Ian Simplification

Percentage Accurate: 6.8% → 8.3%

Time: 21.1s

Alternatives: 6

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: 6.8% 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: 8.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(\pi \cdot \pi\right)\\ t_1 := \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right) \cdot -2\\ t_2 := {\left(\pi + -2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{3}\\ \frac{-1}{\frac{\pi \cdot \left(\pi \cdot 0.25\right) + \left(\pi + t\_1\right) \cdot \left(t\_1 + \pi \cdot 1.5\right)}{\frac{{\left(0 - t\_2\right)}^{2} - \left(t\_0 \cdot t\_0\right) \cdot 0.015625}{t\_2 + \frac{t\_0}{8}}}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* PI (* PI PI)))
        (t_1 (* (acos (sqrt (+ 0.5 (* -0.5 x)))) -2.0))
        (t_2 (pow (+ PI (* -2.0 (acos (pow (+ 0.5 (/ x -2.0)) 0.5)))) 3.0)))
   (/
    -1.0
    (/
     (+ (* PI (* PI 0.25)) (* (+ PI t_1) (+ t_1 (* PI 1.5))))
     (/
      (- (pow (- 0.0 t_2) 2.0) (* (* t_0 t_0) 0.015625))
      (+ t_2 (/ t_0 8.0)))))))

C

double code(double x) {
	double t_0 = ((double) M_PI) * (((double) M_PI) * ((double) M_PI));
	double t_1 = acos(sqrt((0.5 + (-0.5 * x)))) * -2.0;
	double t_2 = pow((((double) M_PI) + (-2.0 * acos(pow((0.5 + (x / -2.0)), 0.5)))), 3.0);
	return -1.0 / (((((double) M_PI) * (((double) M_PI) * 0.25)) + ((((double) M_PI) + t_1) * (t_1 + (((double) M_PI) * 1.5)))) / ((pow((0.0 - t_2), 2.0) - ((t_0 * t_0) * 0.015625)) / (t_2 + (t_0 / 8.0))));
}

Java

public static double code(double x) {
	double t_0 = Math.PI * (Math.PI * Math.PI);
	double t_1 = Math.acos(Math.sqrt((0.5 + (-0.5 * x)))) * -2.0;
	double t_2 = Math.pow((Math.PI + (-2.0 * Math.acos(Math.pow((0.5 + (x / -2.0)), 0.5)))), 3.0);
	return -1.0 / (((Math.PI * (Math.PI * 0.25)) + ((Math.PI + t_1) * (t_1 + (Math.PI * 1.5)))) / ((Math.pow((0.0 - t_2), 2.0) - ((t_0 * t_0) * 0.015625)) / (t_2 + (t_0 / 8.0))));
}

Python

def code(x):
	t_0 = math.pi * (math.pi * math.pi)
	t_1 = math.acos(math.sqrt((0.5 + (-0.5 * x)))) * -2.0
	t_2 = math.pow((math.pi + (-2.0 * math.acos(math.pow((0.5 + (x / -2.0)), 0.5)))), 3.0)
	return -1.0 / (((math.pi * (math.pi * 0.25)) + ((math.pi + t_1) * (t_1 + (math.pi * 1.5)))) / ((math.pow((0.0 - t_2), 2.0) - ((t_0 * t_0) * 0.015625)) / (t_2 + (t_0 / 8.0))))

Julia

function code(x)
	t_0 = Float64(pi * Float64(pi * pi))
	t_1 = Float64(acos(sqrt(Float64(0.5 + Float64(-0.5 * x)))) * -2.0)
	t_2 = Float64(pi + Float64(-2.0 * acos((Float64(0.5 + Float64(x / -2.0)) ^ 0.5)))) ^ 3.0
	return Float64(-1.0 / Float64(Float64(Float64(pi * Float64(pi * 0.25)) + Float64(Float64(pi + t_1) * Float64(t_1 + Float64(pi * 1.5)))) / Float64(Float64((Float64(0.0 - t_2) ^ 2.0) - Float64(Float64(t_0 * t_0) * 0.015625)) / Float64(t_2 + Float64(t_0 / 8.0)))))
end

