Ian Simplification

Percentage Accurate: 7.0% → 8.5%

Time: 12.8s

Alternatives: 8

Speedup: 1.1×

Specification

Math

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

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: 7.0% accurate, 1.0× speedup

Math

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

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: 8.5% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (- PI (* 0.5 PI)))
       (t_1 (* t_0 t_0))
       (t_2 (* (* -0.5 PI) (* -0.5 PI)))
       (t_3 (acos (sqrt (* (- 1.0 x) 0.5))))
       (t_4 (pow t_3 2.0))
       (t_5 (* t_4 4.0)))
  (/
   (/
    (- (* t_1 t_1) (* t_5 t_5))
    (* (- 1.0 (/ (* -4.0 t_4) t_2)) t_2))
   (+ (- (* 0.5 PI) PI) (* t_3 -2.0)))))

C

double code(double x) {
	double t_0 = ((double) M_PI) - (0.5 * ((double) M_PI));
	double t_1 = t_0 * t_0;
	double t_2 = (-0.5 * ((double) M_PI)) * (-0.5 * ((double) M_PI));
	double t_3 = acos(sqrt(((1.0 - x) * 0.5)));
	double t_4 = pow(t_3, 2.0);
	double t_5 = t_4 * 4.0;
	return (((t_1 * t_1) - (t_5 * t_5)) / ((1.0 - ((-4.0 * t_4) / t_2)) * t_2)) / (((0.5 * ((double) M_PI)) - ((double) M_PI)) + (t_3 * -2.0));
}

Java

public static double code(double x) {
	double t_0 = Math.PI - (0.5 * Math.PI);
	double t_1 = t_0 * t_0;
	double t_2 = (-0.5 * Math.PI) * (-0.5 * Math.PI);
	double t_3 = Math.acos(Math.sqrt(((1.0 - x) * 0.5)));
	double t_4 = Math.pow(t_3, 2.0);
	double t_5 = t_4 * 4.0;
	return (((t_1 * t_1) - (t_5 * t_5)) / ((1.0 - ((-4.0 * t_4) / t_2)) * t_2)) / (((0.5 * Math.PI) - Math.PI) + (t_3 * -2.0));
}

Python

def code(x):
	t_0 = math.pi - (0.5 * math.pi)
	t_1 = t_0 * t_0
	t_2 = (-0.5 * math.pi) * (-0.5 * math.pi)
	t_3 = math.acos(math.sqrt(((1.0 - x) * 0.5)))
	t_4 = math.pow(t_3, 2.0)
	t_5 = t_4 * 4.0
	return (((t_1 * t_1) - (t_5 * t_5)) / ((1.0 - ((-4.0 * t_4) / t_2)) * t_2)) / (((0.5 * math.pi) - math.pi) + (t_3 * -2.0))

Julia

function code(x)
	t_0 = Float64(pi - Float64(0.5 * pi))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(Float64(-0.5 * pi) * Float64(-0.5 * pi))
	t_3 = acos(sqrt(Float64(Float64(1.0 - x) * 0.5)))
	t_4 = t_3 ^ 2.0
	t_5 = Float64(t_4 * 4.0)
	return Float64(Float64(Float64(Float64(t_1 * t_1) - Float64(t_5 * t_5)) / Float64(Float64(1.0 - Float64(Float64(-4.0 * t_4) / t_2)) * t_2)) / Float64(Float64(Float64(0.5 * pi) - pi) + Float64(t_3 * -2.0)))
end

MATLAB

function tmp = code(x)
	t_0 = pi - (0.5 * pi);
	t_1 = t_0 * t_0;
	t_2 = (-0.5 * pi) * (-0.5 * pi);
	t_3 = acos(sqrt(((1.0 - x) * 0.5)));
	t_4 = t_3 ^ 2.0;
	t_5 = t_4 * 4.0;
	tmp = (((t_1 * t_1) - (t_5 * t_5)) / ((1.0 - ((-4.0 * t_4) / t_2)) * t_2)) / (((0.5 * pi) - pi) + (t_3 * -2.0));
end

