Hyperbolic arcsine

Average Error: 52.8 → 0.2

Time: 6.0s

Precision: binary64

Math

\[\log \left(x + \sqrt{x \cdot x + 1}\right) \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.0730541910957112:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.0010906364138287723:\\ \;\;\;\;x - 0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(0 + \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1.0730541910957112)
   (log (- (/ 0.125 (pow x 3.0)) (/ 0.5 x)))
   (if (<= x 0.0010906364138287723)
     (- x (* 0.16666666666666666 (pow x 3.0)))
     (log (+ 0.0 (+ x (hypot 1.0 x)))))))

C

double code(double x) {
	return log((x + sqrt(((x * x) + 1.0))));
}

↓

double code(double x) {
	double tmp;
	if (x <= -1.0730541910957112) {
		tmp = log(((0.125 / pow(x, 3.0)) - (0.5 / x)));
	} else if (x <= 0.0010906364138287723) {
		tmp = x - (0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log((0.0 + (x + hypot(1.0, x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	return Math.log((x + Math.sqrt(((x * x) + 1.0))));
}

↓

public static double code(double x) {
	double tmp;
	if (x <= -1.0730541910957112) {
		tmp = Math.log(((0.125 / Math.pow(x, 3.0)) - (0.5 / x)));
	} else if (x <= 0.0010906364138287723) {
		tmp = x - (0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log((0.0 + (x + Math.hypot(1.0, x))));
	}
	return tmp;
}

Python

def code(x):
	return math.log((x + math.sqrt(((x * x) + 1.0))))

↓

def code(x):
	tmp = 0
	if x <= -1.0730541910957112:
		tmp = math.log(((0.125 / math.pow(x, 3.0)) - (0.5 / x)))
	elif x <= 0.0010906364138287723:
		tmp = x - (0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log((0.0 + (x + math.hypot(1.0, x))))
	return tmp

Julia

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) + 1.0))))
end

↓

function code(x)
	tmp = 0.0
	if (x <= -1.0730541910957112)
		tmp = log(Float64(Float64(0.125 / (x ^ 3.0)) - Float64(0.5 / x)));
	elseif (x <= 0.0010906364138287723)
		tmp = Float64(x - Float64(0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(0.0 + Float64(x + hypot(1.0, x))));
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) + 1.0))));
end

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0730541910957112)
		tmp = log(((0.125 / (x ^ 3.0)) - (0.5 / x)));
	elseif (x <= 0.0010906364138287723)
		tmp = x - (0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log((0.0 + (x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[Log[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := If[LessEqual[x, -1.0730541910957112], N[Log[N[(N[(0.125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.0010906364138287723], N[(x - N[(0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(0.0 + N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Error

Bits error versus x

Target

Original	52.8
Target	45.4
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if x < -1.0730541910957112

   1. Initial program 62.8

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]

   2. Simplified 62.9

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   3. Taylor expanded in x around -inf 0.3

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - 0.5 \cdot \frac{1}{x}\right)} \]

   4. Simplified 0.3

      \[\leadsto \log \color{blue}{\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right)} \]

   if -1.0730541910957112 < x < 0.0010906364138287723

   1. Initial program 59.0

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]

   2. Simplified 59.0

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   3. Taylor expanded in x around 0 0.2

      \[\leadsto \color{blue}{x - 0.16666666666666666 \cdot {x}^{3}} \]

   if 0.0010906364138287723 < x

   1. Initial program 30.9

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]

   2. Simplified 0.1

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   3. Applied egg-rr 0.1

      \[\leadsto \log \color{blue}{\left({\left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)}^{2}\right)} \]

   4. Applied egg-rr 0.1

      \[\leadsto \log \color{blue}{\left(0 + \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0730541910957112:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.0010906364138287723:\\ \;\;\;\;x - 0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(0 + \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :herbie-target
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))