MATLAB

function tmp = code(x)
	t_0 = pi * (pi * pi);
	t_1 = acos(sqrt((0.5 + (-0.5 * x)))) * -2.0;
	t_2 = (pi + (-2.0 * acos(((0.5 + (x / -2.0)) ^ 0.5)))) ^ 3.0;
	tmp = -1.0 / (((pi * (pi * 0.25)) + ((pi + t_1) * (t_1 + (pi * 1.5)))) / ((((0.0 - t_2) ^ 2.0) - ((t_0 * t_0) * 0.015625)) / (t_2 + (t_0 / 8.0))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcCos[N[Sqrt[N[(0.5 + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(Pi + N[(-2.0 * N[ArcCos[N[Power[N[(0.5 + N[(x / -2.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]}, N[(-1.0 / N[(N[(N[(Pi * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision] + N[(N[(Pi + t$95$1), $MachinePrecision] * N[(t$95$1 + N[(Pi * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[(0.0 - t$95$2), $MachinePrecision], 2.0], $MachinePrecision] - N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.015625), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 + N[(t$95$0 / 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 6.9%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 8.3%

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

6. Applied egg-rr 8.3%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{4} \cdot {\mathsf{PI}\left(\right)}^{2} + \left(\mathsf{PI}\left(\right) - 2 \cdot \cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) - 2 \cdot \cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)\right)}{\frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{3} - {\left(\mathsf{PI}\left(\right) - 2 \cdot \cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)\right)}^{3}}\right)}\right) \]

9. Simplified 8.3%

   \[\leadsto \frac{1}{\color{blue}{\frac{\pi \cdot \left(\pi \cdot 0.25\right) + \left(\pi + \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right) \cdot -2\right) \cdot \left(\pi \cdot 1.5 + \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right) \cdot -2\right)}{\pi \cdot \left(0.125 \cdot \left(\pi \cdot \pi\right)\right) - {\left(\pi + \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right) \cdot -2\right)}^{3}}}} \]

11. Applied egg-rr 8.3%

    \[\leadsto \frac{1}{\frac{\pi \cdot \left(\pi \cdot 0.25\right) + \left(\pi + \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right) \cdot -2\right) \cdot \left(\pi \cdot 1.5 + \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right) \cdot -2\right)}{\color{blue}{\frac{\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot 0.015625 - {\left(0 - {\left(\pi + \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) \cdot -2\right)}^{3}\right)}^{2}}{\frac{\pi \cdot \left(\pi \cdot \pi\right)}{8} - \left(0 - {\left(\pi + \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) \cdot -2\right)}^{3}\right)}}}} \]

12. Final simplification 8.3%

    \[\leadsto \frac{-1}{\frac{\pi \cdot \left(\pi \cdot 0.25\right) + \left(\pi + \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right) \cdot -2\right) \cdot \left(\cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right) \cdot -2 + \pi \cdot 1.5\right)}{\frac{{\left(0 - {\left(\pi + -2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{3}\right)}^{2} - \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot 0.015625}{{\left(\pi + -2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{3} + \frac{\pi \cdot \left(\pi \cdot \pi\right)}{8}}}} \]
13. Add Preprocessing

Alternative 2: 8.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\\ t_1 := \pi + t\_0\\ \frac{\left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot 0.0625 - {t\_1}^{4}}{\left(\frac{\pi \cdot \pi}{4} + {t\_1}^{2}\right) \cdot \left(t\_0 + \left(\pi + \frac{\pi}{2}\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* -2.0 (acos (pow (+ 0.5 (/ x -2.0)) 0.5)))) (t_1 (+ PI t_0)))
   (/
    (- (* (* PI (* PI (* PI PI))) 0.0625) (pow t_1 4.0))
    (* (+ (/ (* PI PI) 4.0) (pow t_1 2.0)) (+ t_0 (+ PI (/ PI 2.0)))))))

C

double code(double x) {
	double t_0 = -2.0 * acos(pow((0.5 + (x / -2.0)), 0.5));
	double t_1 = ((double) M_PI) + t_0;
	return (((((double) M_PI) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))) * 0.0625) - pow(t_1, 4.0)) / ((((((double) M_PI) * ((double) M_PI)) / 4.0) + pow(t_1, 2.0)) * (t_0 + (((double) M_PI) + (((double) M_PI) / 2.0))));
}

Java

public static double code(double x) {
	double t_0 = -2.0 * Math.acos(Math.pow((0.5 + (x / -2.0)), 0.5));
	double t_1 = Math.PI + t_0;
	return (((Math.PI * (Math.PI * (Math.PI * Math.PI))) * 0.0625) - Math.pow(t_1, 4.0)) / ((((Math.PI * Math.PI) / 4.0) + Math.pow(t_1, 2.0)) * (t_0 + (Math.PI + (Math.PI / 2.0))));
}