Wolfram

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

Derivation

1. Initial program 7.0%

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

   1. lift-*.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. asin-acos N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. sub-flip-reverse N/A

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

   7. acos-asin N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-asin.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. distribute-rgt-in N/A

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

3. Applied rewrites 8.5%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--r+ N/A

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

   4. flip-- N/A

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

   5. lower-unsound-/.f64 N/A

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

5. Applied rewrites 8.5%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-unsound-/.f64 N/A

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

7. Applied rewrites 8.5%

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

   1. lift-+.f64 N/A

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

   2. add-flip N/A

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

   3. sub-to-mult N/A

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

   4. lower-unsound-*.f64 N/A

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

9. Applied rewrites 8.5%

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

Alternative 2: 8.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

   1. lift-*.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. asin-acos N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. sub-flip-reverse N/A

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

   7. acos-asin N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-asin.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. distribute-rgt-in N/A

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

3. Applied rewrites 8.5%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--r+ N/A

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

   4. flip-- N/A

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

   5. lower-unsound-/.f64 N/A

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

5. Applied rewrites 8.5%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-unsound-/.f64 N/A

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

7. Applied rewrites 8.5%

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

Alternative 3: 8.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\\ \left(\left(0.125 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \left(0.5 \cdot \pi\right) - 16 \cdot {t\_0}^{4}\right) \cdot \frac{-1}{\left(4 \cdot {t\_0}^{2} - \left(-0.5 \cdot \pi\right) \cdot \left(0.5 \cdot \pi\right)\right) \cdot \left(2 \cdot t\_0 - -0.5 \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (acos (sqrt (* (- 1.0 x) 0.5)))))
  (*
   (-
    (* (* 0.125 (* (* PI PI) PI)) (* 0.5 PI))
    (* 16.0 (pow t_0 4.0)))
   (/
    -1.0
    (*
     (- (* 4.0 (pow t_0 2.0)) (* (* -0.5 PI) (* 0.5 PI)))
     (- (* 2.0 t_0) (* -0.5 PI)))))))

C

double code(double x) {
	double t_0 = acos(sqrt(((1.0 - x) * 0.5)));
	return (((0.125 * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI))) * (0.5 * ((double) M_PI))) - (16.0 * pow(t_0, 4.0))) * (-1.0 / (((4.0 * pow(t_0, 2.0)) - ((-0.5 * ((double) M_PI)) * (0.5 * ((double) M_PI)))) * ((2.0 * t_0) - (-0.5 * ((double) M_PI)))));
}

Java

public static double code(double x) {
	double t_0 = Math.acos(Math.sqrt(((1.0 - x) * 0.5)));
	return (((0.125 * ((Math.PI * Math.PI) * Math.PI)) * (0.5 * Math.PI)) - (16.0 * Math.pow(t_0, 4.0))) * (-1.0 / (((4.0 * Math.pow(t_0, 2.0)) - ((-0.5 * Math.PI) * (0.5 * Math.PI))) * ((2.0 * t_0) - (-0.5 * Math.PI))));
}

Python

def code(x):
	t_0 = math.acos(math.sqrt(((1.0 - x) * 0.5)))
	return (((0.125 * ((math.pi * math.pi) * math.pi)) * (0.5 * math.pi)) - (16.0 * math.pow(t_0, 4.0))) * (-1.0 / (((4.0 * math.pow(t_0, 2.0)) - ((-0.5 * math.pi) * (0.5 * math.pi))) * ((2.0 * t_0) - (-0.5 * math.pi))))

Julia

function code(x)
	t_0 = acos(sqrt(Float64(Float64(1.0 - x) * 0.5)))
	return Float64(Float64(Float64(Float64(0.125 * Float64(Float64(pi * pi) * pi)) * Float64(0.5 * pi)) - Float64(16.0 * (t_0 ^ 4.0))) * Float64(-1.0 / Float64(Float64(Float64(4.0 * (t_0 ^ 2.0)) - Float64(Float64(-0.5 * pi) * Float64(0.5 * pi))) * Float64(Float64(2.0 * t_0) - Float64(-0.5 * pi)))))
end

