Ian Simplification

Percentage Accurate: 6.8% → 8.3%

Time: 5.7s

Alternatives: 9

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 8.3% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (sqrt (/ (- 1.0 x) 2.0))))
        (t_1 (* t_0 2.0))
        (t_2 (* t_0 t_0))
        (t_3 (* t_2 4.0))
        (t_4 (* PI (/ PI 4.0))))
   (/
    (-
     (* (/ PI 2.0) (/ PI 2.0))
     (* (* (asin (/ (sqrt (- 1.0 x)) (sqrt 2.0))) 2.0) t_1))
    (/ (/ (- (* t_3 t_3) (* t_4 t_4)) (fma t_2 4.0 t_4)) (- t_1 (/ PI 2.0))))))

C

double code(double x) {
	double t_0 = asin(sqrt(((1.0 - x) / 2.0)));
	double t_1 = t_0 * 2.0;
	double t_2 = t_0 * t_0;
	double t_3 = t_2 * 4.0;
	double t_4 = ((double) M_PI) * (((double) M_PI) / 4.0);
	return (((((double) M_PI) / 2.0) * (((double) M_PI) / 2.0)) - ((asin((sqrt((1.0 - x)) / sqrt(2.0))) * 2.0) * t_1)) / ((((t_3 * t_3) - (t_4 * t_4)) / fma(t_2, 4.0, t_4)) / (t_1 - (((double) M_PI) / 2.0)));
}

Julia

function code(x)
	t_0 = asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))
	t_1 = Float64(t_0 * 2.0)
	t_2 = Float64(t_0 * t_0)
	t_3 = Float64(t_2 * 4.0)
	t_4 = Float64(pi * Float64(pi / 4.0))
	return Float64(Float64(Float64(Float64(pi / 2.0) * Float64(pi / 2.0)) - Float64(Float64(asin(Float64(sqrt(Float64(1.0 - x)) / sqrt(2.0))) * 2.0) * t_1)) / Float64(Float64(Float64(Float64(t_3 * t_3) - Float64(t_4 * t_4)) / fma(t_2, 4.0, t_4)) / Float64(t_1 - Float64(pi / 2.0))))
end

Wolfram

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

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

7. Applied rewrites 8.3%

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

9. Applied rewrites 8.3%

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

Alternative 2: 8.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

7. Applied rewrites 8.3%

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

8. Taylor expanded in x around 0

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

   1. frac-times N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

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

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

10. Applied rewrites 8.3%

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

Alternative 3: 8.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 6.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-asin.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. flip-- N/A

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

   8. lower-/.f64 N/A

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

3. Applied rewrites 6.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sqrt-div N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-/.f64 8.3

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

5. Applied rewrites 8.3%

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

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

7. Applied rewrites 8.3%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 8.3%

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

Alternative 4: 8.3% accurate, 0.3× speedup

Math

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

FPCore

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

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

Alternative 5: 8.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-/.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. asin-acos-rev N/A

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

   4. sqrt-div N/A

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

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

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

   6. sqrt-div N/A

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

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

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

   8. lift-PI.f64 N/A

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

   9. lift-/.f64 N/A

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

6. Applied rewrites 8.3%

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

Alternative 6: 6.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sqrt-div N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lower-sqrt.f64 6.7

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

3. Applied rewrites 6.7%

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

Alternative 7: 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 6.8%

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

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 8: 5.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \pi, -2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{1}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (fma 0.5 PI (* -2.0 (asin (sqrt 0.5))))
   (- (/ PI 2.0) (* 2.0 (asin (/ 1.0 (sqrt 2.0)))))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fma(0.5, ((double) M_PI), (-2.0 * asin(sqrt(0.5))));
	} else {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin((1.0 / sqrt(2.0))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = fma(0.5, pi, Float64(-2.0 * asin(sqrt(0.5))));
	else
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(Float64(1.0 / sqrt(2.0)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(0.5 * Pi + N[(-2.0 * N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 8.3%

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

   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 8.3

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

   4. Applied rewrites 8.3%

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

   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 5.9%

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

   if -4.999999999999985e-310 < x

   1. Initial program 5.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-sqrt.f64 8.2

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

   3. Applied rewrites 8.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 5.8%

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

Alternative 9: 4.1% accurate, 1.5× 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 2025114 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64
  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))