Ian Simplification

Percentage Accurate: 6.8% → 8.3%

Time: 1.8min

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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{2} - \pi\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\\ t_3 := 2 \cdot t\_2\\ t_4 := t\_3 - \pi\\ t_5 := \left(0 - 2\right) \cdot t\_2\\ t_6 := {t\_5}^{2}\\ \frac{t\_1 - t\_6}{\frac{t\_1 \cdot t\_1 - {t\_5}^{4}}{\frac{\frac{\pi \cdot \left(\pi \cdot \pi\right)}{8} + {t\_4}^{3}}{\frac{\pi \cdot \pi}{4} + \left({t\_4}^{2} + \frac{\pi}{2} \cdot \left(\pi - t\_3\right)\right)} \cdot \left(t\_1 + t\_6\right)}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ PI 2.0) PI))
        (t_1 (* t_0 t_0))
        (t_2 (acos (pow (+ 0.5 (/ x -2.0)) 0.5)))
        (t_3 (* 2.0 t_2))
        (t_4 (- t_3 PI))
        (t_5 (* (- 0.0 2.0) t_2))
        (t_6 (pow t_5 2.0)))
   (/
    (- t_1 t_6)
    (/
     (- (* t_1 t_1) (pow t_5 4.0))
     (*
      (/
       (+ (/ (* PI (* PI PI)) 8.0) (pow t_4 3.0))
       (+ (/ (* PI PI) 4.0) (+ (pow t_4 2.0) (* (/ PI 2.0) (- PI t_3)))))
      (+ t_1 t_6))))))

C

double code(double x) {
	double t_0 = (((double) M_PI) / 2.0) - ((double) M_PI);
	double t_1 = t_0 * t_0;
	double t_2 = acos(pow((0.5 + (x / -2.0)), 0.5));
	double t_3 = 2.0 * t_2;
	double t_4 = t_3 - ((double) M_PI);
	double t_5 = (0.0 - 2.0) * t_2;
	double t_6 = pow(t_5, 2.0);
	return (t_1 - t_6) / (((t_1 * t_1) - pow(t_5, 4.0)) / (((((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) / 8.0) + pow(t_4, 3.0)) / (((((double) M_PI) * ((double) M_PI)) / 4.0) + (pow(t_4, 2.0) + ((((double) M_PI) / 2.0) * (((double) M_PI) - t_3))))) * (t_1 + t_6)));
}

Java

public static double code(double x) {
	double t_0 = (Math.PI / 2.0) - Math.PI;
	double t_1 = t_0 * t_0;
	double t_2 = Math.acos(Math.pow((0.5 + (x / -2.0)), 0.5));
	double t_3 = 2.0 * t_2;
	double t_4 = t_3 - Math.PI;
	double t_5 = (0.0 - 2.0) * t_2;
	double t_6 = Math.pow(t_5, 2.0);
	return (t_1 - t_6) / (((t_1 * t_1) - Math.pow(t_5, 4.0)) / (((((Math.PI * (Math.PI * Math.PI)) / 8.0) + Math.pow(t_4, 3.0)) / (((Math.PI * Math.PI) / 4.0) + (Math.pow(t_4, 2.0) + ((Math.PI / 2.0) * (Math.PI - t_3))))) * (t_1 + t_6)));
}

Python

def code(x):
	t_0 = (math.pi / 2.0) - math.pi
	t_1 = t_0 * t_0
	t_2 = math.acos(math.pow((0.5 + (x / -2.0)), 0.5))
	t_3 = 2.0 * t_2
	t_4 = t_3 - math.pi
	t_5 = (0.0 - 2.0) * t_2
	t_6 = math.pow(t_5, 2.0)
	return (t_1 - t_6) / (((t_1 * t_1) - math.pow(t_5, 4.0)) / (((((math.pi * (math.pi * math.pi)) / 8.0) + math.pow(t_4, 3.0)) / (((math.pi * math.pi) / 4.0) + (math.pow(t_4, 2.0) + ((math.pi / 2.0) * (math.pi - t_3))))) * (t_1 + t_6)))

Julia

function code(x)
	t_0 = Float64(Float64(pi / 2.0) - pi)
	t_1 = Float64(t_0 * t_0)
	t_2 = acos((Float64(0.5 + Float64(x / -2.0)) ^ 0.5))
	t_3 = Float64(2.0 * t_2)
	t_4 = Float64(t_3 - pi)
	t_5 = Float64(Float64(0.0 - 2.0) * t_2)
	t_6 = t_5 ^ 2.0
	return Float64(Float64(t_1 - t_6) / Float64(Float64(Float64(t_1 * t_1) - (t_5 ^ 4.0)) / Float64(Float64(Float64(Float64(Float64(pi * Float64(pi * pi)) / 8.0) + (t_4 ^ 3.0)) / Float64(Float64(Float64(pi * pi) / 4.0) + Float64((t_4 ^ 2.0) + Float64(Float64(pi / 2.0) * Float64(pi - t_3))))) * Float64(t_1 + t_6))))
end

