Ian Simplification

Percentage Accurate: 7.0% → 8.6%

Time: 4.0s

Alternatives: 5

Speedup: 1.0×

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: 7.0% 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: 8.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

   1. lift--.f64 N/A

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

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

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

   8. lift-PI.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. mult-flip N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. lift-/.f64 N/A

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

   17. lift--.f64 N/A

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

   18. lift-sqrt.f64 N/A

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

3. Applied rewrites 7.0%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. metadata-eval N/A

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

   5. mult-flip N/A

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

   6. sqrt-div N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-/.f64 7.1

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

5. Applied rewrites 7.1%

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

   1. lift-asin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. sqrt-div N/A

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

   7. asin-acos N/A

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

   8. lift-PI.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lift-PI.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. sqrt-div N/A

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

   17. lower-acos.f64 N/A

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

   18. sqrt-div N/A

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

   19. lower-sqrt.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lift--.f64 8.6

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

7. Applied rewrites 8.6%

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

8. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. mult-flip N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 8.6

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

10. Applied rewrites 8.6%

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

Alternative 2: 7.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   (fma PI 0.5 (* -2.0 (asin (sqrt 0.5))))
   (fma PI 0.5 (* -2.0 (asin (/ 1.0 (sqrt 2.0)))))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = fma(((double) M_PI), 0.5, (-2.0 * asin(sqrt(0.5))));
	} else {
		tmp = fma(((double) M_PI), 0.5, (-2.0 * asin((1.0 / sqrt(2.0)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = fma(pi, 0.5, Float64(-2.0 * asin(sqrt(0.5))));
	else
		tmp = fma(pi, 0.5, Float64(-2.0 * asin(Float64(1.0 / sqrt(2.0)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[(Pi * 0.5 + N[(-2.0 * N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * 0.5 + 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 < -1.999999999999994e-310

   1. Initial program 7.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 4.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 4.1

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

   6. Applied rewrites 4.1%

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

   if -1.999999999999994e-310 < x

   1. Initial program 7.0%

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

      1. lift--.f64 N/A

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

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

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

      8. lift-PI.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-/.f64 N/A

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

      17. lift--.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

   3. Applied rewrites 7.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. metadata-eval N/A

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

      5. mult-flip N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-/.f64 7.1

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

   5. Applied rewrites 7.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 4.1%

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

Alternative 3: 5.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 4.1%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 4.1

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

6. Applied rewrites 4.1%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lift-PI.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-acos.f64 5.4

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

8. Applied rewrites 5.4%

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

Alternative 4: 5.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

   1. lift--.f64 N/A

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

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

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

   8. lift-PI.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. mult-flip N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. lift-/.f64 N/A

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

   17. lift--.f64 N/A

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

   18. lift-sqrt.f64 N/A

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

3. Applied rewrites 7.0%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. mult-flip N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 7.0

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

6. Applied rewrites 7.0%

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

Alternative 5: 4.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 4.1%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 4.1

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

6. Applied rewrites 4.1%

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

Reproduce

herbie shell --seed 2025130 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64
  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))