code(x)
	t_0 = Float64(Float64(pi * pi) * 0.25)
	t_1 = acos(sqrt(Float64(0.5 + Float64(-0.5 * x))))
	t_2 = t_1 ^ 2.0
	return Float64(Float64(Float64(t_0 + Float64(t_1 * Float64(Float64(t_1 * 4.0) - pi))) / Float64((Float64(t_0 + Float64(4.0 * t_2)) ^ 2.0) - Float64(Float64(pi * pi) * t_2))) * Float64(Float64(Float64(pi * Float64(pi * pi)) * -0.125) + Float64((t_1 ^ 3.0) * 8.0)))
end

MATLAB

function tmp = code(x)
	t_0 = (pi * pi) * 0.25;
	t_1 = acos(sqrt((0.5 + (-0.5 * x))));
	t_2 = t_1 ^ 2.0;
	tmp = ((t_0 + (t_1 * ((t_1 * 4.0) - pi))) / (((t_0 + (4.0 * t_2)) ^ 2.0) - ((pi * pi) * t_2))) * (((pi * (pi * pi)) * -0.125) + ((t_1 ^ 3.0) * 8.0));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision]}, Block[{t$95$1 = N[ArcCos[N[Sqrt[N[(0.5 + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, N[(N[(N[(t$95$0 + N[(t$95$1 * N[(N[(t$95$1 * 4.0), $MachinePrecision] - Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(t$95$0 + N[(4.0 * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[(N[(Pi * Pi), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] + N[(N[Power[t$95$1, 3.0], $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 8.2%

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

   1. asin-acos N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)}\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot 2 + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2}\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right) \cdot 2\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot 2\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot 1 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   8. *-rgt-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)}\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right), \color{blue}{2}\right)\right)\right) \]

4. Applied egg-rr 9.3%

   \[\leadsto \frac{\pi}{2} - \color{blue}{\left(\pi + \left(0 - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right) \cdot 2\right)} \]

6. Applied egg-rr 9.3%

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

8. Applied egg-rr 9.3%

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

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\left(\left(\frac{1}{4} \cdot {\mathsf{PI}\left(\right)}^{2} + 4 \cdot {\cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)}^{2}\right) - \mathsf{PI}\left(\right) \cdot \cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)\right) \cdot \left({\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \mathsf{PI}\left(\right)\right)}^{3} - -8 \cdot {\cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)}^{3}\right)}{{\left(\frac{1}{4} \cdot {\mathsf{PI}\left(\right)}^{2} + 4 \cdot {\cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)}^{2}\right)}^{2} - {\mathsf{PI}\left(\right)}^{2} \cdot {\cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)}^{2}}} \]

11. Simplified 9.3%

    \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.125 + {\cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)}^{3} \cdot 8\right) \cdot \frac{0.25 \cdot \left(\pi \cdot \pi\right) + \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right) \cdot \left(4 \cdot \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right) - \pi\right)}{{\left(0.25 \cdot \left(\pi \cdot \pi\right) + 4 \cdot {\cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)}^{2}\right)}^{2} - \left(\pi \cdot \pi\right) \cdot {\cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)}^{2}}} \]

12. Final simplification 9.3%

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

Alternative 2: 8.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi \cdot \pi}{4}\\ t_1 := 0.5 + \frac{x}{-2}\\ t_2 := -2 \cdot \cos^{-1} \left(\sqrt{t\_1}\right)\\ t_3 := 2 \cdot \cos^{-1} \left({t\_1}^{0.5}\right)\\ t_4 := \pi - t\_3\\ \frac{\pi \cdot \pi}{t\_0 + \left(\pi + t\_2\right) \cdot \left(t\_2 + \pi \cdot 1.5\right)} \cdot \frac{\pi}{8} + \frac{{t\_4}^{3}}{t\_4 \cdot \left(\left(t\_3 - \pi\right) - \frac{\pi}{2}\right) - t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (* PI PI) 4.0))
        (t_1 (+ 0.5 (/ x -2.0)))
        (t_2 (* -2.0 (acos (sqrt t_1))))
        (t_3 (* 2.0 (acos (pow t_1 0.5))))
        (t_4 (- PI t_3)))
   (+
    (* (/ (* PI PI) (+ t_0 (* (+ PI t_2) (+ t_2 (* PI 1.5))))) (/ PI 8.0))
    (/ (pow t_4 3.0) (- (* t_4 (- (- t_3 PI) (/ PI 2.0))) t_0)))))