Python

def code(x):
	t_0 = -2.0 * math.acos(math.pow((0.5 + (x / -2.0)), 0.5))
	t_1 = math.pi + t_0
	return (((math.pi * (math.pi * (math.pi * math.pi))) * 0.0625) - math.pow(t_1, 4.0)) / ((((math.pi * math.pi) / 4.0) + math.pow(t_1, 2.0)) * (t_0 + (math.pi + (math.pi / 2.0))))

Julia

function code(x)
	t_0 = Float64(-2.0 * acos((Float64(0.5 + Float64(x / -2.0)) ^ 0.5)))
	t_1 = Float64(pi + t_0)
	return Float64(Float64(Float64(Float64(pi * Float64(pi * Float64(pi * pi))) * 0.0625) - (t_1 ^ 4.0)) / Float64(Float64(Float64(Float64(pi * pi) / 4.0) + (t_1 ^ 2.0)) * Float64(t_0 + Float64(pi + Float64(pi / 2.0)))))
end

MATLAB

function tmp = code(x)
	t_0 = -2.0 * acos(((0.5 + (x / -2.0)) ^ 0.5));
	t_1 = pi + t_0;
	tmp = (((pi * (pi * (pi * pi))) * 0.0625) - (t_1 ^ 4.0)) / ((((pi * pi) / 4.0) + (t_1 ^ 2.0)) * (t_0 + (pi + (pi / 2.0))));
end

Wolfram

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

Derivation

1. Initial program 6.9%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 8.3%

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

6. Applied egg-rr 8.3%

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

8. Applied egg-rr 8.2%

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

10. Applied egg-rr 8.3%

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

11. Final simplification 8.3%

    \[\leadsto \frac{\left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot 0.0625 - {\left(\pi + -2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{4}}{\left(\frac{\pi \cdot \pi}{4} + {\left(\pi + -2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{2}\right) \cdot \left(-2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) + \left(\pi + \frac{\pi}{2}\right)\right)} \]
12. Add Preprocessing

Alternative 3: 8.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\\ \frac{\pi + \left(2 \cdot \left(t\_0 - \pi\right) + \frac{8 \cdot {t\_0}^{2}}{\pi}\right)}{\frac{2}{\pi} \cdot \left(\pi + 2 \cdot \cos^{-1} \left(\sqrt{0.5 + \frac{x}{-2}}\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (sqrt (+ 0.5 (* -0.5 x))))))
   (/
    (+ PI (+ (* 2.0 (- t_0 PI)) (/ (* 8.0 (pow t_0 2.0)) PI)))
    (* (/ 2.0 PI) (+ PI (* 2.0 (acos (sqrt (+ 0.5 (/ x -2.0))))))))))

C

double code(double x) {
	double t_0 = acos(sqrt((0.5 + (-0.5 * x))));
	return (((double) M_PI) + ((2.0 * (t_0 - ((double) M_PI))) + ((8.0 * pow(t_0, 2.0)) / ((double) M_PI)))) / ((2.0 / ((double) M_PI)) * (((double) M_PI) + (2.0 * acos(sqrt((0.5 + (x / -2.0)))))));
}

Java

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

Python

def code(x):
	t_0 = math.acos(math.sqrt((0.5 + (-0.5 * x))))
	return (math.pi + ((2.0 * (t_0 - math.pi)) + ((8.0 * math.pow(t_0, 2.0)) / math.pi))) / ((2.0 / math.pi) * (math.pi + (2.0 * math.acos(math.sqrt((0.5 + (x / -2.0)))))))

Julia

function code(x)
	t_0 = acos(sqrt(Float64(0.5 + Float64(-0.5 * x))))
	return Float64(Float64(pi + Float64(Float64(2.0 * Float64(t_0 - pi)) + Float64(Float64(8.0 * (t_0 ^ 2.0)) / pi))) / Float64(Float64(2.0 / pi) * Float64(pi + Float64(2.0 * acos(sqrt(Float64(0.5 + Float64(x / -2.0))))))))
end

