Ian Simplification

Percentage Accurate: 6.8% → 8.3%

Time: 16.9s

Alternatives: 3

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 8.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = 2.0 * Math.acos(Math.pow((0.5 + (x / -2.0)), 0.5));
	double t_1 = t_0 + (Math.PI / 2.0);
	return (Math.pow(t_0, 2.0) / t_1) - ((Math.PI / (4.0 / Math.PI)) / t_1);
}

Python

def code(x):
	t_0 = 2.0 * math.acos(math.pow((0.5 + (x / -2.0)), 0.5))
	t_1 = t_0 + (math.pi / 2.0)
	return (math.pow(t_0, 2.0) / t_1) - ((math.pi / (4.0 / math.pi)) / t_1)

Julia

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

MATLAB

function tmp = code(x)
	t_0 = 2.0 * acos(((0.5 + (x / -2.0)) ^ 0.5));
	t_1 = t_0 + (pi / 2.0);
	tmp = ((t_0 ^ 2.0) / t_1) - ((pi / (4.0 / pi)) / t_1);
end

Wolfram

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

Derivation

1. Initial program 6.4%

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

   1. asin-acos N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)}\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot 2 + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2}\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right) \cdot 2\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot 2\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot 1 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   8. *-rgt-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)}\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right), \color{blue}{2}\right)\right)\right) \]

4. Applied egg-rr 8.1%

   \[\leadsto \frac{\pi}{2} - \color{blue}{\left(\pi + \left(0 - \cos^{-1} \left({\left(0.5 - \frac{x}{2}\right)}^{0.5}\right)\right) \cdot 2\right)} \]
5. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \color{blue}{\left(\mathsf{PI}\left(\right) + \left(0 - \cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \left(\color{blue}{\mathsf{PI}\left(\right)} + \left(0 - \cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)\right) \]

   3. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) + \left(0 - \cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)\right) \]

   4. *-un-lft-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(1 \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(0 - \cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)\right)} \cdot 2\right)\right) \]

   5. fma-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{fma}\left(1, \color{blue}{\mathsf{PI}\left(\right)}, \left(0 - \cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)\right)\right) \]

   6. sub0-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{fma}\left(1, \mathsf{PI}\left(\right), \left(\mathsf{neg}\left(\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)\right)\right) \cdot 2\right)\right)\right) \]

   7. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{fma}\left(1, \mathsf{PI}\left(\right), \mathsf{neg}\left(\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)\right)\right)\right) \]

   8. fmm-undef N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(1 \cdot \mathsf{PI}\left(\right) - \color{blue}{\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right) \cdot 2}\right)\right) \]

   9. *-un-lft-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) - \color{blue}{\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)} \cdot 2\right)\right) \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{\_.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)}\right)\right) \]

   11. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)} \cdot 2\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right), \color{blue}{2}\right)\right)\right) \]

6. Applied egg-rr 8.1%

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

7. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. associate-+r+ N/A

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

   6. neg-mul-1 N/A

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

   7. distribute-rgt-out N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

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

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

   11. *-commutative N/A

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

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\left(2 \cdot \cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

9. Simplified 8.1%

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

11. Applied egg-rr 8.1%

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

Alternative 2: 8.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (2.0 * Math.acos(Math.sqrt((0.5 - (0.5 * x))))) + (Math.PI * -0.5);
}

Python

def code(x):
	return (2.0 * math.acos(math.sqrt((0.5 - (0.5 * x))))) + (math.pi * -0.5)

Julia

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

MATLAB

function tmp = code(x)
	tmp = (2.0 * acos(sqrt((0.5 - (0.5 * x))))) + (pi * -0.5);
end

Wolfram

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

Derivation

1. Initial program 6.4%

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

   1. asin-acos N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)}\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot 2 + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2}\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right) \cdot 2\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot 2\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot 1 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   8. *-rgt-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)}\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right), \color{blue}{2}\right)\right)\right) \]

4. Applied egg-rr 8.1%

   \[\leadsto \frac{\pi}{2} - \color{blue}{\left(\pi + \left(0 - \cos^{-1} \left({\left(0.5 - \frac{x}{2}\right)}^{0.5}\right)\right) \cdot 2\right)} \]
5. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \color{blue}{\left(\mathsf{PI}\left(\right) + \left(0 - \cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \left(\color{blue}{\mathsf{PI}\left(\right)} + \left(0 - \cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)\right) \]

   3. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) + \left(0 - \cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)\right) \]