MATLAB

function tmp = code(x)
	t_0 = (pi / 2.0) - pi;
	t_1 = t_0 * t_0;
	t_2 = acos(((0.5 + (x / -2.0)) ^ 0.5));
	t_3 = 2.0 * t_2;
	t_4 = t_3 - pi;
	t_5 = (0.0 - 2.0) * t_2;
	t_6 = t_5 ^ 2.0;
	tmp = (t_1 - t_6) / (((t_1 * t_1) - (t_5 ^ 4.0)) / (((((pi * (pi * pi)) / 8.0) + (t_4 ^ 3.0)) / (((pi * pi) / 4.0) + ((t_4 ^ 2.0) + ((pi / 2.0) * (pi - t_3))))) * (t_1 + t_6)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(Pi / 2.0), $MachinePrecision] - Pi), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[ArcCos[N[Power[N[(0.5 + N[(x / -2.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - Pi), $MachinePrecision]}, Block[{t$95$5 = N[(N[(0.0 - 2.0), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$6 = N[Power[t$95$5, 2.0], $MachinePrecision]}, N[(N[(t$95$1 - t$95$6), $MachinePrecision] / N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[Power[t$95$5, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] / 8.0), $MachinePrecision] + N[Power[t$95$4, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * Pi), $MachinePrecision] / 4.0), $MachinePrecision] + N[(N[Power[t$95$4, 2.0], $MachinePrecision] + N[(N[(Pi / 2.0), $MachinePrecision] * N[(Pi - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 7.8%

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

   1. associate--r+ N/A

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

   2. flip-- N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right) \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right) - \left(\left(0 - \cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right) \cdot \left(\left(0 - \cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)\right), \color{blue}{\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right) + \left(0 - \cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)}\right) \]

6. Applied egg-rr 9.3%

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

8. Applied egg-rr 9.3%

   \[\leadsto \frac{\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right) - {\left(2 \cdot \left(0 - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)\right)}^{2}}{\color{blue}{\frac{\left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right)\right) \cdot \left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right)\right) - {\left(-2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{4}}{\left(\left(\frac{\pi}{2} - \pi\right) + 2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right) \cdot \left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right) + {\left(-2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{2}\right)}}} \]

10. Applied egg-rr 9.4%

    \[\leadsto \frac{\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right) - {\left(2 \cdot \left(0 - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)\right)}^{2}}{\frac{\left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right)\right) \cdot \left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right)\right) - {\left(-2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{4}}{\color{blue}{\frac{\frac{\pi \cdot \left(\pi \cdot \pi\right)}{8} + {\left(2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) + \left(0 - \pi\right)\right)}^{3}}{\frac{\pi \cdot \pi}{4} + \left({\left(2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) + \left(0 - \pi\right)\right)}^{2} - \frac{\pi}{2} \cdot \left(2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) + \left(0 - \pi\right)\right)\right)}} \cdot \left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right) + {\left(-2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{2}\right)}} \]

11. Final simplification 9.4%

    \[\leadsto \frac{\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right) - {\left(\left(0 - 2\right) \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{2}}{\frac{\left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right)\right) \cdot \left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right)\right) - {\left(\left(0 - 2\right) \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{4}}{\frac{\frac{\pi \cdot \left(\pi \cdot \pi\right)}{8} + {\left(2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) - \pi\right)}^{3}}{\frac{\pi \cdot \pi}{4} + \left({\left(2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right) - \pi\right)}^{2} + \frac{\pi}{2} \cdot \left(\pi - 2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)\right)} \cdot \left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right) + {\left(\left(0 - 2\right) \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{2}\right)}} \]
12. Add Preprocessing

