Ian Simplification

Percentage Accurate: 6.8% → 8.3%

Time: 5.2s

Alternatives: 9

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 8.3% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (sqrt (/ (- 1.0 x) 2.0)))) (t_1 (* t_0 2.0)))
   (/
    (-
     (* (/ PI 2.0) (/ PI 2.0))
     (* (* (asin (/ (sqrt (- 1.0 x)) (sqrt 2.0))) 2.0) t_1))
    (/ (- (pow (/ PI 2.0) 2.0) (* (pow t_0 2.0) 4.0)) (- (/ PI 2.0) t_1)))))

C

double code(double x) {
	double t_0 = asin(sqrt(((1.0 - x) / 2.0)));
	double t_1 = t_0 * 2.0;
	return (((((double) M_PI) / 2.0) * (((double) M_PI) / 2.0)) - ((asin((sqrt((1.0 - x)) / sqrt(2.0))) * 2.0) * t_1)) / ((pow((((double) M_PI) / 2.0), 2.0) - (pow(t_0, 2.0) * 4.0)) / ((((double) M_PI) / 2.0) - t_1));
}

Java

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

Python

def code(x):
	t_0 = math.asin(math.sqrt(((1.0 - x) / 2.0)))
	t_1 = t_0 * 2.0
	return (((math.pi / 2.0) * (math.pi / 2.0)) - ((math.asin((math.sqrt((1.0 - x)) / math.sqrt(2.0))) * 2.0) * t_1)) / ((math.pow((math.pi / 2.0), 2.0) - (math.pow(t_0, 2.0) * 4.0)) / ((math.pi / 2.0) - t_1))

Julia

function code(x)
	t_0 = asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))
	t_1 = Float64(t_0 * 2.0)
	return Float64(Float64(Float64(Float64(pi / 2.0) * Float64(pi / 2.0)) - Float64(Float64(asin(Float64(sqrt(Float64(1.0 - x)) / sqrt(2.0))) * 2.0) * t_1)) / Float64(Float64((Float64(pi / 2.0) ^ 2.0) - Float64((t_0 ^ 2.0) * 4.0)) / Float64(Float64(pi / 2.0) - t_1)))
end

MATLAB

function tmp = code(x)
	t_0 = asin(sqrt(((1.0 - x) / 2.0)));
	t_1 = t_0 * 2.0;
	tmp = (((pi / 2.0) * (pi / 2.0)) - ((asin((sqrt((1.0 - x)) / sqrt(2.0))) * 2.0) * t_1)) / ((((pi / 2.0) ^ 2.0) - ((t_0 ^ 2.0) * 4.0)) / ((pi / 2.0) - t_1));
end

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-asin.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. flip-- N/A

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

   8. lower-/.f64 N/A

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

3. Applied rewrites 6.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sqrt-div N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-/.f64 8.3

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

5. Applied rewrites 8.3%

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

7. Applied rewrites 8.3%

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

Alternative 2: 8.3% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (sqrt (* 0.5 (- 1.0 x))))))
   (*
    (fma
     0.25
     (* PI PI)
     (* -4.0 (* (asin (* (pow (sqrt 2.0) -1.0) (sqrt (- 1.0 x)))) t_0)))
    (/
     (fma PI 0.5 (* -2.0 t_0))
     (fma 0.25 (* PI PI) (* -4.0 (pow t_0 2.0)))))))

C

double code(double x) {
	double t_0 = asin(sqrt((0.5 * (1.0 - x))));
	return fma(0.25, (((double) M_PI) * ((double) M_PI)), (-4.0 * (asin((pow(sqrt(2.0), -1.0) * sqrt((1.0 - x)))) * t_0))) * (fma(((double) M_PI), 0.5, (-2.0 * t_0)) / fma(0.25, (((double) M_PI) * ((double) M_PI)), (-4.0 * pow(t_0, 2.0))));
}

Julia

function code(x)
	t_0 = asin(sqrt(Float64(0.5 * Float64(1.0 - x))))
	return Float64(fma(0.25, Float64(pi * pi), Float64(-4.0 * Float64(asin(Float64((sqrt(2.0) ^ -1.0) * sqrt(Float64(1.0 - x)))) * t_0))) * Float64(fma(pi, 0.5, Float64(-2.0 * t_0)) / fma(0.25, Float64(pi * pi), Float64(-4.0 * (t_0 ^ 2.0)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[Sqrt[N[(0.5 * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[(0.25 * N[(Pi * Pi), $MachinePrecision] + N[(-4.0 * N[(N[ArcSin[N[(N[Power[N[Sqrt[2.0], $MachinePrecision], -1.0], $MachinePrecision] * N[Sqrt[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * 0.5 + N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(0.25 * N[(Pi * Pi), $MachinePrecision] + N[(-4.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-asin.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. flip-- N/A

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

   8. lower-/.f64 N/A

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

3. Applied rewrites 6.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sqrt-div N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-/.f64 8.3

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

5. Applied rewrites 8.3%

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

7. Applied rewrites 8.3%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 8.3%

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

Alternative 3: 8.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (asin (sqrt (/ (- 1.0 x) 2.0))) 2.0)))
   (/
    (-
     (* (/ PI 2.0) (/ PI 2.0))
     (* (* (asin (/ (sqrt (- 1.0 x)) (sqrt 2.0))) 2.0) t_0))
    (+ (/ PI 2.0) t_0))))