   4. *-un-lft-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(1 \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(0 - \cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)\right)} \cdot 2\right)\right) \]

   5. fma-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{fma}\left(1, \color{blue}{\mathsf{PI}\left(\right)}, \left(0 - \cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)\right)\right) \]

   6. sub0-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{fma}\left(1, \mathsf{PI}\left(\right), \left(\mathsf{neg}\left(\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)\right)\right) \cdot 2\right)\right)\right) \]

   7. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{fma}\left(1, \mathsf{PI}\left(\right), \mathsf{neg}\left(\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)\right)\right)\right) \]

   8. fmm-undef N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(1 \cdot \mathsf{PI}\left(\right) - \color{blue}{\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right) \cdot 2}\right)\right) \]

   9. *-un-lft-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) - \color{blue}{\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)} \cdot 2\right)\right) \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{\_.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)}\right)\right) \]

   11. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)} \cdot 2\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right), \color{blue}{2}\right)\right)\right) \]

6. Applied egg-rr 8.1%

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

7. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. associate-+r+ N/A

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

   6. neg-mul-1 N/A

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

   7. distribute-rgt-out N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

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

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

   11. *-commutative N/A

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

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\left(2 \cdot \cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

9. Simplified 8.1%

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

Alternative 3: 5.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (Math.PI * -0.5) + (2.0 * Math.acos(Math.sqrt(0.5)));
}

Python

def code(x):
	return (math.pi * -0.5) + (2.0 * math.acos(math.sqrt(0.5)))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (pi * -0.5) + (2.0 * acos(sqrt(0.5)));
end

Wolfram

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

Derivation

1. Initial program 6.4%

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

   1. asin-acos N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(2 \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)}\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot 2 + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2}\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\color{blue}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right) \cdot 2\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot 2 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot 2\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) \cdot 1 + \left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)\right) \]

   8. *-rgt-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) + \color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot 2\right)}\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)} \cdot 2\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right), \color{blue}{2}\right)\right)\right) \]

4. Applied egg-rr 8.1%

   \[\leadsto \frac{\pi}{2} - \color{blue}{\left(\pi + \left(0 - \cos^{-1} \left({\left(0.5 - \frac{x}{2}\right)}^{0.5}\right)\right) \cdot 2\right)} \]
5. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \color{blue}{\left(\mathsf{PI}\left(\right) + \left(0 - \cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \left(\color{blue}{\mathsf{PI}\left(\right)} + \left(0 - \cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)\right) \]

   3. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) + \left(0 - \cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)\right) \]

   4. *-un-lft-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(1 \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(0 - \cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)\right)} \cdot 2\right)\right) \]

   5. fma-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{fma}\left(1, \color{blue}{\mathsf{PI}\left(\right)}, \left(0 - \cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)\right) \cdot 2\right)\right)\right) \]

   6. sub0-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{fma}\left(1, \mathsf{PI}\left(\right), \left(\mathsf{neg}\left(\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)\right)\right) \cdot 2\right)\right)\right) \]

   7. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{fma}\left(1, \mathsf{PI}\left(\right), \mathsf{neg}\left(\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)\right)\right)\right) \]

   8. fmm-undef N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(1 \cdot \mathsf{PI}\left(\right) - \color{blue}{\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right) \cdot 2}\right)\right) \]

   9. *-un-lft-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\mathsf{PI}\left(\right) - \color{blue}{\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)} \cdot 2\right)\right) \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{\_.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right) \cdot 2\right)}\right)\right) \]

   11. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right)} \cdot 2\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\cos^{-1} \left({\left(\frac{1}{2} - \frac{x}{2}\right)}^{\frac{1}{2}}\right), \color{blue}{2}\right)\right)\right) \]

6. Applied egg-rr 8.1%

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

7. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. associate-+r+ N/A

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

   6. neg-mul-1 N/A

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

   7. distribute-rgt-out N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

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

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

   11. *-commutative N/A

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

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\left(2 \cdot \cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

9. Simplified 8.1%

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

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{acos.f64}\left(\color{blue}{\left(\sqrt{\frac{1}{2}}\right)}\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]
11. Step-by-step derivation

    1. sqrt-lowering-sqrt.f64 5.5%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{acos.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

12. Simplified 5.5%

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

13. Final simplification 5.5%

    \[\leadsto \pi \cdot -0.5 + 2 \cdot \cos^{-1} \left(\sqrt{0.5}\right) \]
14. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.1× 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 2024160 
(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))))))