Ian Simplification

Percentage Accurate: 6.8% → 8.3%

Time: 5.8s

Alternatives: 6

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 8.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (-
  (acos (/ (sin (acos x)) (sqrt (fma x 2.0 2.0))))
  (asin (sqrt (/ (- 1.0 x) 2.0)))))

C

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

Julia

function code(x)
	return Float64(acos(Float64(sin(acos(x)) / sqrt(fma(x, 2.0, 2.0)))) - asin(sqrt(Float64(Float64(1.0 - x) / 2.0))))
end

Wolfram

code[x_] := N[(N[ArcCos[N[(N[Sin[N[ArcCos[x], $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(x * 2.0 + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 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. count-2-rev N/A

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

   4. associate--r+ N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-asin.f64 N/A

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

   8. acos-asin N/A

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

   9. lower--.f64 N/A

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

   10. lower-acos.f64 6.8

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

3. Applied rewrites 6.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. metadata-eval N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. associate-/r* N/A

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

   11. lift-*.f64 N/A

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

   12. sqrt-div N/A

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

   13. lower-/.f64 N/A

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

   14. lift--.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. sin-acos-rev N/A

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

   17. lower-sin.f64 N/A

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

   18. lower-acos.f64 N/A

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

   19. lower-sqrt.f64 8.3

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lift-+.f64 N/A

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

   23. distribute-rgt-in N/A

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

   24. metadata-eval N/A

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

   25. lower-fma.f64 8.3

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

5. Applied rewrites 8.3%

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

Alternative 2: 8.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[ArcCos[N[(N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(x * 2.0 + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[ArcSin[N[Sqrt[N[(-0.5 * x + 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. 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. count-2-rev N/A

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

   4. associate--r+ N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-asin.f64 N/A

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

   8. acos-asin N/A

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

   9. lower--.f64 N/A

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

   10. lower-acos.f64 6.8

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

3. Applied rewrites 6.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. metadata-eval N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. associate-/r* N/A

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

   11. lift-*.f64 N/A

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

   12. sqrt-div N/A

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

   13. lower-/.f64 N/A

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

   14. lift--.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. sin-acos-rev N/A

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

   17. lower-sin.f64 N/A

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

   18. lower-acos.f64 N/A

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

   19. lower-sqrt.f64 8.3

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lift-+.f64 N/A

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

   23. distribute-rgt-in N/A

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

   24. metadata-eval N/A

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

   25. lower-fma.f64 8.3

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

5. Applied rewrites 8.3%

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

   1. lift-sin.f64 N/A

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

   2. lift-acos.f64 N/A

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

   3. sin-acos N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 8.3

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

7. Applied rewrites 8.3%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 8.3

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

10. Applied rewrites 8.3%

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

Alternative 3: 8.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $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. 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. lower-acos.f64 8.3

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

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. Add Preprocessing

Alternative 4: 6.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. lift--.f64 N/A

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

   2. sub-negate 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. metadata-eval 6.8

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

3. Applied rewrites 6.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. sqrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 6.7

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

5. Applied rewrites 6.7%

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

Alternative 5: 6.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[ArcSin[N[Sqrt[N[(0.5 * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * -2.0 + N[(Pi * 0.5), $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. flip3-- N/A

      \[\leadsto \color{blue}{\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \cdot \left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) + \frac{\pi}{2} \cdot \left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \cdot \left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) + \frac{\pi}{2} \cdot \left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)}} \]

3. Applied rewrites 6.8%

   \[\leadsto \color{blue}{\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}^{3} \cdot 8}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2, \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right), 2, \frac{\pi}{2}\right), {\left(\frac{\pi}{2}\right)}^{2}\right)}} \]
4. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(\frac{\pi}{2}\right)}^{3} - {\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}^{3} \cdot 8}}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2, \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right), 2, \frac{\pi}{2}\right), {\left(\frac{\pi}{2}\right)}^{2}\right)} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(\frac{\pi}{2}\right)}^{3}} - {\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}^{3} \cdot 8}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2, \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right), 2, \frac{\pi}{2}\right), {\left(\frac{\pi}{2}\right)}^{2}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - \color{blue}{{\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}^{3} \cdot 8}}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2, \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right), 2, \frac{\pi}{2}\right), {\left(\frac{\pi}{2}\right)}^{2}\right)} \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - \color{blue}{{\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}^{3}} \cdot 8}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2, \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right), 2, \frac{\pi}{2}\right), {\left(\frac{\pi}{2}\right)}^{2}\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}^{3} \cdot \color{blue}{{2}^{3}}}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2, \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right), 2, \frac{\pi}{2}\right), {\left(\frac{\pi}{2}\right)}^{2}\right)} \]

   6. unpow-prod-down N/A

      \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - \color{blue}{{\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2\right)}^{3}}}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2, \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right), 2, \frac{\pi}{2}\right), {\left(\frac{\pi}{2}\right)}^{2}\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\color{blue}{\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2\right)}}^{3}}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2, \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right), 2, \frac{\pi}{2}\right), {\left(\frac{\pi}{2}\right)}^{2}\right)} \]

   8. difference-cubes N/A

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

5. Applied rewrites 6.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 6.8

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

7. Applied rewrites 6.8%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

10. Applied rewrites 6.8%

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

Alternative 6: 4.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. lift--.f64 N/A

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

   2. sub-negate 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. metadata-eval 6.8

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

3. Applied rewrites 6.8%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. metadata-eval N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. associate-/r* N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 6.8

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. distribute-rgt-in N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f64 6.8

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

5. Applied rewrites 6.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 4.1%

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

Developer Target 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = asin(x)
end function

Java

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

Python

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

Julia

function code(x)
	return asin(x)
end

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025106 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

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

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