C

double code(double x) {
	double t_0 = asin(sqrt(((1.0 - x) / 2.0))) * 2.0;
	return (((((double) M_PI) / 2.0) * (((double) M_PI) / 2.0)) - ((asin((sqrt((1.0 - x)) / sqrt(2.0))) * 2.0) * t_0)) / ((((double) M_PI) / 2.0) + t_0);
}

Java

public static double code(double x) {
	double t_0 = Math.asin(Math.sqrt(((1.0 - x) / 2.0))) * 2.0;
	return (((Math.PI / 2.0) * (Math.PI / 2.0)) - ((Math.asin((Math.sqrt((1.0 - x)) / Math.sqrt(2.0))) * 2.0) * t_0)) / ((Math.PI / 2.0) + t_0);
}

Python

def code(x):
	t_0 = math.asin(math.sqrt(((1.0 - x) / 2.0))) * 2.0
	return (((math.pi / 2.0) * (math.pi / 2.0)) - ((math.asin((math.sqrt((1.0 - x)) / math.sqrt(2.0))) * 2.0) * t_0)) / ((math.pi / 2.0) + t_0)

Julia

function code(x)
	t_0 = Float64(asin(sqrt(Float64(Float64(1.0 - x) / 2.0))) * 2.0)
	return Float64(Float64(Float64(Float64(pi / 2.0) * Float64(pi / 2.0)) - Float64(Float64(asin(Float64(sqrt(Float64(1.0 - x)) / sqrt(2.0))) * 2.0) * t_0)) / Float64(Float64(pi / 2.0) + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = asin(sqrt(((1.0 - x) / 2.0))) * 2.0;
	tmp = (((pi / 2.0) * (pi / 2.0)) - ((asin((sqrt((1.0 - x)) / sqrt(2.0))) * 2.0) * t_0)) / ((pi / 2.0) + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]}, N[(N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[ArcSin[N[(N[Sqrt[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(Pi / 2.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-asin.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. flip-- N/A

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

   8. lower-/.f64 N/A

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

3. Applied rewrites 6.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sqrt-div N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-/.f64 8.3

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

5. Applied rewrites 8.3%

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

Alternative 4: 8.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (sqrt (* 0.5 (- 1.0 x))))))
   (/
    (fma
     (* PI PI)
     0.25
     (* -4.0 (* (asin (* (pow (sqrt 2.0) -1.0) (sqrt (- 1.0 x)))) t_0)))
    (fma t_0 2.0 (* PI 0.5)))))

C

double code(double x) {
	double t_0 = asin(sqrt((0.5 * (1.0 - x))));
	return fma((((double) M_PI) * ((double) M_PI)), 0.25, (-4.0 * (asin((pow(sqrt(2.0), -1.0) * sqrt((1.0 - x)))) * t_0))) / fma(t_0, 2.0, (((double) M_PI) * 0.5));
}

Julia

function code(x)
	t_0 = asin(sqrt(Float64(0.5 * Float64(1.0 - x))))
	return Float64(fma(Float64(pi * pi), 0.25, Float64(-4.0 * Float64(asin(Float64((sqrt(2.0) ^ -1.0) * sqrt(Float64(1.0 - x)))) * t_0))) / fma(t_0, 2.0, Float64(pi * 0.5)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[Sqrt[N[(0.5 * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.25 + N[(-4.0 * N[(N[ArcSin[N[(N[Power[N[Sqrt[2.0], $MachinePrecision], -1.0], $MachinePrecision] * N[Sqrt[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * 2.0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-asin.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. flip-- N/A

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

   8. lower-/.f64 N/A

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

3. Applied rewrites 6.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sqrt-div N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-/.f64 8.3

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

5. Applied rewrites 8.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 8.3%

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

Alternative 5: 8.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 6.8

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

4. Applied rewrites 6.8%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, x, 0.5\right)}}\right) \]
5. Step-by-step derivation

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-/.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-acos.f64 8.3

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

6. Applied rewrites 8.3%

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

Alternative 6: 6.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sqrt-div N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lower-sqrt.f64 6.8

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

3. Applied rewrites 6.8%

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

Alternative 7: 6.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 6.8

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

4. Applied rewrites 6.8%

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

Alternative 8: 5.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{1}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e-310)
   (- (/ PI 2.0) (* 2.0 (asin (sqrt 0.5))))
   (- (/ PI 2.0) (* 2.0 (asin (/ 1.0 (sqrt 2.0)))))))

C

double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(0.5)));
	} else {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin((1.0 / sqrt(2.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(0.5)));
	} else {
		tmp = (Math.PI / 2.0) - (2.0 * Math.asin((1.0 / Math.sqrt(2.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4e-310:
		tmp = (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(0.5)))
	else:
		tmp = (math.pi / 2.0) - (2.0 * math.asin((1.0 / math.sqrt(2.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e-310)
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(0.5))));
	else
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(Float64(1.0 / sqrt(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4e-310)
		tmp = (pi / 2.0) - (2.0 * asin(sqrt(0.5)));
	else
		tmp = (pi / 2.0) - (2.0 * asin((1.0 / sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4e-310], N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.999999999999988e-310

   1. Initial program 8.6%

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

   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 5.9%

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

   if -3.999999999999988e-310 < x

   1. Initial program 5.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-sqrt.f64 8.0

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

   3. Applied rewrites 8.0%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 5.8%

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

Alternative 9: 4.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

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. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64
  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))