C

double code(double x) {
	double t_0 = (((double) M_PI) * ((double) M_PI)) / 4.0;
	double t_1 = 0.5 + (x / -2.0);
	double t_2 = -2.0 * acos(sqrt(t_1));
	double t_3 = 2.0 * acos(pow(t_1, 0.5));
	double t_4 = ((double) M_PI) - t_3;
	return (((((double) M_PI) * ((double) M_PI)) / (t_0 + ((((double) M_PI) + t_2) * (t_2 + (((double) M_PI) * 1.5))))) * (((double) M_PI) / 8.0)) + (pow(t_4, 3.0) / ((t_4 * ((t_3 - ((double) M_PI)) - (((double) M_PI) / 2.0))) - t_0));
}

Java

public static double code(double x) {
	double t_0 = (Math.PI * Math.PI) / 4.0;
	double t_1 = 0.5 + (x / -2.0);
	double t_2 = -2.0 * Math.acos(Math.sqrt(t_1));
	double t_3 = 2.0 * Math.acos(Math.pow(t_1, 0.5));
	double t_4 = Math.PI - t_3;
	return (((Math.PI * Math.PI) / (t_0 + ((Math.PI + t_2) * (t_2 + (Math.PI * 1.5))))) * (Math.PI / 8.0)) + (Math.pow(t_4, 3.0) / ((t_4 * ((t_3 - Math.PI) - (Math.PI / 2.0))) - t_0));
}

Python

def code(x):
	t_0 = (math.pi * math.pi) / 4.0
	t_1 = 0.5 + (x / -2.0)
	t_2 = -2.0 * math.acos(math.sqrt(t_1))
	t_3 = 2.0 * math.acos(math.pow(t_1, 0.5))
	t_4 = math.pi - t_3
	return (((math.pi * math.pi) / (t_0 + ((math.pi + t_2) * (t_2 + (math.pi * 1.5))))) * (math.pi / 8.0)) + (math.pow(t_4, 3.0) / ((t_4 * ((t_3 - math.pi) - (math.pi / 2.0))) - t_0))

Julia

function code(x)
	t_0 = Float64(Float64(pi * pi) / 4.0)
	t_1 = Float64(0.5 + Float64(x / -2.0))
	t_2 = Float64(-2.0 * acos(sqrt(t_1)))
	t_3 = Float64(2.0 * acos((t_1 ^ 0.5)))
	t_4 = Float64(pi - t_3)
	return Float64(Float64(Float64(Float64(pi * pi) / Float64(t_0 + Float64(Float64(pi + t_2) * Float64(t_2 + Float64(pi * 1.5))))) * Float64(pi / 8.0)) + Float64((t_4 ^ 3.0) / Float64(Float64(t_4 * Float64(Float64(t_3 - pi) - Float64(pi / 2.0))) - t_0)))
end

MATLAB

function tmp = code(x)
	t_0 = (pi * pi) / 4.0;
	t_1 = 0.5 + (x / -2.0);
	t_2 = -2.0 * acos(sqrt(t_1));
	t_3 = 2.0 * acos((t_1 ^ 0.5));
	t_4 = pi - t_3;
	tmp = (((pi * pi) / (t_0 + ((pi + t_2) * (t_2 + (pi * 1.5))))) * (pi / 8.0)) + ((t_4 ^ 3.0) / ((t_4 * ((t_3 - pi) - (pi / 2.0))) - t_0));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(Pi * Pi), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + N[(x / -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 * N[ArcCos[N[Sqrt[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[ArcCos[N[Power[t$95$1, 0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(Pi - t$95$3), $MachinePrecision]}, N[(N[(N[(N[(Pi * Pi), $MachinePrecision] / N[(t$95$0 + N[(N[(Pi + t$95$2), $MachinePrecision] * N[(t$95$2 + N[(Pi * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi / 8.0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[t$95$4, 3.0], $MachinePrecision] / N[(N[(t$95$4 * N[(N[(t$95$3 - Pi), $MachinePrecision] - N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 8.2%

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

   1. asin-acos N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)}\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot 2 + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2}\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right) \cdot 2\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot 2\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot 1 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   8. *-rgt-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)}\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right), \color{blue}{2}\right)\right)\right) \]