MATLAB

function tmp = code(x)
	t_0 = acos(sqrt(((1.0 - x) * 0.5)));
	tmp = (((0.125 * ((pi * pi) * pi)) * (0.5 * pi)) - (16.0 * (t_0 ^ 4.0))) * (-1.0 / (((4.0 * (t_0 ^ 2.0)) - ((-0.5 * pi) * (0.5 * pi))) * ((2.0 * t_0) - (-0.5 * pi))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(0.125 * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision] - N[(16.0 * N[Power[t$95$0, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(N[(N[(4.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(-0.5 * Pi), $MachinePrecision] * N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * t$95$0), $MachinePrecision] - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.0%

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

   1. lift-*.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. asin-acos N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. sub-flip-reverse N/A

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

   7. acos-asin N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-asin.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. distribute-rgt-in N/A

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

3. Applied rewrites 8.5%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--r+ N/A

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

   4. flip-- N/A

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

   5. lower-unsound-/.f64 N/A

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

5. Applied rewrites 8.5%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-unsound-/.f64 N/A

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

7. Applied rewrites 8.5%

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

9. Applied rewrites 8.5%

   \[\leadsto \color{blue}{\left(\left(0.125 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \left(0.5 \cdot \pi\right) - 16 \cdot {\cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)}^{4}\right) \cdot \frac{-1}{\left(4 \cdot {\cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)}^{2} - \left(-0.5 \cdot \pi\right) \cdot \left(0.5 \cdot \pi\right)\right) \cdot \left(2 \cdot \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right) - -0.5 \cdot \pi\right)}} \]
10. Add Preprocessing

Alternative 4: 8.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\\ \frac{\left(0.125 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \left(0.5 \cdot \pi\right) - 16 \cdot {t\_0}^{4}}{\left(-2 \cdot t\_0 - 0.5 \cdot \pi\right) \cdot \left(4 \cdot {t\_0}^{2} - \left(-0.5 \cdot \pi\right) \cdot \left(0.5 \cdot \pi\right)\right)} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (acos (sqrt (* (- 1.0 x) 0.5)))))
  (/
   (-
    (* (* 0.125 (* (* PI PI) PI)) (* 0.5 PI))
    (* 16.0 (pow t_0 4.0)))
   (*
    (- (* -2.0 t_0) (* 0.5 PI))
    (- (* 4.0 (pow t_0 2.0)) (* (* -0.5 PI) (* 0.5 PI)))))))

C

double code(double x) {
	double t_0 = acos(sqrt(((1.0 - x) * 0.5)));
	return (((0.125 * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI))) * (0.5 * ((double) M_PI))) - (16.0 * pow(t_0, 4.0))) / (((-2.0 * t_0) - (0.5 * ((double) M_PI))) * ((4.0 * pow(t_0, 2.0)) - ((-0.5 * ((double) M_PI)) * (0.5 * ((double) M_PI)))));
}

Java

public static double code(double x) {
	double t_0 = Math.acos(Math.sqrt(((1.0 - x) * 0.5)));
	return (((0.125 * ((Math.PI * Math.PI) * Math.PI)) * (0.5 * Math.PI)) - (16.0 * Math.pow(t_0, 4.0))) / (((-2.0 * t_0) - (0.5 * Math.PI)) * ((4.0 * Math.pow(t_0, 2.0)) - ((-0.5 * Math.PI) * (0.5 * Math.PI))));
}

Python

def code(x):
	t_0 = math.acos(math.sqrt(((1.0 - x) * 0.5)))
	return (((0.125 * ((math.pi * math.pi) * math.pi)) * (0.5 * math.pi)) - (16.0 * math.pow(t_0, 4.0))) / (((-2.0 * t_0) - (0.5 * math.pi)) * ((4.0 * math.pow(t_0, 2.0)) - ((-0.5 * math.pi) * (0.5 * math.pi))))

