Ian Simplification

Percentage Accurate: 7.0% → 8.5%

Time: 6.1s

Alternatives: 5

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 - \frac{\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}{\pi}\right) \cdot 4\\ t_1 := 1 + t\_0\\ \left(\frac{1}{t\_1} - \frac{{t\_0}^{2}}{t\_1}\right) \cdot \left(\pi \cdot 0.5\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (- 0.5 (/ (acos (sqrt (fma -0.5 x 0.5))) PI)) 4.0))
        (t_1 (+ 1.0 t_0)))
   (* (- (/ 1.0 t_1) (/ (pow t_0 2.0) t_1)) (* PI 0.5))))

C

double code(double x) {
	double t_0 = (0.5 - (acos(sqrt(fma(-0.5, x, 0.5))) / ((double) M_PI))) * 4.0;
	double t_1 = 1.0 + t_0;
	return ((1.0 / t_1) - (pow(t_0, 2.0) / t_1)) * (((double) M_PI) * 0.5);
}

Julia

function code(x)
	t_0 = Float64(Float64(0.5 - Float64(acos(sqrt(fma(-0.5, x, 0.5))) / pi)) * 4.0)
	t_1 = Float64(1.0 + t_0)
	return Float64(Float64(Float64(1.0 / t_1) - Float64((t_0 ^ 2.0) / t_1)) * Float64(pi * 0.5))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 - N[(N[ArcCos[N[Sqrt[N[(-0.5 * x + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / t$95$1), $MachinePrecision] - N[(N[Power[t$95$0, 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.5), $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. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. asin-acos N/A

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

   6. mult-flip N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-acos.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. mult-flip N/A

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

   15. metadata-eval N/A

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

   16. *-commutative N/A

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

   17. distribute-lft-out-- N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

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

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

   21. +-commutative N/A

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

   22. lower-fma.f64 8.5

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

3. Applied rewrites 8.5%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 8.5

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

5. Applied rewrites 8.5%

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

7. Applied rewrites 8.5%

   \[\leadsto \color{blue}{\left(1 - \frac{\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}{\pi} \cdot 4\right) \cdot \left(\pi \cdot 0.5\right)} \]

9. Applied rewrites 8.5%

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

Alternative 2: 8.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(Pi * 0.5 + N[(-2.0 * N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(x * -0.5 + 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-asin.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. asin-acos N/A

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

   6. mult-flip N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-acos.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. mult-flip N/A

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

   15. metadata-eval N/A

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

   16. *-commutative N/A

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

   17. distribute-lft-out-- N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

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

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

   21. +-commutative N/A

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

   22. lower-fma.f64 8.5

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

3. Applied rewrites 8.5%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 8.5

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

5. Applied rewrites 8.5%

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

   1. lift--.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-acos.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

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

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

   12. lower-fma.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

7. Applied rewrites 8.5%

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

Alternative 3: 7.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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

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

   3. lift-/.f64 N/A

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

   4. sqrt-div N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lower-sqrt.f64 6.9

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

3. Applied rewrites 6.9%

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

Alternative 4: 6.9% 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. sub-flip N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. mult-flip N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. distribute-lft-neg-out 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. lower-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. lift-/.f64 N/A

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

   18. lift--.f64 N/A

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

3. Applied rewrites 7.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 0.5, -2 \cdot \sin^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)} \]
4. 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 2025142 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64
  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))