4. Applied egg-rr 9.3%

   \[\leadsto \frac{\pi}{2} - \color{blue}{\left(\pi + \left(0 - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right) \cdot 2\right)} \]

6. Applied egg-rr 9.3%

   \[\leadsto \color{blue}{\frac{\frac{\pi \cdot \left(\pi \cdot \pi\right)}{8}}{\frac{\pi \cdot \pi}{4} + \left(\pi - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) \cdot 2\right) \cdot \left(\frac{\pi}{2} + \left(\pi - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) \cdot 2\right)\right)} - \frac{{\left(\pi - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) \cdot 2\right)}^{3}}{\frac{\pi \cdot \pi}{4} + \left(\pi - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) \cdot 2\right) \cdot \left(\frac{\pi}{2} + \left(\pi - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) \cdot 2\right)\right)}} \]

8. Applied egg-rr 9.3%

   \[\leadsto \color{blue}{\frac{\pi \cdot \pi}{\frac{\pi \cdot \pi}{4} + \left(\pi + -2 \cdot \cos^{-1} \left(\sqrt{0.5 + \frac{x}{-2}}\right)\right) \cdot \left(\pi \cdot 1.5 + -2 \cdot \cos^{-1} \left(\sqrt{0.5 + \frac{x}{-2}}\right)\right)} \cdot \frac{\pi}{8}} - \frac{{\left(\pi - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) \cdot 2\right)}^{3}}{\frac{\pi \cdot \pi}{4} + \left(\pi - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) \cdot 2\right) \cdot \left(\frac{\pi}{2} + \left(\pi - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) \cdot 2\right)\right)} \]

9. Final simplification 9.3%

   \[\leadsto \frac{\pi \cdot \pi}{\frac{\pi \cdot \pi}{4} + \left(\pi + -2 \cdot \cos^{-1} \left(\sqrt{0.5 + \frac{x}{-2}}\right)\right) \cdot \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + \frac{x}{-2}}\right) + \pi \cdot 1.5\right)} \cdot \frac{\pi}{8} + \frac{{\left(\pi - 2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{3}}{\left(\pi - 2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right) \cdot \left(\left(2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) - \pi\right) - \frac{\pi}{2}\right) - \frac{\pi \cdot \pi}{4}} \]
10. Add Preprocessing

Alternative 3: 8.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\\ \frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.125 - {\left(-2 \cdot t\_0\right)}^{3}}{{\left(2 \cdot t\_0\right)}^{2} + \left(\frac{\pi \cdot \pi}{4} + \pi \cdot t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (pow (+ 0.5 (/ x -2.0)) 0.5))))
   (/
    (- (* (* PI (* PI PI)) -0.125) (pow (* -2.0 t_0) 3.0))
    (+ (pow (* 2.0 t_0) 2.0) (+ (/ (* PI PI) 4.0) (* PI t_0))))))

C

double code(double x) {
	double t_0 = acos(pow((0.5 + (x / -2.0)), 0.5));
	return (((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * -0.125) - pow((-2.0 * t_0), 3.0)) / (pow((2.0 * t_0), 2.0) + (((((double) M_PI) * ((double) M_PI)) / 4.0) + (((double) M_PI) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = Math.acos(Math.pow((0.5 + (x / -2.0)), 0.5));
	return (((Math.PI * (Math.PI * Math.PI)) * -0.125) - Math.pow((-2.0 * t_0), 3.0)) / (Math.pow((2.0 * t_0), 2.0) + (((Math.PI * Math.PI) / 4.0) + (Math.PI * t_0)));
}

Python

def code(x):
	t_0 = math.acos(math.pow((0.5 + (x / -2.0)), 0.5))
	return (((math.pi * (math.pi * math.pi)) * -0.125) - math.pow((-2.0 * t_0), 3.0)) / (math.pow((2.0 * t_0), 2.0) + (((math.pi * math.pi) / 4.0) + (math.pi * t_0)))

Julia

function code(x)
	t_0 = acos((Float64(0.5 + Float64(x / -2.0)) ^ 0.5))
	return Float64(Float64(Float64(Float64(pi * Float64(pi * pi)) * -0.125) - (Float64(-2.0 * t_0) ^ 3.0)) / Float64((Float64(2.0 * t_0) ^ 2.0) + Float64(Float64(Float64(pi * pi) / 4.0) + Float64(pi * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = acos(((0.5 + (x / -2.0)) ^ 0.5));
	tmp = (((pi * (pi * pi)) * -0.125) - ((-2.0 * t_0) ^ 3.0)) / (((2.0 * t_0) ^ 2.0) + (((pi * pi) / 4.0) + (pi * t_0)));
end