Alternative 2: 8.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{2} - \pi\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\\ t_3 := \left(0 - 2\right) \cdot t\_2\\ t_4 := {t\_3}^{2}\\ \frac{t\_1 - t\_4}{\frac{t\_1 \cdot t\_1 - {t\_3}^{4}}{\left(t\_1 + t\_4\right) \cdot \frac{-1}{\frac{-1}{t\_0 + 2 \cdot t\_2}}}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ PI 2.0) PI))
        (t_1 (* t_0 t_0))
        (t_2 (acos (pow (+ 0.5 (/ x -2.0)) 0.5)))
        (t_3 (* (- 0.0 2.0) t_2))
        (t_4 (pow t_3 2.0)))
   (/
    (- t_1 t_4)
    (/
     (- (* t_1 t_1) (pow t_3 4.0))
     (* (+ t_1 t_4) (/ -1.0 (/ -1.0 (+ t_0 (* 2.0 t_2)))))))))

C

double code(double x) {
	double t_0 = (((double) M_PI) / 2.0) - ((double) M_PI);
	double t_1 = t_0 * t_0;
	double t_2 = acos(pow((0.5 + (x / -2.0)), 0.5));
	double t_3 = (0.0 - 2.0) * t_2;
	double t_4 = pow(t_3, 2.0);
	return (t_1 - t_4) / (((t_1 * t_1) - pow(t_3, 4.0)) / ((t_1 + t_4) * (-1.0 / (-1.0 / (t_0 + (2.0 * t_2))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.PI / 2.0) - Math.PI;
	double t_1 = t_0 * t_0;
	double t_2 = Math.acos(Math.pow((0.5 + (x / -2.0)), 0.5));
	double t_3 = (0.0 - 2.0) * t_2;
	double t_4 = Math.pow(t_3, 2.0);
	return (t_1 - t_4) / (((t_1 * t_1) - Math.pow(t_3, 4.0)) / ((t_1 + t_4) * (-1.0 / (-1.0 / (t_0 + (2.0 * t_2))))));
}

Python

def code(x):
	t_0 = (math.pi / 2.0) - math.pi
	t_1 = t_0 * t_0
	t_2 = math.acos(math.pow((0.5 + (x / -2.0)), 0.5))
	t_3 = (0.0 - 2.0) * t_2
	t_4 = math.pow(t_3, 2.0)
	return (t_1 - t_4) / (((t_1 * t_1) - math.pow(t_3, 4.0)) / ((t_1 + t_4) * (-1.0 / (-1.0 / (t_0 + (2.0 * t_2))))))

Julia

function code(x)
	t_0 = Float64(Float64(pi / 2.0) - pi)
	t_1 = Float64(t_0 * t_0)
	t_2 = acos((Float64(0.5 + Float64(x / -2.0)) ^ 0.5))
	t_3 = Float64(Float64(0.0 - 2.0) * t_2)
	t_4 = t_3 ^ 2.0
	return Float64(Float64(t_1 - t_4) / Float64(Float64(Float64(t_1 * t_1) - (t_3 ^ 4.0)) / Float64(Float64(t_1 + t_4) * Float64(-1.0 / Float64(-1.0 / Float64(t_0 + Float64(2.0 * t_2)))))))
end

MATLAB

function tmp = code(x)
	t_0 = (pi / 2.0) - pi;
	t_1 = t_0 * t_0;
	t_2 = acos(((0.5 + (x / -2.0)) ^ 0.5));
	t_3 = (0.0 - 2.0) * t_2;
	t_4 = t_3 ^ 2.0;
	tmp = (t_1 - t_4) / (((t_1 * t_1) - (t_3 ^ 4.0)) / ((t_1 + t_4) * (-1.0 / (-1.0 / (t_0 + (2.0 * t_2))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(Pi / 2.0), $MachinePrecision] - Pi), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[ArcCos[N[Power[N[(0.5 + N[(x / -2.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.0 - 2.0), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$3, 2.0], $MachinePrecision]}, N[(N[(t$95$1 - t$95$4), $MachinePrecision] / N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[Power[t$95$3, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 + t$95$4), $MachinePrecision] * N[(-1.0 / N[(-1.0 / N[(t$95$0 + N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 7.8%

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

   1. associate--r+ N/A

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

   2. flip-- N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right) \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right) - \left(\left(0 - \cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right) \cdot \left(\left(0 - \cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)\right), \color{blue}{\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right) + \left(0 - \cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)}\right) \]

6. Applied egg-rr 9.3%

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

8. Applied egg-rr 9.3%

   \[\leadsto \frac{\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right) - {\left(2 \cdot \left(0 - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)\right)}^{2}}{\color{blue}{\frac{\left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right)\right) \cdot \left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right)\right) - {\left(-2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{4}}{\left(\left(\frac{\pi}{2} - \pi\right) + 2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right) \cdot \left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right) + {\left(-2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{2}\right)}}} \]