MATLAB

function tmp = code(x)
	t_0 = acos(sqrt((0.5 + (-0.5 * x))));
	tmp = (pi + ((2.0 * (t_0 - pi)) + ((8.0 * (t_0 ^ 2.0)) / pi))) / ((2.0 / pi) * (pi + (2.0 * acos(sqrt((0.5 + (x / -2.0)))))));
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[(Pi + N[(N[(2.0 * N[(t$95$0 - Pi), $MachinePrecision]), $MachinePrecision] + N[(N[(8.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 / Pi), $MachinePrecision] * N[(Pi + N[(2.0 * N[ArcCos[N[Sqrt[N[(0.5 + N[(x / -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.9%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 8.3%

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

6. Applied egg-rr 8.3%

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

8. Applied egg-rr 8.3%

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

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) + 2 \cdot \cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)\right) - 2 \cdot \frac{{\mathsf{PI}\left(\right)}^{2} - 4 \cdot {\cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)}^{2}}{\mathsf{PI}\left(\right)}\right)}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{PI.f64}\left(\right)\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(x, -2\right)\right)\right)\right), 2\right)\right)\right)\right) \]

11. Simplified 8.3%

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

12. Final simplification 8.3%

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

Alternative 4: 8.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/
  -1.0
  (/ -1.0 (+ (* 2.0 (acos (sqrt (+ 0.5 (/ x -2.0))))) (- (/ PI 2.0) PI)))))

C

double code(double x) {
	return -1.0 / (-1.0 / ((2.0 * acos(sqrt((0.5 + (x / -2.0))))) + ((((double) M_PI) / 2.0) - ((double) M_PI))));
}

Java

public static double code(double x) {
	return -1.0 / (-1.0 / ((2.0 * Math.acos(Math.sqrt((0.5 + (x / -2.0))))) + ((Math.PI / 2.0) - Math.PI)));
}

Python

def code(x):
	return -1.0 / (-1.0 / ((2.0 * math.acos(math.sqrt((0.5 + (x / -2.0))))) + ((math.pi / 2.0) - math.pi)))

Julia

function code(x)
	return Float64(-1.0 / Float64(-1.0 / Float64(Float64(2.0 * acos(sqrt(Float64(0.5 + Float64(x / -2.0))))) + Float64(Float64(pi / 2.0) - pi))))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 / (-1.0 / ((2.0 * acos(sqrt((0.5 + (x / -2.0))))) + ((pi / 2.0) - pi)));
end

Wolfram

code[x_] := N[(-1.0 / N[(-1.0 / N[(N[(2.0 * N[ArcCos[N[Sqrt[N[(0.5 + N[(x / -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(Pi / 2.0), $MachinePrecision] - Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 8.3%

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

6. Applied egg-rr 8.3%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4} + \left(\mathsf{PI}\left(\right) - \cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right) \cdot 2\right) \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{PI}\left(\right) - \cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)\right)}{\frac{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{8} - {\left(\mathsf{PI}\left(\right) - \cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)}^{3}}\right)}\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\frac{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{8} - {\left(\mathsf{PI}\left(\right) - \cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)}^{3}}{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4} + \left(\mathsf{PI}\left(\right) - \cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right) \cdot 2\right) \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{PI}\left(\right) - \cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)\right)}}}\right)\right) \]

8. Applied egg-rr 8.3%

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

9. Final simplification 8.3%

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

Alternative 5: 8.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 8.3%

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

5. Taylor expanded in x around 0

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

   1. associate--r+ N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. *-commutative N/A

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

   6. sub-neg N/A

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

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

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

   8. *-lft-identity N/A

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

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

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

   10. metadata-eval N/A

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

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

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

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

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

   13. *-commutative N/A

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

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

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

   15. metadata-eval N/A

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

7. Simplified 8.3%

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

Alternative 6: 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 6.9%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 8.3%

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

5. Taylor expanded in x around 0

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

   1. associate--r+ N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. *-commutative N/A

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

   6. sub-neg N/A

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

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

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

   8. *-lft-identity N/A

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

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

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

   10. metadata-eval N/A

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

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

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

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

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

   13. *-commutative N/A

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

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

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

   15. metadata-eval N/A

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

7. Simplified 8.3%

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

8. Taylor expanded in x around 0

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

   1. sqrt-lowering-sqrt.f64 5.2%

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

10. Simplified 5.2%

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

11. Final simplification 5.2%

    \[\leadsto \pi \cdot -0.5 + 2 \cdot \cos^{-1} \left(\sqrt{0.5}\right) \]
12. 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 2024140 
(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))))))