Wolfram

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

Derivation

1. Initial program 8.2%

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

   1. asin-acos N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)}\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot 2 + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2}\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right) \cdot 2\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot 2\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot 1 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   8. *-rgt-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)}\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right), \color{blue}{2}\right)\right)\right) \]

4. Applied egg-rr 9.3%

   \[\leadsto \frac{\pi}{2} - \color{blue}{\left(\pi + \left(0 - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right) \cdot 2\right)} \]

6. Applied egg-rr 9.3%

   \[\leadsto \color{blue}{\frac{{\left(\frac{\pi}{2} - \pi\right)}^{3} - {\left(-\cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) \cdot 2\right)}^{3}}{\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right) + \left({\left(-\cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) \cdot 2\right)}^{2} + \left(\frac{\pi}{2} - \pi\right) \cdot \left(-\cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) \cdot 2\right)\right)}} \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right)}^{3} - {\left(\mathsf{neg}\left(\cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)\right)}^{3}}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right) \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right) + \left(\left(\mathsf{neg}\left(\cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)\right) \cdot \left(\mathsf{neg}\left(\cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)\right) + \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right)} \cdot \left(\mathsf{neg}\left(\cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)\right)\right)} \]

   2. flip3-- N/A

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

8. Applied egg-rr 9.3%

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

10. Applied egg-rr 9.3%

    \[\leadsto \color{blue}{\frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.125 - {\left(-2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{3}}{{\left(2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{2} + \left(\frac{\pi \cdot \pi}{4} + \pi \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}} \]
11. Add Preprocessing

Alternative 4: 8.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\\ \frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.125 + {t\_0}^{3} \cdot 8}{t\_0 \cdot \left(\pi + t\_0 \cdot 4\right) + \pi \cdot \left(\pi \cdot 0.25\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (sqrt (+ 0.5 (* -0.5 x))))))
   (/
    (+ (* (* PI (* PI PI)) -0.125) (* (pow t_0 3.0) 8.0))
    (+ (* t_0 (+ PI (* t_0 4.0))) (* PI (* PI 0.25))))))

C

double code(double x) {
	double t_0 = acos(sqrt((0.5 + (-0.5 * x))));
	return (((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * -0.125) + (pow(t_0, 3.0) * 8.0)) / ((t_0 * (((double) M_PI) + (t_0 * 4.0))) + (((double) M_PI) * (((double) M_PI) * 0.25)));
}

Java

public static double code(double x) {
	double t_0 = Math.acos(Math.sqrt((0.5 + (-0.5 * x))));
	return (((Math.PI * (Math.PI * Math.PI)) * -0.125) + (Math.pow(t_0, 3.0) * 8.0)) / ((t_0 * (Math.PI + (t_0 * 4.0))) + (Math.PI * (Math.PI * 0.25)));
}

Python

def code(x):
	t_0 = math.acos(math.sqrt((0.5 + (-0.5 * x))))
	return (((math.pi * (math.pi * math.pi)) * -0.125) + (math.pow(t_0, 3.0) * 8.0)) / ((t_0 * (math.pi + (t_0 * 4.0))) + (math.pi * (math.pi * 0.25)))

Julia

function code(x)
	t_0 = acos(sqrt(Float64(0.5 + Float64(-0.5 * x))))
	return Float64(Float64(Float64(Float64(pi * Float64(pi * pi)) * -0.125) + Float64((t_0 ^ 3.0) * 8.0)) / Float64(Float64(t_0 * Float64(pi + Float64(t_0 * 4.0))) + Float64(pi * Float64(pi * 0.25))))
end

MATLAB

function tmp = code(x)
	t_0 = acos(sqrt((0.5 + (-0.5 * x))));
	tmp = (((pi * (pi * pi)) * -0.125) + ((t_0 ^ 3.0) * 8.0)) / ((t_0 * (pi + (t_0 * 4.0))) + (pi * (pi * 0.25)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[Sqrt[N[(0.5 + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] + N[(N[Power[t$95$0, 3.0], $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * N[(Pi + N[(t$95$0 * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 8.2%

