Ian Simplification

Percentage Accurate: 6.8% → 8.3%

Time: 5.6s

Alternatives: 6

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: 6.8% 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

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

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 0.25 (* PI PI) (* -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(0.25, (((double) M_PI) * ((double) M_PI)), (-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(0.25, Float64(pi * pi), 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[(0.25 * N[(Pi * Pi), $MachinePrecision] + 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 6.8%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

7. Applied rewrites 8.3%

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

8. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

10. Applied rewrites 8.3%

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

Alternative 2: 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 3: 8.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (sqrt (* 0.5 (- 1.0 x))))))
   (/
    (fma
     (* PI PI)
     0.25
     (* -4.0 (* (asin (* (/ 1.0 (sqrt 2.0)) (sqrt (- 1.0 x)))) t_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 fma((((double) M_PI) * ((double) M_PI)), 0.25, (-4.0 * (asin(((1.0 / sqrt(2.0)) * sqrt((1.0 - x)))) * t_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(fma(Float64(pi * pi), 0.25, Float64(-4.0 * Float64(asin(Float64(Float64(1.0 / sqrt(2.0)) * sqrt(Float64(1.0 - x)))) * t_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[(Pi * Pi), $MachinePrecision] * 0.25 + N[(-4.0 * N[(N[ArcSin[N[(N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * 2.0 + N[(0.5 * Pi), $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%

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

Alternative 4: 8.3% 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 6.8%

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

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

3. Applied rewrites 8.3%

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

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 8.3%

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

Alternative 5: 6.8% 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 6.8%

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

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 6.8

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

4. Applied rewrites 6.8%

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

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 6.8

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

7. Applied rewrites 6.8%

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

Alternative 6: 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 6.8%

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

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 6.8

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

4. Applied rewrites 6.8%

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

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

Reproduce

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