10. Applied egg-rr 9.3%

    \[\leadsto \frac{\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right) - {\left(2 \cdot \left(0 - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)\right)}^{2}}{\frac{\left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right)\right) \cdot \left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right)\right) - {\left(-2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{4}}{\color{blue}{\frac{1}{\frac{1}{\left(\frac{\pi}{2} - \pi\right) + 2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)}}} \cdot \left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right) + {\left(-2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{2}\right)}} \]

11. Final simplification 9.3%

    \[\leadsto \frac{\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right) - {\left(\left(0 - 2\right) \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{2}}{\frac{\left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right)\right) \cdot \left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right)\right) - {\left(\left(0 - 2\right) \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{4}}{\left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right) + {\left(\left(0 - 2\right) \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{2}\right) \cdot \frac{-1}{\frac{-1}{\left(\frac{\pi}{2} - \pi\right) + 2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)}}}} \]
12. Add Preprocessing

Alternative 3: 8.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{2} - \pi\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\\ t_3 := \left(0 - 2\right) \cdot t\_2\\ \frac{\pi \cdot \left(\pi \cdot 0.25\right) + {\cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2} \cdot -4}{\frac{t\_1 \cdot t\_1 - {t\_3}^{4}}{\left(t\_1 + {t\_3}^{2}\right) \cdot \left(t\_0 + 2 \cdot t\_2\right)}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ PI 2.0) PI))
        (t_1 (* t_0 t_0))
        (t_2 (acos (pow (+ 0.5 (/ x -2.0)) 0.5)))
        (t_3 (* (- 0.0 2.0) t_2)))
   (/
    (+ (* PI (* PI 0.25)) (* (pow (acos (sqrt (+ 0.5 (* x -0.5)))) 2.0) -4.0))
    (/
     (- (* t_1 t_1) (pow t_3 4.0))
     (* (+ t_1 (pow t_3 2.0)) (+ t_0 (* 2.0 t_2)))))))

C

double code(double x) {
	double t_0 = (((double) M_PI) / 2.0) - ((double) M_PI);
	double t_1 = t_0 * t_0;
	double t_2 = acos(pow((0.5 + (x / -2.0)), 0.5));
	double t_3 = (0.0 - 2.0) * t_2;
	return ((((double) M_PI) * (((double) M_PI) * 0.25)) + (pow(acos(sqrt((0.5 + (x * -0.5)))), 2.0) * -4.0)) / (((t_1 * t_1) - pow(t_3, 4.0)) / ((t_1 + pow(t_3, 2.0)) * (t_0 + (2.0 * t_2))));
}

Java

public static double code(double x) {
	double t_0 = (Math.PI / 2.0) - Math.PI;
	double t_1 = t_0 * t_0;
	double t_2 = Math.acos(Math.pow((0.5 + (x / -2.0)), 0.5));
	double t_3 = (0.0 - 2.0) * t_2;
	return ((Math.PI * (Math.PI * 0.25)) + (Math.pow(Math.acos(Math.sqrt((0.5 + (x * -0.5)))), 2.0) * -4.0)) / (((t_1 * t_1) - Math.pow(t_3, 4.0)) / ((t_1 + Math.pow(t_3, 2.0)) * (t_0 + (2.0 * t_2))));
}

Python

def code(x):
	t_0 = (math.pi / 2.0) - math.pi
	t_1 = t_0 * t_0
	t_2 = math.acos(math.pow((0.5 + (x / -2.0)), 0.5))
	t_3 = (0.0 - 2.0) * t_2
	return ((math.pi * (math.pi * 0.25)) + (math.pow(math.acos(math.sqrt((0.5 + (x * -0.5)))), 2.0) * -4.0)) / (((t_1 * t_1) - math.pow(t_3, 4.0)) / ((t_1 + math.pow(t_3, 2.0)) * (t_0 + (2.0 * t_2))))

Julia

function code(x)
	t_0 = Float64(Float64(pi / 2.0) - pi)
	t_1 = Float64(t_0 * t_0)
	t_2 = acos((Float64(0.5 + Float64(x / -2.0)) ^ 0.5))
	t_3 = Float64(Float64(0.0 - 2.0) * t_2)
	return Float64(Float64(Float64(pi * Float64(pi * 0.25)) + Float64((acos(sqrt(Float64(0.5 + Float64(x * -0.5)))) ^ 2.0) * -4.0)) / Float64(Float64(Float64(t_1 * t_1) - (t_3 ^ 4.0)) / Float64(Float64(t_1 + (t_3 ^ 2.0)) * Float64(t_0 + Float64(2.0 * t_2)))))
end