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

   1. asin-acos N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)}\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot 2 + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2}\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right) \cdot 2\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot 2\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot 1 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   8. *-rgt-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)}\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right), \color{blue}{2}\right)\right)\right) \]

4. Applied egg-rr 9.3%

   \[\leadsto \frac{\pi}{2} - \color{blue}{\left(\pi + \left(0 - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right) \cdot 2\right)} \]

6. Applied egg-rr 9.3%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \mathsf{PI}\left(\right)\right)}^{3} - -8 \cdot {\cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)}^{3}}{-2 \cdot \left(\cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right) \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \mathsf{PI}\left(\right)\right)\right) + \left(4 \cdot {\cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)}^{2} + {\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \mathsf{PI}\left(\right)\right)}^{2}\right)}} \]

9. Simplified 9.3%

   \[\leadsto \color{blue}{\frac{{\cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)}^{3} \cdot 8 + \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.125}{\cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right) \cdot \left(\pi + 4 \cdot \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right) + \pi \cdot \left(\pi \cdot 0.25\right)}} \]

10. Final simplification 9.3%

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

Alternative 5: 8.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (Math.PI + (-2.0 * Math.acos(Math.sqrt((0.5 - (0.5 * x))))));
}

Python

def code(x):
	return (math.pi / 2.0) - (math.pi + (-2.0 * math.acos(math.sqrt((0.5 - (0.5 * x))))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(pi + Float64(-2.0 * acos(sqrt(Float64(0.5 - Float64(0.5 * x)))))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (pi + (-2.0 * acos(sqrt((0.5 - (0.5 * x))))));
end

Wolfram

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

Derivation

1. Initial program 8.2%

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

   1. asin-acos N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)}\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot 2 + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2}\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right) \cdot 2\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot 2\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot 1 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   8. *-rgt-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)}\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right), \color{blue}{2}\right)\right)\right) \]

4. Applied egg-rr 9.3%

   \[\leadsto \frac{\pi}{2} - \color{blue}{\left(\pi + \left(0 - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right) \cdot 2\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \color{blue}{\left(\mathsf{PI}\left(\right) + -2 \cdot \cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)\right)}\right) \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) + -2 \cdot \cos^{-1} \left(\sqrt{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x}\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) + -2 \cdot \cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(-2 \cdot \cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)}\right)\right) \]

   4. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{-2} \cdot \cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(-2, \color{blue}{\cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)}\right)\right)\right) \]

   6. acos-lowering-acos.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(-2, \mathsf{acos.f64}\left(\left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right)\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(-2, \mathsf{acos.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot x\right)\right)\right)\right)\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(-2, \mathsf{acos.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot x\right)\right)\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 9.3%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(-2, \mathsf{acos.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, x\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 9.3%

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

Alternative 6: 8.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (Math.PI * -0.5) - (-2.0 * Math.acos(Math.sqrt((0.5 + (x / -2.0)))));
}

Python

def code(x):
	return (math.pi * -0.5) - (-2.0 * math.acos(math.sqrt((0.5 + (x / -2.0)))))

Julia

function code(x)
	return Float64(Float64(pi * -0.5) - Float64(-2.0 * acos(sqrt(Float64(0.5 + Float64(x / -2.0))))))
end

MATLAB

function tmp = code(x)
	tmp = (pi * -0.5) - (-2.0 * acos(sqrt((0.5 + (x / -2.0)))));
end

Wolfram

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

Derivation

1. Initial program 8.2%

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

   1. asin-acos N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)}\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot 2 + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2}\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right) \cdot 2\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot 2\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot 1 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   8. *-rgt-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)}\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right), \color{blue}{2}\right)\right)\right) \]

4. Applied egg-rr 9.3%

   \[\leadsto \frac{\pi}{2} - \color{blue}{\left(\pi + \left(0 - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right) \cdot 2\right)} \]

6. Applied egg-rr 9.3%

   \[\leadsto \color{blue}{\frac{{\left(\frac{\pi}{2} - \pi\right)}^{3} - {\left(-\cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) \cdot 2\right)}^{3}}{\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right) + \left({\left(-\cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) \cdot 2\right)}^{2} + \left(\frac{\pi}{2} - \pi\right) \cdot \left(-\cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) \cdot 2\right)\right)}} \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right)}^{3} - {\left(\mathsf{neg}\left(\cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)\right)}^{3}}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right) \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right) + \left(\left(\mathsf{neg}\left(\cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)\right) \cdot \left(\mathsf{neg}\left(\cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)\right) + \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right)} \cdot \left(\mathsf{neg}\left(\cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)\right)\right)} \]