Julia

function code(x)
	t_0 = acos(sqrt(Float64(Float64(1.0 - x) * 0.5)))
	return Float64(Float64(Float64(Float64(0.125 * Float64(Float64(pi * pi) * pi)) * Float64(0.5 * pi)) - Float64(16.0 * (t_0 ^ 4.0))) / Float64(Float64(Float64(-2.0 * t_0) - Float64(0.5 * pi)) * Float64(Float64(4.0 * (t_0 ^ 2.0)) - Float64(Float64(-0.5 * pi) * Float64(0.5 * pi)))))
end

MATLAB

function tmp = code(x)
	t_0 = acos(sqrt(((1.0 - x) * 0.5)));
	tmp = (((0.125 * ((pi * pi) * pi)) * (0.5 * pi)) - (16.0 * (t_0 ^ 4.0))) / (((-2.0 * t_0) - (0.5 * pi)) * ((4.0 * (t_0 ^ 2.0)) - ((-0.5 * pi) * (0.5 * pi))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(0.125 * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision] - N[(16.0 * N[Power[t$95$0, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(-2.0 * t$95$0), $MachinePrecision] - N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision] * N[(N[(4.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(-0.5 * Pi), $MachinePrecision] * N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.0%

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

   1. lift-*.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. asin-acos N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. sub-flip-reverse N/A

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

   7. acos-asin N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-asin.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. distribute-rgt-in N/A

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

3. Applied rewrites 8.5%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--r+ N/A

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

   4. flip-- N/A

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

   5. lower-unsound-/.f64 N/A

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

5. Applied rewrites 8.5%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-unsound-/.f64 N/A

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

7. Applied rewrites 8.5%

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

9. Applied rewrites 8.5%

   \[\leadsto \color{blue}{\frac{\left(0.125 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \left(0.5 \cdot \pi\right) - 16 \cdot {\cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)}^{4}}{\left(-2 \cdot \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right) - 0.5 \cdot \pi\right) \cdot \left(4 \cdot {\cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)}^{2} - \left(-0.5 \cdot \pi\right) \cdot \left(0.5 \cdot \pi\right)\right)}} \]
10. Add Preprocessing

Alternative 5: 8.5% accurate, 1.1× speedup

Math

\[\left(1.5707963267948966 - \pi\right) - -2 \cdot \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right) \]

FPCore

(FPCore (x)
  :precision binary64
  (- (- 1.5707963267948966 PI) (* -2.0 (acos (sqrt (* (- 1.0 x) 0.5))))))

C

double code(double x) {
	return (1.5707963267948966 - ((double) M_PI)) - (-2.0 * acos(sqrt(((1.0 - x) * 0.5))));
}

Java

public static double code(double x) {
	return (1.5707963267948966 - Math.PI) - (-2.0 * Math.acos(Math.sqrt(((1.0 - x) * 0.5))));
}

Python

def code(x):
	return (1.5707963267948966 - math.pi) - (-2.0 * math.acos(math.sqrt(((1.0 - x) * 0.5))))

Julia

function code(x)
	return Float64(Float64(1.5707963267948966 - pi) - Float64(-2.0 * acos(sqrt(Float64(Float64(1.0 - x) * 0.5)))))
end

MATLAB

function tmp = code(x)
	tmp = (1.5707963267948966 - pi) - (-2.0 * acos(sqrt(((1.0 - x) * 0.5))));
end

Wolfram

code[x_] := N[(N[(1.5707963267948966 - Pi), $MachinePrecision] - N[(-2.0 * N[ArcCos[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

   1. lift-*.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. asin-acos N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. sub-flip-reverse N/A

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

   7. acos-asin N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-asin.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. distribute-rgt-in N/A