MATLAB

function tmp = code(x)
	t_0 = (pi / 2.0) - pi;
	t_1 = t_0 * t_0;
	t_2 = acos(((0.5 + (x / -2.0)) ^ 0.5));
	t_3 = (0.0 - 2.0) * t_2;
	tmp = ((pi * (pi * 0.25)) + ((acos(sqrt((0.5 + (x * -0.5)))) ^ 2.0) * -4.0)) / (((t_1 * t_1) - (t_3 ^ 4.0)) / ((t_1 + (t_3 ^ 2.0)) * (t_0 + (2.0 * t_2))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(Pi / 2.0), $MachinePrecision] - Pi), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[ArcCos[N[Power[N[(0.5 + N[(x / -2.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.0 - 2.0), $MachinePrecision] * t$95$2), $MachinePrecision]}, N[(N[(N[(Pi * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[ArcCos[N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[Power[t$95$3, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 7.8%

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

   1. associate--r+ N/A

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

   2. flip-- N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right) \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right) - \left(\left(0 - \cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right) \cdot \left(\left(0 - \cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)\right), \color{blue}{\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right) + \left(0 - \cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)}\right) \]

6. Applied egg-rr 9.3%

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

8. Applied egg-rr 9.3%

   \[\leadsto \frac{\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right) - {\left(2 \cdot \left(0 - \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)\right)}^{2}}{\color{blue}{\frac{\left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right)\right) \cdot \left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right)\right) - {\left(-2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{4}}{\left(\left(\frac{\pi}{2} - \pi\right) + 2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right) \cdot \left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right) + {\left(-2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{2}\right)}}} \]

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \mathsf{PI}\left(\right)\right)}^{2} - 4 \cdot {\cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)}^{2}\right)}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(2, \mathsf{acos.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(x, -2\right)\right), \frac{1}{2}\right)\right)\right)\right), 4\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(2, \mathsf{acos.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(x, -2\right)\right), \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(2, \mathsf{acos.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(x, -2\right)\right), \frac{1}{2}\right)\right)\right)\right), 2\right)\right)\right)\right)\right) \]
10. Step-by-step derivation

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

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

    2. metadata-eval N/A

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

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

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

    4. distribute-lft-neg-in N/A

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

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

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

11. Simplified 9.3%

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

12. Final simplification 9.3%

    \[\leadsto \frac{\pi \cdot \left(\pi \cdot 0.25\right) + {\cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2} \cdot -4}{\frac{\left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right)\right) \cdot \left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right)\right) - {\left(\left(0 - 2\right) \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{4}}{\left(\left(\frac{\pi}{2} - \pi\right) \cdot \left(\frac{\pi}{2} - \pi\right) + {\left(\left(0 - 2\right) \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}^{2}\right) \cdot \left(\left(\frac{\pi}{2} - \pi\right) + 2 \cdot \cos^{-1} \left({\left(0.5 + \frac{x}{-2}\right)}^{0.5}\right)\right)}} \]
13. Add Preprocessing

Alternative 4: 8.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.8%

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

   1. associate--r+ N/A

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

   2. flip-- N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right) \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right) - \left(\left(0 - \cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right) \cdot \left(\left(0 - \cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)\right), \color{blue}{\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \mathsf{PI}\left(\right)\right) + \left(0 - \cos^{-1} \left({\left(\frac{1}{2} + \frac{x}{-2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)}\right) \]

6. Applied egg-rr 9.3%

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

7. Taylor expanded in x around 0

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

9. Simplified 9.3%

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

10. Final simplification 9.3%

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

Alternative 5: 8.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.8%

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

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

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

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

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

   4. sub-neg N/A

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

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

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

   6. sub-neg N/A

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

   7. neg-mul-1 N/A

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

   8. distribute-rgt-out N/A

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

   9. metadata-eval N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

   14. metadata-eval N/A

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

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

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

7. Simplified 9.3%

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

8. Final simplification 9.3%

   \[\leadsto \pi \cdot -0.5 + 2 \cdot \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \]
9. 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 7.8%

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

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

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

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

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

   4. sub-neg N/A

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

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

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

   6. sub-neg N/A

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

   7. neg-mul-1 N/A

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

   8. distribute-rgt-out N/A

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

   9. metadata-eval N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

   14. metadata-eval N/A

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

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

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

7. Simplified 9.3%

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

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 5.4%

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

10. Simplified 5.4%

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

11. Final simplification 5.4%

    \[\leadsto \pi \cdot -0.5 + 2 \cdot \cos^{-1} \left(\sqrt{0.5}\right) \]
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 2024144 
(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))))))