   2. flip3-- N/A

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

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right), \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)\right)}\right) \]

8. Applied egg-rr 9.3%

   \[\leadsto \color{blue}{\pi \cdot -0.5 - -2 \cdot \cos^{-1} \left(\sqrt{0.5 + \frac{x}{-2}}\right)} \]
9. Add Preprocessing

Alternative 7: 5.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (Math.PI * -0.5) + (2.0 * Math.acos(Math.sqrt(0.5)));
}

Python

def code(x):
	return (math.pi * -0.5) + (2.0 * math.acos(math.sqrt(0.5)))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (pi * -0.5) + (2.0 * acos(sqrt(0.5)));
end

Wolfram

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

Derivation

1. Initial program 8.2%

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

   1. asin-acos N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)}\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot 2 + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2}\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right) \cdot 2\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot 2\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot 1 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   8. *-rgt-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)}\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right), \color{blue}{2}\right)\right)\right) \]

4. Applied egg-rr 9.3%

   \[\leadsto \frac{\pi}{2} - \color{blue}{\left(\pi + \left(0 - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right) \cdot 2\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{acos.f64}\left(\color{blue}{\left(\sqrt{\frac{1}{2}}\right)}\right)\right), 2\right)\right)\right) \]
6. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 5.2%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{acos.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right), 2\right)\right)\right) \]

7. Simplified 5.2%

   \[\leadsto \frac{\pi}{2} - \left(\pi + \left(0 - \cos^{-1} \color{blue}{\left(\sqrt{0.5}\right)}\right) \cdot 2\right) \]
8. Step-by-step derivation

   1. associate--r+ N/A

      \[\leadsto \left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right) - \color{blue}{\left(0 - \cos^{-1} \left(\sqrt{\frac{1}{2}}\right)\right) \cdot 2} \]

   2. sub0-neg N/A

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

   3. cancel-sign-sub N/A

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

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right), \color{blue}{\left(\cos^{-1} \left(\sqrt{\frac{1}{2}}\right) \cdot 2\right)}\right) \]

   5. div-inv N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} - \mathsf{PI}\left(\right)\right), \left(\cos^{-1} \color{blue}{\left(\sqrt{\frac{1}{2}}\right)} \cdot 2\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} - \mathsf{PI}\left(\right)\right), \left(\cos^{-1} \left(\sqrt{\frac{1}{2}}\right) \cdot 2\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \mathsf{PI}\left(\right)\right), \left(\cos^{-1} \color{blue}{\left(\sqrt{\frac{1}{2}}\right)} \cdot 2\right)\right) \]

   8. *-un-lft-identity N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - 1 \cdot \mathsf{PI}\left(\right)\right), \left(\cos^{-1} \left(\sqrt{\frac{1}{2}}\right) \cdot 2\right)\right) \]

   9. distribute-rgt-out-- N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} - 1\right)\right), \left(\color{blue}{\cos^{-1} \left(\sqrt{\frac{1}{2}}\right)} \cdot 2\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{-1}{2}\right), \left(\cos^{-1} \left(\sqrt{\frac{1}{2}}\right) \cdot 2\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{-2}\right), \left(\cos^{-1} \left(\sqrt{\frac{1}{2}}\right) \cdot 2\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{1}{-2}\right)\right), \left(\color{blue}{\cos^{-1} \left(\sqrt{\frac{1}{2}}\right)} \cdot 2\right)\right) \]

   13. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{1}{-2}\right)\right), \left(\cos^{-1} \color{blue}{\left(\sqrt{\frac{1}{2}}\right)} \cdot 2\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(\cos^{-1} \left(\sqrt{\frac{1}{2}}\right) \cdot 2\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\cos^{-1} \left(\sqrt{\frac{1}{2}}\right), \color{blue}{2}\right)\right) \]

   16. acos-lowering-acos.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\sqrt{\frac{1}{2}}\right)\right), 2\right)\right) \]

   17. sqrt-lowering-sqrt.f64 5.2%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), 2\right)\right) \]

9. Applied egg-rr 5.2%

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

10. Final simplification 5.2%

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

Developer Target 1: 100.0% accurate, 2.1× 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 2024148 
(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))))))