Ian Simplification

Percentage Accurate: 7.0% → 8.5%

Time: 7.0s

Alternatives: 7

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 8.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(\sin^{-1} \left(\sqrt{0.5 \cdot \left(1 - x\right)}\right), -2, 0.5 \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - 2 \cdot \sin^{-1} \left(\frac{\sqrt{1 - x}}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-309)
   (fma (asin (sqrt (* 0.5 (- 1.0 x)))) -2.0 (* 0.5 PI))
   (- (* PI 0.5) (* 2.0 (asin (/ (sqrt (- 1.0 x)) (sqrt 2.0)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = fma(asin(sqrt((0.5 * (1.0 - x)))), -2.0, (0.5 * ((double) M_PI)));
	} else {
		tmp = (((double) M_PI) * 0.5) - (2.0 * asin((sqrt((1.0 - x)) / sqrt(2.0))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = fma(asin(sqrt(Float64(0.5 * Float64(1.0 - x)))), -2.0, Float64(0.5 * pi));
	else
		tmp = Float64(Float64(pi * 0.5) - Float64(2.0 * asin(Float64(sqrt(Float64(1.0 - x)) / sqrt(2.0)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], N[(N[ArcSin[N[Sqrt[N[(0.5 * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * -2.0 + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $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. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-flip N/A

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

      10. associate-/r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 7.0

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

      14. lift-/.f64 N/A

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

      15. div-flip N/A

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

      16. associate-/r/ N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 7.0

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

   3. Applied rewrites 7.0%

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

   if -1.000000000000002e-309 < 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-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

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 6.9

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

   3. Applied rewrites 6.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 6.9

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

   5. Applied rewrites 6.9%

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

Alternative 2: 8.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (Math.PI * 0.5) - (2.0 * ((0.5 * Math.PI) - Math.acos(Math.sqrt((0.5 * (1.0 - x))))));
}

Python

def code(x):
	return (math.pi * 0.5) - (2.0 * ((0.5 * math.pi) - math.acos(math.sqrt((0.5 * (1.0 - x))))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (pi * 0.5) - (2.0 * ((0.5 * pi) - acos(sqrt((0.5 * (1.0 - x))))));
end

Wolfram

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

Derivation

1. Initial program 7.0%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. div-flip N/A

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

   8. associate-/r/ N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 N/A

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

   11. lower-acos.f64 8.5

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

   12. lift-/.f64 N/A

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

   13. div-flip N/A

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

   14. associate-/r/ N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f64 8.5

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

3. Applied rewrites 8.5%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 8.5

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

5. Applied rewrites 8.5%

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

Alternative 3: 7.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. div-flip N/A

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

   10. associate-/r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval 7.0

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

   14. lift-/.f64 N/A

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

   15. div-flip N/A

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

   16. associate-/r/ N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 7.0

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

3. Applied rewrites 7.0%

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

Alternative 4: 5.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-308}:\\ \;\;\;\;\cos^{-1} \left(\sqrt{0.5}\right) - \sin^{-1} \left(\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{1}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e-308)
   (- (acos (sqrt 0.5)) (asin (sqrt 0.5)))
   (- (/ PI 2.0) (* 2.0 (asin (/ 1.0 (sqrt 2.0)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1e-308) {
		tmp = acos(sqrt(0.5)) - asin(sqrt(0.5));
	} else {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin((1.0 / sqrt(2.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e-308) {
		tmp = Math.acos(Math.sqrt(0.5)) - Math.asin(Math.sqrt(0.5));
	} else {
		tmp = (Math.PI / 2.0) - (2.0 * Math.asin((1.0 / Math.sqrt(2.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e-308:
		tmp = math.acos(math.sqrt(0.5)) - math.asin(math.sqrt(0.5))
	else:
		tmp = (math.pi / 2.0) - (2.0 * math.asin((1.0 / math.sqrt(2.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e-308)
		tmp = Float64(acos(sqrt(0.5)) - asin(sqrt(0.5)));
	else
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(Float64(1.0 / sqrt(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e-308)
		tmp = acos(sqrt(0.5)) - asin(sqrt(0.5));
	else
		tmp = (pi / 2.0) - (2.0 * asin((1.0 / sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e-308], N[(N[ArcCos[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision] - N[ArcSin[N[Sqrt[0.5], $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 < 9.9999999999999991e-309

   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. count-2-rev N/A

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

      4. associate--r+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-asin.f64 N/A

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

      8. acos-asin N/A

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

      9. lower--.f64 N/A

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

      10. lower-acos.f64 4.1

          \[\leadsto \color{blue}{\cos^{-1} \left(\sqrt{0.5}\right)} - \sin^{-1} \left(\sqrt{0.5}\right) \]

   6. Applied rewrites 4.1%

      \[\leadsto \color{blue}{\cos^{-1} \left(\sqrt{0.5}\right) - \sin^{-1} \left(\sqrt{0.5}\right)} \]

   if 9.9999999999999991e-309 < 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-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

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 6.9

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

   3. Applied rewrites 6.9%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 4.1

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

   6. Applied rewrites 4.1%

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

Alternative 5: 5.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval 4.1

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

   9. lift-/.f64 N/A

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

   10. mult-flip N/A

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

   11. metadata-eval N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 4.1

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

6. Applied rewrites 4.1%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-acos.f64 5.3

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

8. Applied rewrites 5.3%

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

Alternative 6: 4.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\sqrt{0.5}\right) - \sin^{-1} \left(\sqrt{0.5}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (acos (sqrt 0.5)) (asin (sqrt 0.5))))

C

double code(double x) {
	return acos(sqrt(0.5)) - asin(sqrt(0.5));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = acos(sqrt(0.5d0)) - asin(sqrt(0.5d0))
end function

Java

public static double code(double x) {
	return Math.acos(Math.sqrt(0.5)) - Math.asin(Math.sqrt(0.5));
}

Python

def code(x):
	return math.acos(math.sqrt(0.5)) - math.asin(math.sqrt(0.5))

Julia

function code(x)
	return Float64(acos(sqrt(0.5)) - asin(sqrt(0.5)))
end

MATLAB

function tmp = code(x)
	tmp = acos(sqrt(0.5)) - asin(sqrt(0.5));
end

Wolfram

code[x_] := N[(N[ArcCos[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision] - N[ArcSin[N[Sqrt[0.5], $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. count-2-rev N/A

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

   4. associate--r+ N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-asin.f64 N/A

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

   8. acos-asin N/A

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

   9. lower--.f64 N/A

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

   10. lower-acos.f64 4.1

       \[\leadsto \color{blue}{\cos^{-1} \left(\sqrt{0.5}\right)} - \sin^{-1} \left(\sqrt{0.5}\right) \]

6. Applied rewrites 4.1%

   \[\leadsto \color{blue}{\cos^{-1} \left(\sqrt{0.5}\right) - \sin^{-1} \left(\sqrt{0.5}\right)} \]
7. Add Preprocessing

Alternative 7: 4.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval 4.1

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

   9. lift-/.f64 N/A

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

   10. mult-flip N/A

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

   11. metadata-eval N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 4.1

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

6. Applied rewrites 4.1%

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

Reproduce

herbie shell --seed 2025156 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64
  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))