Ian Simplification

Average Error: 59.7 → 58.3

Time: 18.8s

Precision: binary64

Cost: 26312

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-162}:\\ \;\;\;\;\frac{1}{\frac{1}{0.5 \cdot \pi + \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right) \cdot -2}}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\left(\sin^{-1} \left(\sqrt{0.5}\right) \cdot -2 + \pi \cdot -0.5\right) + \pi\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1.65e-162)
   (/ 1.0 (/ 1.0 (+ (* 0.5 PI) (* (asin (sqrt (+ 0.5 (* x -0.5)))) -2.0))))
   (if (<= x 5.5e-17)
     (+ (+ (* (asin (sqrt 0.5)) -2.0) (* PI -0.5)) PI)
     (- (/ 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))));
}

↓

double code(double x) {
	double tmp;
	if (x <= -1.65e-162) {
		tmp = 1.0 / (1.0 / ((0.5 * ((double) M_PI)) + (asin(sqrt((0.5 + (x * -0.5)))) * -2.0)));
	} else if (x <= 5.5e-17) {
		tmp = ((asin(sqrt(0.5)) * -2.0) + (((double) M_PI) * -0.5)) + ((double) M_PI);
	} else {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
	}
	return tmp;
}

Java

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

↓

public static double code(double x) {
	double tmp;
	if (x <= -1.65e-162) {
		tmp = 1.0 / (1.0 / ((0.5 * Math.PI) + (Math.asin(Math.sqrt((0.5 + (x * -0.5)))) * -2.0)));
	} else if (x <= 5.5e-17) {
		tmp = ((Math.asin(Math.sqrt(0.5)) * -2.0) + (Math.PI * -0.5)) + Math.PI;
	} else {
		tmp = (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -1.65e-162:
		tmp = 1.0 / (1.0 / ((0.5 * math.pi) + (math.asin(math.sqrt((0.5 + (x * -0.5)))) * -2.0)))
	elif x <= 5.5e-17:
		tmp = ((math.asin(math.sqrt(0.5)) * -2.0) + (math.pi * -0.5)) + math.pi
	else:
		tmp = (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -1.65e-162)
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(0.5 * pi) + Float64(asin(sqrt(Float64(0.5 + Float64(x * -0.5)))) * -2.0))));
	elseif (x <= 5.5e-17)
		tmp = Float64(Float64(Float64(asin(sqrt(0.5)) * -2.0) + Float64(pi * -0.5)) + pi);
	else
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.65e-162)
		tmp = 1.0 / (1.0 / ((0.5 * pi) + (asin(sqrt((0.5 + (x * -0.5)))) * -2.0)));
	elseif (x <= 5.5e-17)
		tmp = ((asin(sqrt(0.5)) * -2.0) + (pi * -0.5)) + pi;
	else
		tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
	end
	tmp_2 = tmp;
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]

↓

code[x_] := If[LessEqual[x, -1.65e-162], N[(1.0 / N[(1.0 / N[(N[(0.5 * Pi), $MachinePrecision] + N[(N[ArcSin[N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e-17], N[(N[(N[(N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] + N[(Pi * -0.5), $MachinePrecision]), $MachinePrecision] + Pi), $MachinePrecision], 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]]]

Target

Original	59.7
Target	0.0
Herbie	58.3

\[\sin^{-1} x \]

Derivation

1. Split input into 3 regimes

2. if x < -1.65000000000000007e-162

   1. Initial program 56.5

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

   2. Applied egg-rr 56.5

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

   if -1.65000000000000007e-162 < x < 5.50000000000000001e-17

   1. Initial program 62.2

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

   2. Applied egg-rr 60.2

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

   3. Taylor expanded in x around 0 60.2

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

   if 5.50000000000000001e-17 < x

   1. Initial program 25.7

      \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-162}:\\ \;\;\;\;\frac{1}{\frac{1}{0.5 \cdot \pi + \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right) \cdot -2}}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\left(\sin^{-1} \left(\sqrt{0.5}\right) \cdot -2 + \pi \cdot -0.5\right) + \pi\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	58.8
Cost	26304

\[\left(\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right) \cdot -2 + \pi \cdot -0.5\right) + \pi \]

Alternative 2
Error	59.7
Cost	20096

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

Alternative 3
Error	59.7
Cost	19840

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

Alternative 4
Error	61.4
Cost	19584

\[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \]

Reproduce

herbie shell --seed 2023066 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))