Ian Simplification

Percentage Accurate: 6.8% → 8.3%

Time: 5.7s

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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.125 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\\ t_1 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\\ t_2 := {\left(0.5 \cdot \pi - t\_1\right)}^{3} \cdot -8\\ t_3 := \left(\pi \cdot 0.5 - t\_1\right) \cdot -2\\ \frac{\frac{t\_2 \cdot t\_2 - t\_0 \cdot t\_0}{t\_2 - t\_0}}{\mathsf{fma}\left(\pi \cdot 0.5, \pi \cdot 0.5, {t\_3}^{2} - \left(\pi \cdot 0.5\right) \cdot t\_3\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.125 (* (* PI PI) PI)))
        (t_1 (acos (sqrt (fma x -0.5 0.5))))
        (t_2 (* (pow (- (* 0.5 PI) t_1) 3.0) -8.0))
        (t_3 (* (- (* PI 0.5) t_1) -2.0)))
   (/
    (/ (- (* t_2 t_2) (* t_0 t_0)) (- t_2 t_0))
    (fma (* PI 0.5) (* PI 0.5) (- (pow t_3 2.0) (* (* PI 0.5) t_3))))))

C

double code(double x) {
	double t_0 = 0.125 * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI));
	double t_1 = acos(sqrt(fma(x, -0.5, 0.5)));
	double t_2 = pow(((0.5 * ((double) M_PI)) - t_1), 3.0) * -8.0;
	double t_3 = ((((double) M_PI) * 0.5) - t_1) * -2.0;
	return (((t_2 * t_2) - (t_0 * t_0)) / (t_2 - t_0)) / fma((((double) M_PI) * 0.5), (((double) M_PI) * 0.5), (pow(t_3, 2.0) - ((((double) M_PI) * 0.5) * t_3)));
}

Julia

function code(x)
	t_0 = Float64(0.125 * Float64(Float64(pi * pi) * pi))
	t_1 = acos(sqrt(fma(x, -0.5, 0.5)))
	t_2 = Float64((Float64(Float64(0.5 * pi) - t_1) ^ 3.0) * -8.0)
	t_3 = Float64(Float64(Float64(pi * 0.5) - t_1) * -2.0)
	return Float64(Float64(Float64(Float64(t_2 * t_2) - Float64(t_0 * t_0)) / Float64(t_2 - t_0)) / fma(Float64(pi * 0.5), Float64(pi * 0.5), Float64((t_3 ^ 2.0) - Float64(Float64(pi * 0.5) * t_3))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.125 * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[ArcCos[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(N[(0.5 * Pi), $MachinePrecision] - t$95$1), $MachinePrecision], 3.0], $MachinePrecision] * -8.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(Pi * 0.5), $MachinePrecision] - t$95$1), $MachinePrecision] * -2.0), $MachinePrecision]}, N[(N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(Pi * 0.5), $MachinePrecision] * N[(Pi * 0.5), $MachinePrecision] + N[(N[Power[t$95$3, 2.0], $MachinePrecision] - N[(N[(Pi * 0.5), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 6.8%

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

   1. lift-asin.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. asin-acos N/A

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

   6. lift-/.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-acos.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lift-sqrt.f64 8.3

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

   13. lift--.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. div-sub N/A

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

   16. metadata-eval N/A

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

   17. *-lft-identity N/A

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

   18. associate-*l/ N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

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

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

   22. +-commutative N/A

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

   23. lower-fma.f64 8.3

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

3. Applied rewrites 8.3%

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

4. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

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

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

6. Applied rewrites 8.3%

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

8. Applied rewrites 8.3%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3} \cdot -8\right)}{\color{blue}{\mathsf{fma}\left(\pi \cdot 0.5, \pi \cdot 0.5, {\left(\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot -2\right)}^{2} - \left(\pi \cdot 0.5\right) \cdot \left(\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot -2\right)\right)}} \]

10. Applied rewrites 8.3%

    \[\leadsto \frac{\frac{\left({\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3} \cdot -8\right) \cdot \left({\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3} \cdot -8\right) - \left(0.125 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \left(0.125 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)}{{\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3} \cdot -8 - 0.125 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}}{\mathsf{fma}\left(\color{blue}{\pi \cdot 0.5}, \pi \cdot 0.5, {\left(\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot -2\right)}^{2} - \left(\pi \cdot 0.5\right) \cdot \left(\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot -2\right)\right)} \]
11. Add Preprocessing

Alternative 2: 8.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\\ t_1 := 0.5 \cdot \pi - t\_0\\ \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.125, {\left(\pi \cdot 0.5 - t\_0\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left({t\_1}^{2}, 4, 0.25 \cdot \left(\pi \cdot \pi\right)\right) - \left(-t\_1 \cdot \pi\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (sqrt (fma x -0.5 0.5)))) (t_1 (- (* 0.5 PI) t_0)))
   (/
    (fma (* (* PI PI) PI) 0.125 (* (pow (- (* PI 0.5) t_0) 3.0) -8.0))
    (- (fma (pow t_1 2.0) 4.0 (* 0.25 (* PI PI))) (- (* t_1 PI))))))

