Ian Simplification

Average Error: 59.6 → 59.2

Time: 45.8s

Precision: binary64

Cost: 19972

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-162}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \left(\left(1 + \sin^{-1} \left(\sqrt{0.5}\right)\right) - 1\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1.7e-162)
   (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0)))))
   (- (/ PI 2.0) (* 2.0 (- (+ 1.0 (asin (sqrt 0.5))) 1.0)))))

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 <= -1.7e-162) {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
	} else {
		tmp = (((double) M_PI) / 2.0) - (2.0 * ((1.0 + asin(sqrt(0.5))) - 1.0));
	}
	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 <= -1.7e-162) {
		tmp = (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
	} else {
		tmp = (Math.PI / 2.0) - (2.0 * ((1.0 + Math.asin(Math.sqrt(0.5))) - 1.0));
	}
	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 <= -1.7e-162:
		tmp = (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))
	else:
		tmp = (math.pi / 2.0) - (2.0 * ((1.0 + math.asin(math.sqrt(0.5))) - 1.0))
	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 <= -1.7e-162)
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))));
	else
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * Float64(Float64(1.0 + asin(sqrt(0.5))) - 1.0)));
	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 <= -1.7e-162)
		tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
	else
		tmp = (pi / 2.0) - (2.0 * ((1.0 + asin(sqrt(0.5))) - 1.0));
	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, -1.7e-162], 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], N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[(N[(1.0 + N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	59.6
Target	0
Herbie	59.2

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

Derivation

1. Split input into 2 regimes

2. if x < -1.7e-162

   1. Initial program 56.3

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

   if -1.7e-162 < x

   1. Initial program 60.8

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

   2. Taylor expanded in x around 0 62.2

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

   3. Applied egg-rr 60.2

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left(\left(1 + \sin^{-1} \left(\sqrt{0.5}\right)\right) - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-162}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \left(\left(1 + \sin^{-1} \left(\sqrt{0.5}\right)\right) - 1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	58.7
Cost	39616

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

Alternative 2
Error	58.8
Cost	20096

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

Alternative 3
Error	59.6
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 2023077 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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