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

3. Applied rewrites 8.5%

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

4. Evaluated real constant 8.5%

   \[\leadsto \color{blue}{1.5707963267948966} - \left(\pi + \left(-\cos^{-1} \left(\sqrt{0.5 \cdot \left(1 - x\right)}\right)\right) \cdot 2\right) \]
5. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\frac{884279719003555}{562949953421312} - \left(\pi + \left(-\cos^{-1} \left(\sqrt{\frac{1}{2} \cdot \left(1 - x\right)}\right)\right) \cdot 2\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{884279719003555}{562949953421312} - \color{blue}{\left(\pi + \left(-\cos^{-1} \left(\sqrt{\frac{1}{2} \cdot \left(1 - x\right)}\right)\right) \cdot 2\right)} \]

   3. associate--r+ N/A

      \[\leadsto \color{blue}{\left(\frac{884279719003555}{562949953421312} - \pi\right) - \left(-\cos^{-1} \left(\sqrt{\frac{1}{2} \cdot \left(1 - x\right)}\right)\right) \cdot 2} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\frac{884279719003555}{562949953421312} - \pi\right) - \color{blue}{\left(-\cos^{-1} \left(\sqrt{\frac{1}{2} \cdot \left(1 - x\right)}\right)\right) \cdot 2} \]

   5. lift-neg.f64 N/A

      \[\leadsto \left(\frac{884279719003555}{562949953421312} - \pi\right) - \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} \cdot \left(1 - x\right)}\right)\right)\right)} \cdot 2 \]

   6. distribute-lft-neg-out N/A

      \[\leadsto \left(\frac{884279719003555}{562949953421312} - \pi\right) - \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2} \cdot \left(1 - x\right)}\right) \cdot 2\right)\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\frac{884279719003555}{562949953421312} - \pi\right) - \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - x\right)}}\right) \cdot 2\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \left(\frac{884279719003555}{562949953421312} - \pi\right) - \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\color{blue}{\left(1 - x\right) \cdot \frac{1}{2}}}\right) \cdot 2\right)\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\frac{884279719003555}{562949953421312} - \pi\right) - \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\color{blue}{\left(1 - x\right) \cdot \frac{1}{2}}}\right) \cdot 2\right)\right) \]

   10. distribute-rgt-neg-out N/A

       \[\leadsto \left(\frac{884279719003555}{562949953421312} - \pi\right) - \color{blue}{\cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot \frac{1}{2}}\right) \cdot \left(\mathsf{neg}\left(2\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \left(\frac{884279719003555}{562949953421312} - \pi\right) - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot \frac{1}{2}}\right) \cdot \color{blue}{-2} \]

   12. lift-*.f64 N/A

       \[\leadsto \left(\frac{884279719003555}{562949953421312} - \pi\right) - \color{blue}{\cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot \frac{1}{2}}\right) \cdot -2} \]

6. Applied rewrites 8.5%

   \[\leadsto \color{blue}{\left(1.5707963267948966 - \pi\right) - -2 \cdot \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)} \]
7. Add Preprocessing

Alternative 6: 7.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-160}:\\ \;\;\;\;1.5707963267948966 - 2 \cdot \sin^{-1} \left(\sqrt{0.5 \cdot \left(1 - x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1.5707963267948966 - \cos^{-1} \left(\sqrt{0.5}\right) \cdot -2\right) - \pi\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x -1.15e-160)
  (- 1.5707963267948966 (* 2.0 (asin (sqrt (* 0.5 (- 1.0 x))))))
  (- (- 1.5707963267948966 (* (acos (sqrt 0.5)) -2.0)) PI)))

