Ian Simplification

Percentage Accurate: 6.7% → 8.2%

Time: 13.5s

Alternatives: 5

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: 6.7% 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: 8.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. sub-neg N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-acos.f64 7.7

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

   12. lift-/.f64 N/A

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

   13. lift--.f64 N/A

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

   14. div-sub N/A

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

   15. metadata-eval N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. div-inv N/A

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

   19. metadata-eval N/A

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

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

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. lower-fma.f64 N/A

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

4. Applied rewrites 7.7%

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

5. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

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

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

   6. distribute-lft-in N/A

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

   7. associate-+r+ N/A

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

   8. distribute-rgt1-in N/A

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

   9. metadata-eval N/A

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

   10. neg-mul-1 N/A

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

   11. +-commutative N/A

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

7. Applied rewrites 7.7%

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

9. Applied rewrites 7.7%

   \[\leadsto \frac{\mathsf{fma}\left(4, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{2}, \left(\pi \cdot \pi\right) \cdot -0.25\right)}{\color{blue}{\mathsf{fma}\left(2, \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 0.5 \cdot \pi\right)}} \]
10. Step-by-step derivation

11. Applied rewrites 7.7%

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

Alternative 2: 8.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. sub-neg N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-acos.f64 8.2

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

   12. lift-/.f64 N/A

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

   13. lift--.f64 N/A

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

   14. div-sub N/A

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

   15. metadata-eval N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. div-inv N/A

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

   19. metadata-eval N/A

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

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

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. lower-fma.f64 N/A

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

4. Applied rewrites 8.2%

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

5. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

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

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

   6. distribute-lft-in N/A

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

   7. associate-+r+ N/A

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

   8. distribute-rgt1-in N/A

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

   9. metadata-eval N/A

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

   10. neg-mul-1 N/A

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

   11. +-commutative N/A

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

7. Applied rewrites 8.2%

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

9. Applied rewrites 8.2%

   \[\leadsto \frac{\mathsf{fma}\left(4, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{2}, \left(\pi \cdot \pi\right) \cdot -0.25\right)}{\color{blue}{\mathsf{fma}\left(2, \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 0.5 \cdot \pi\right)}} \]
10. 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

real(8) function code(x)
    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 2024229 
(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))))))