C

double code(double x) {
	double t_0 = acos(sqrt(fma(x, -0.5, 0.5)));
	double t_1 = (0.5 * ((double) M_PI)) - t_0;
	return fma(((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)), 0.125, (pow(((((double) M_PI) * 0.5) - t_0), 3.0) * -8.0)) / (fma(pow(t_1, 2.0), 4.0, (0.25 * (((double) M_PI) * ((double) M_PI)))) - -(t_1 * ((double) M_PI)));
}

Julia

function code(x)
	t_0 = acos(sqrt(fma(x, -0.5, 0.5)))
	t_1 = Float64(Float64(0.5 * pi) - t_0)
	return Float64(fma(Float64(Float64(pi * pi) * pi), 0.125, Float64((Float64(Float64(pi * 0.5) - t_0) ^ 3.0) * -8.0)) / Float64(fma((t_1 ^ 2.0), 4.0, Float64(0.25 * Float64(pi * pi))) - Float64(-Float64(t_1 * pi))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * Pi), $MachinePrecision] - t$95$0), $MachinePrecision]}, N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * 0.125 + N[(N[Power[N[(N[(Pi * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision], 3.0], $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] * 4.0 + N[(0.25 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - (-N[(t$95$1 * Pi), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 6.8%

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

   1. lift-asin.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. asin-acos N/A

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

   6. lift-/.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-acos.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lift-sqrt.f64 8.3

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

   13. lift--.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. div-sub N/A

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

   16. metadata-eval N/A

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

   17. *-lft-identity N/A

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

   18. associate-*l/ N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

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

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

   22. +-commutative N/A

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

   23. lower-fma.f64 8.3

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

3. Applied rewrites 8.3%

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

4. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

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

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

6. Applied rewrites 8.3%

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

8. Applied rewrites 8.3%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3} \cdot -8\right)}{\color{blue}{\mathsf{fma}\left(\pi \cdot 0.5, \pi \cdot 0.5, {\left(\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot -2\right)}^{2} - \left(\pi \cdot 0.5\right) \cdot \left(\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot -2\right)\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{8}, {\left(\pi \cdot \frac{1}{2} - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right)}^{3} \cdot -8\right)}{\left(\frac{1}{4} \cdot {\mathsf{PI}\left(\right)}^{2} + 4 \cdot {\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)\right)}^{2}\right) - \color{blue}{-1 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)\right)\right)}} \]

11. Applied rewrites 8.3%

    \[\leadsto \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left({\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{2}, 4, 0.25 \cdot \left(\pi \cdot \pi\right)\right) - \color{blue}{\left(-\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot \pi\right)}} \]
12. Add Preprocessing

Alternative 3: 8.3% accurate, 0.9× speedup

Math

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

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 6.8%

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

   1. lift-asin.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. asin-acos N/A

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

   6. lift-/.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-acos.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lift-sqrt.f64 8.3

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

   13. lift--.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. div-sub N/A

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

   16. metadata-eval N/A

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

   17. *-lft-identity N/A

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

   18. associate-*l/ N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

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

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

   22. +-commutative N/A

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

   23. lower-fma.f64 8.3

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

3. Applied rewrites 8.3%

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

4. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

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

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

6. Applied rewrites 8.3%

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

Alternative 4: 6.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in x around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

   8. metadata-eval N/A

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

   9. div-sub N/A

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

   10. lower-fma.f64 N/A

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

4. Applied rewrites 6.8%

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

Alternative 5: 5.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-310)
   (fma (asin (sqrt 0.5)) -2.0 (* PI 0.5))
   (- (/ PI 2.0) (* 2.0 (asin (/ 1.0 (sqrt 2.0)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-310) {
		tmp = fma(asin(sqrt(0.5)), -2.0, (((double) M_PI) * 0.5));
	} else {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin((1.0 / sqrt(2.0))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-310)
		tmp = fma(asin(sqrt(0.5)), -2.0, Float64(pi * 0.5));
	else
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(Float64(1.0 / sqrt(2.0)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-310], N[(N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision] * -2.0 + N[(Pi * 0.5), $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 < -9.999999999999969e-311

   1. Initial program 6.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-asin.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

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

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

      8. lift-PI.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. add-sqr-sqrt N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 6.7%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 6.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 4.1%

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

   if -9.999999999999969e-311 < x

   1. Initial program 6.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-sqrt.f64 6.8

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

   3. Applied rewrites 6.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 4.1%

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

Alternative 6: 4.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[ArcSin[N[Sqrt[0.5], $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. lift-*.f64 N/A

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

   3. lift-asin.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

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

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

   8. lift-PI.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. add-sqr-sqrt N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

3. Applied rewrites 6.7%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 6.8%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 4.1%

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

Reproduce

herbie shell --seed 2025143 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64
  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))