C

double code(double x) {
	double tmp;
	if (x <= -1.15e-160) {
		tmp = 1.5707963267948966 - (2.0 * asin(sqrt((0.5 * (1.0 - x)))));
	} else {
		tmp = (1.5707963267948966 - (acos(sqrt(0.5)) * -2.0)) - ((double) M_PI);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.15e-160) {
		tmp = 1.5707963267948966 - (2.0 * Math.asin(Math.sqrt((0.5 * (1.0 - x)))));
	} else {
		tmp = (1.5707963267948966 - (Math.acos(Math.sqrt(0.5)) * -2.0)) - Math.PI;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.15e-160:
		tmp = 1.5707963267948966 - (2.0 * math.asin(math.sqrt((0.5 * (1.0 - x)))))
	else:
		tmp = (1.5707963267948966 - (math.acos(math.sqrt(0.5)) * -2.0)) - math.pi
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.15e-160)
		tmp = Float64(1.5707963267948966 - Float64(2.0 * asin(sqrt(Float64(0.5 * Float64(1.0 - x))))));
	else
		tmp = Float64(Float64(1.5707963267948966 - Float64(acos(sqrt(0.5)) * -2.0)) - pi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.15e-160)
		tmp = 1.5707963267948966 - (2.0 * asin(sqrt((0.5 * (1.0 - x)))));
	else
		tmp = (1.5707963267948966 - (acos(sqrt(0.5)) * -2.0)) - pi;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.15e-160], N[(1.5707963267948966 - N[(2.0 * N[ArcSin[N[Sqrt[N[(0.5 * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.5707963267948966 - N[(N[ArcCos[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] - Pi), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1499999999999999e-160

   1. Initial program 7.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 4.1%

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

   5. Evaluated real constant 4.1%

      \[\leadsto \color{blue}{1.5707963267948966} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto 1.5707963267948966 - 2 \cdot \color{blue}{\sin^{-1} \left(\sqrt{\frac{1}{2} \cdot \left(1 - x\right)}\right)} \]
   7. Step-by-step derivation

      1. lower-asin.f64 N/A

         \[\leadsto \frac{884279719003555}{562949953421312} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1}{2} \cdot \left(1 - x\right)}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{884279719003555}{562949953421312} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1}{2} \cdot \left(1 - x\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{884279719003555}{562949953421312} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1}{2} \cdot \left(1 - x\right)}\right) \]

      4. lower--.f64 7.0%

         \[\leadsto 1.5707963267948966 - 2 \cdot \sin^{-1} \left(\sqrt{0.5 \cdot \left(1 - x\right)}\right) \]

   8. Applied rewrites 7.0%

      \[\leadsto 1.5707963267948966 - 2 \cdot \color{blue}{\sin^{-1} \left(\sqrt{0.5 \cdot \left(1 - x\right)}\right)} \]

   if -1.1499999999999999e-160 < x

   1. Initial program 7.0%

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

      1. lift-*.f64 N/A

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

      2. lift-asin.f64 N/A

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

      3. asin-acos N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-flip-reverse N/A

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

      7. acos-asin N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-asin.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. distribute-rgt-in N/A

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

   3. Applied rewrites 8.5%

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

   4. Evaluated real constant 8.5%

      \[\leadsto \color{blue}{1.5707963267948966} - \left(\pi + \left(-\cos^{-1} \left(\sqrt{0.5 \cdot \left(1 - x\right)}\right)\right) \cdot 2\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto 1.5707963267948966 - \left(\pi + \left(-\cos^{-1} \left(\sqrt{\color{blue}{\frac{1}{2}}}\right)\right) \cdot 2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 5.4%

      \[\leadsto 1.5707963267948966 - \left(\pi + \left(-\cos^{-1} \left(\sqrt{\color{blue}{0.5}}\right)\right) \cdot 2\right) \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\frac{884279719003555}{562949953421312} - \left(\pi + \left(-\cos^{-1} \left(\sqrt{\frac{1}{2}}\right)\right) \cdot 2\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{884279719003555}{562949953421312} - \color{blue}{\left(\pi + \left(-\cos^{-1} \left(\sqrt{\frac{1}{2}}\right)\right) \cdot 2\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{884279719003555}{562949953421312} - \color{blue}{\left(\left(-\cos^{-1} \left(\sqrt{\frac{1}{2}}\right)\right) \cdot 2 + \pi\right)} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\frac{884279719003555}{562949953421312} - \left(-\cos^{-1} \left(\sqrt{\frac{1}{2}}\right)\right) \cdot 2\right) - \pi} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{884279719003555}{562949953421312} - \left(-\cos^{-1} \left(\sqrt{\frac{1}{2}}\right)\right) \cdot 2\right) - \pi} \]

