Ian Simplification

Percentage Accurate: 6.7% → 8.2%

Time: 14.2s

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, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (sqrt (fma -0.5 x 0.5)))))
   (/
    1.0
    (/ (fma t_0 2.0 (* PI 0.5)) (fma (pow t_0 2.0) 4.0 (* (* PI PI) -0.25))))))

C

double code(double x) {
	double t_0 = acos(sqrt(fma(-0.5, x, 0.5)));
	return 1.0 / (fma(t_0, 2.0, (((double) M_PI) * 0.5)) / fma(pow(t_0, 2.0), 4.0, ((((double) M_PI) * ((double) M_PI)) * -0.25)));
}

Julia

function code(x)
	t_0 = acos(sqrt(fma(-0.5, x, 0.5)))
	return Float64(1.0 / Float64(fma(t_0, 2.0, Float64(pi * 0.5)) / fma((t_0 ^ 2.0), 4.0, Float64(Float64(pi * pi) * -0.25))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[Sqrt[N[(-0.5 * x + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(1.0 / N[(N[(t$95$0 * 2.0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] * 4.0 + N[(N[(Pi * Pi), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 8.1%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 9.5

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

   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 9.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 9.5%

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

9. Applied rewrites 9.6%

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

10. Final simplification 9.6%

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

Alternative 2: 8.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (sqrt (fma -0.5 x 0.5)))))
   (*
    (/ 1.0 (fma t_0 2.0 (* PI 0.5)))
    (fma (pow t_0 2.0) 4.0 (* (* PI PI) -0.25)))))

C

double code(double x) {
	double t_0 = acos(sqrt(fma(-0.5, x, 0.5)));
	return (1.0 / fma(t_0, 2.0, (((double) M_PI) * 0.5))) * fma(pow(t_0, 2.0), 4.0, ((((double) M_PI) * ((double) M_PI)) * -0.25));
}

Julia

function code(x)
	t_0 = acos(sqrt(fma(-0.5, x, 0.5)))
	return Float64(Float64(1.0 / fma(t_0, 2.0, Float64(pi * 0.5))) * fma((t_0 ^ 2.0), 4.0, Float64(Float64(pi * pi) * -0.25)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[Sqrt[N[(-0.5 * x + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 / N[(t$95$0 * 2.0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * 4.0 + N[(N[(Pi * Pi), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 8.1%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 9.5

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

   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 9.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 9.5%

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

9. Applied rewrites 9.6%

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

10. Final simplification 9.6%

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

Alternative 3: 8.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.1%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 9.5

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

   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 9.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 9.5%

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

9. Applied rewrites 9.6%

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

10. Final simplification 9.6%

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

Alternative 4: 8.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.1%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 9.5

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

   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 9.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 9.5%

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

8. Final simplification 9.5%

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

Alternative 5: 5.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.1%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 9.5

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

   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 9.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 9.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 5.3%

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

11. Final simplification 5.3%

    \[\leadsto \mathsf{fma}\left(\cos^{-1} \left(\sqrt{0.5}\right), 2, \pi \cdot -0.5\right) \]
12. 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 2024235 
(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))))))