Ian Simplification

Percentage Accurate: 6.7% → 8.2%

Time: 5.2s

Alternatives: 7

Speedup: 1.1×

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: 6.7% 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: 8.2% accurate, 0.2× speedup

Math

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

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)
      (* (asin (sqrt (/ (/ (- 1.0 (* x x)) (+ 1.0 x)) 2.0))) 2.0)))
    (/ (- (pow (/ PI 2.0) 2.0) (pow t_0 2.0)) (- (/ 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) * (asin(sqrt((((1.0 - (x * x)) / (1.0 + x)) / 2.0))) * 2.0))) / ((pow((((double) M_PI) / 2.0), 2.0) - pow(t_0, 2.0)) / ((((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) * (Math.asin(Math.sqrt((((1.0 - (x * x)) / (1.0 + x)) / 2.0))) * 2.0))) / ((Math.pow((Math.PI / 2.0), 2.0) - Math.pow(t_0, 2.0)) / ((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) * (math.asin(math.sqrt((((1.0 - (x * x)) / (1.0 + x)) / 2.0))) * 2.0))) / ((math.pow((math.pi / 2.0), 2.0) - math.pow(t_0, 2.0)) / ((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) * Float64(asin(sqrt(Float64(Float64(Float64(1.0 - Float64(x * x)) / Float64(1.0 + x)) / 2.0))) * 2.0))) / Float64(Float64((Float64(pi / 2.0) ^ 2.0) - (t_0 ^ 2.0)) / 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) * (asin(sqrt((((1.0 - (x * x)) / (1.0 + x)) / 2.0))) * 2.0))) / ((((pi / 2.0) ^ 2.0) - (t_0 ^ 2.0)) / ((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] * N[(N[ArcSin[N[Sqrt[N[(N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[(Pi / 2.0), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(Pi / 2.0), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.7%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 6.7%

   \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2\right) \cdot \left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2\right)}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2}} \]
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 8.2

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

5. Applied rewrites 8.2%

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

7. Applied rewrites 8.2%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. metadata-eval N/A

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

   5. unpow2 N/A

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

   6. lower--.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-+.f64 8.2

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

9. Applied rewrites 8.2%

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

Alternative 2: 8.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = asin(sqrt(Float64(0.5 * Float64(1.0 - x))))
	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) * Float64(asin(sqrt(Float64(Float64(1.0 - x) / 2.0))) * 2.0))) / Float64(fma(Float64(pi * pi), 0.25, Float64(-4.0 * (t_0 ^ 2.0))) / fma(t_0, -2.0, Float64(0.5 * pi))))
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[(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] * N[(N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.25 + N[(-4.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * -2.0 + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.7%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 6.7%

   \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2\right) \cdot \left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2\right)}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2}} \]
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 8.2

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

5. Applied rewrites 8.2%

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

7. Applied rewrites 8.2%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 8.2%

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

Alternative 3: 8.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	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) * Float64(asin(sqrt(Float64(Float64(1.0 - x) / 2.0))) * 2.0))) / fma(Float64(Float64(pi / 2.0) - acos(sqrt(Float64(0.5 * Float64(1.0 - x))))), 2.0, Float64(0.5 * pi)))
end

Wolfram

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

Derivation

1. Initial program 6.7%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 6.7%

   \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2\right) \cdot \left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2\right)}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2}} \]
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 8.2

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

5. Applied rewrites 8.2%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sqrt-prod N/A

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

   5. lower-asin.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-PI.f64 8.2

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

8. Applied rewrites 8.2%

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

   1. lift-asin.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. asin-acos N/A

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

   6. sqrt-prod N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lower--.f64 N/A

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

   10. sqrt-prod N/A

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

   11. lower-acos.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift--.f64 N/A

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

   14. lift-sqrt.f64 8.2

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

10. Applied rewrites 8.2%

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

Alternative 4: 8.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(N[(Pi * Pi), $MachinePrecision] / 4.0), $MachinePrecision] - N[(N[(N[ArcSin[N[(N[Sqrt[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[ArcSin[N[Sqrt[N[(0.5 * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0 + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.7%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 6.7%

   \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2\right) \cdot \left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2\right)}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2}} \]
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 8.2

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

5. Applied rewrites 8.2%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sqrt-prod N/A

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

   5. lower-asin.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-PI.f64 8.2

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

8. Applied rewrites 8.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. unpow2 N/A

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

   6. lift-PI.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 8.2

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

10. Applied rewrites 8.2%

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

Alternative 5: 8.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

   1. lift-asin.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. asin-acos N/A

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

   6. lift-/.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-acos.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lift-sqrt.f64 8.2

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

3. Applied rewrites 8.2%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 8.2%

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

Alternative 6: 6.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

2. Taylor expanded in x around 0

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

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

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

   2. lower-fma.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-asin.f64 N/A

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

   7. sqrt-unprod N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift--.f64 6.7

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

4. Applied rewrites 6.7%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 6.7

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

7. Applied rewrites 6.7%

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

Alternative 7: 4.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

2. Taylor expanded in x around 0

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

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

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

   2. lower-fma.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-asin.f64 N/A

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

   7. sqrt-unprod N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift--.f64 6.7

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

4. Applied rewrites 6.7%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 4.1%

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

Developer Target 1: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (asin x))

C

double code(double x) {
	return asin(x);
}

Fortran

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 = asin(x)
end function

Java

public static double code(double x) {
	return Math.asin(x);
}

Python

def code(x):
	return math.asin(x)

Julia

function code(x)
	return asin(x)
end

MATLAB

function tmp = code(x)
	tmp = asin(x);
end

Wolfram

code[x_] := N[ArcSin[x], $MachinePrecision]

Reproduce

herbie shell --seed 2025095 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :alt
  (! :herbie-platform default (asin x))

  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))