      6. lower--.f64 5.4%

         \[\leadsto \color{blue}{\left(1.5707963267948966 - \left(-\cos^{-1} \left(\sqrt{0.5}\right)\right) \cdot 2\right)} - \pi \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{884279719003555}{562949953421312} - \color{blue}{\left(-\cos^{-1} \left(\sqrt{\frac{1}{2}}\right)\right) \cdot 2}\right) - \pi \]

      8. lift-neg.f64 N/A

         \[\leadsto \left(\frac{884279719003555}{562949953421312} - \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2}}\right)\right)\right)} \cdot 2\right) - \pi \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \left(\frac{884279719003555}{562949953421312} - \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2}}\right) \cdot 2\right)\right)}\right) - \pi \]

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

          \[\leadsto \left(\frac{884279719003555}{562949953421312} - \color{blue}{\cos^{-1} \left(\sqrt{\frac{1}{2}}\right) \cdot \left(\mathsf{neg}\left(2\right)\right)}\right) - \pi \]

      11. metadata-eval N/A

          \[\leadsto \left(\frac{884279719003555}{562949953421312} - \cos^{-1} \left(\sqrt{\frac{1}{2}}\right) \cdot \color{blue}{-2}\right) - \pi \]

      12. lower-*.f64 5.4%

          \[\leadsto \left(1.5707963267948966 - \color{blue}{\cos^{-1} \left(\sqrt{0.5}\right) \cdot -2}\right) - \pi \]

   9. Applied rewrites 5.4%

      \[\leadsto \color{blue}{\left(1.5707963267948966 - \cos^{-1} \left(\sqrt{0.5}\right) \cdot -2\right) - \pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 5.4% accurate, 1.2× speedup

Math

\[\left(1.5707963267948966 - \cos^{-1} \left(\sqrt{0.5}\right) \cdot -2\right) - \pi \]

FPCore

(FPCore (x)
  :precision binary64
  (- (- 1.5707963267948966 (* (acos (sqrt 0.5)) -2.0)) PI))

C

double code(double x) {
	return (1.5707963267948966 - (acos(sqrt(0.5)) * -2.0)) - ((double) M_PI);
}

Java

public static double code(double x) {
	return (1.5707963267948966 - (Math.acos(Math.sqrt(0.5)) * -2.0)) - Math.PI;
}

Python

def code(x):
	return (1.5707963267948966 - (math.acos(math.sqrt(0.5)) * -2.0)) - math.pi

Julia

function code(x)
	return Float64(Float64(1.5707963267948966 - Float64(acos(sqrt(0.5)) * -2.0)) - pi)
end

MATLAB

function tmp = code(x)
	tmp = (1.5707963267948966 - (acos(sqrt(0.5)) * -2.0)) - pi;
end

Wolfram

code[x_] := N[(N[(1.5707963267948966 - N[(N[ArcCos[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] - Pi), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

   1. lift-*.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. asin-acos N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. sub-flip-reverse N/A

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

   7. acos-asin N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-asin.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. distribute-rgt-in N/A

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

3. Applied rewrites 8.5%

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

4. Evaluated real constant 8.5%

   \[\leadsto \color{blue}{1.5707963267948966} - \left(\pi + \left(-\cos^{-1} \left(\sqrt{0.5 \cdot \left(1 - x\right)}\right)\right) \cdot 2\right) \]

5. Taylor expanded in x around 0

   \[\leadsto 1.5707963267948966 - \left(\pi + \left(-\cos^{-1} \left(\sqrt{\color{blue}{\frac{1}{2}}}\right)\right) \cdot 2\right) \]
6. Step-by-step derivation

7. Applied rewrites 5.4%

   \[\leadsto 1.5707963267948966 - \left(\pi + \left(-\cos^{-1} \left(\sqrt{\color{blue}{0.5}}\right)\right) \cdot 2\right) \]
8. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\frac{884279719003555}{562949953421312} - \left(\pi + \left(-\cos^{-1} \left(\sqrt{\frac{1}{2}}\right)\right) \cdot 2\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{884279719003555}{562949953421312} - \color{blue}{\left(\pi + \left(-\cos^{-1} \left(\sqrt{\frac{1}{2}}\right)\right) \cdot 2\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{884279719003555}{562949953421312} - \color{blue}{\left(\left(-\cos^{-1} \left(\sqrt{\frac{1}{2}}\right)\right) \cdot 2 + \pi\right)} \]

   4. associate--r+ N/A

      \[\leadsto \color{blue}{\left(\frac{884279719003555}{562949953421312} - \left(-\cos^{-1} \left(\sqrt{\frac{1}{2}}\right)\right) \cdot 2\right) - \pi} \]

   5. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{884279719003555}{562949953421312} - \left(-\cos^{-1} \left(\sqrt{\frac{1}{2}}\right)\right) \cdot 2\right) - \pi} \]

   6. lower--.f64 5.4%

      \[\leadsto \color{blue}{\left(1.5707963267948966 - \left(-\cos^{-1} \left(\sqrt{0.5}\right)\right) \cdot 2\right)} - \pi \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\frac{884279719003555}{562949953421312} - \color{blue}{\left(-\cos^{-1} \left(\sqrt{\frac{1}{2}}\right)\right) \cdot 2}\right) - \pi \]

   8. lift-neg.f64 N/A

      \[\leadsto \left(\frac{884279719003555}{562949953421312} - \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2}}\right)\right)\right)} \cdot 2\right) - \pi \]

   9. distribute-lft-neg-out N/A

      \[\leadsto \left(\frac{884279719003555}{562949953421312} - \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1}{2}}\right) \cdot 2\right)\right)}\right) - \pi \]

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

       \[\leadsto \left(\frac{884279719003555}{562949953421312} - \color{blue}{\cos^{-1} \left(\sqrt{\frac{1}{2}}\right) \cdot \left(\mathsf{neg}\left(2\right)\right)}\right) - \pi \]

   11. metadata-eval N/A

       \[\leadsto \left(\frac{884279719003555}{562949953421312} - \cos^{-1} \left(\sqrt{\frac{1}{2}}\right) \cdot \color{blue}{-2}\right) - \pi \]

   12. lower-*.f64 5.4%

       \[\leadsto \left(1.5707963267948966 - \color{blue}{\cos^{-1} \left(\sqrt{0.5}\right) \cdot -2}\right) - \pi \]

9. Applied rewrites 5.4%

   \[\leadsto \color{blue}{\left(1.5707963267948966 - \cos^{-1} \left(\sqrt{0.5}\right) \cdot -2\right) - \pi} \]
10. Add Preprocessing

Alternative 8: 4.1% accurate, 1.2× speedup

Math

\[1.5707963267948966 - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \]

FPCore

(FPCore (x)
  :precision binary64
  (- 1.5707963267948966 (* 2.0 (asin (sqrt 0.5)))))

C

double code(double x) {
	return 1.5707963267948966 - (2.0 * asin(sqrt(0.5)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 1.5707963267948966d0 - (2.0d0 * asin(sqrt(0.5d0)))
end function

Java

public static double code(double x) {
	return 1.5707963267948966 - (2.0 * Math.asin(Math.sqrt(0.5)));
}

Python

def code(x):
	return 1.5707963267948966 - (2.0 * math.asin(math.sqrt(0.5)))

Julia

function code(x)
	return Float64(1.5707963267948966 - Float64(2.0 * asin(sqrt(0.5))))
end

MATLAB

function tmp = code(x)
	tmp = 1.5707963267948966 - (2.0 * asin(sqrt(0.5)));
end

Wolfram

code[x_] := N[(1.5707963267948966 - N[(2.0 * N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 4.1%

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

5. Evaluated real constant 4.1%

   \[\leadsto \color{blue}{1.5707963267948966} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64
  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))