Hyperbolic arcsine

Average Error: 83.12% → 0.1%

Time: 12.0s

Precision: binary64

Cost: 20488

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.023:\\ \;\;\;\;\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)\\ \mathbf{elif}\;x \leq 0.0225:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.023)
   (log (/ 1.0 (- (hypot 1.0 x) x)))
   (if (<= x 0.0225)
     (+
      (* -0.16666666666666666 (pow x 3.0))
      (+ (* 0.075 (pow x 5.0)) (+ x (* -0.044642857142857144 (pow x 7.0)))))
     (log (+ 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 <= -0.023) {
		tmp = log((1.0 / (hypot(1.0, x) - x)));
	} else if (x <= 0.0225) {
		tmp = (-0.16666666666666666 * pow(x, 3.0)) + ((0.075 * pow(x, 5.0)) + (x + (-0.044642857142857144 * pow(x, 7.0))));
	} else {
		tmp = log((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 <= -0.023) {
		tmp = Math.log((1.0 / (Math.hypot(1.0, x) - x)));
	} else if (x <= 0.0225) {
		tmp = (-0.16666666666666666 * Math.pow(x, 3.0)) + ((0.075 * Math.pow(x, 5.0)) + (x + (-0.044642857142857144 * Math.pow(x, 7.0))));
	} else {
		tmp = Math.log((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 <= -0.023:
		tmp = math.log((1.0 / (math.hypot(1.0, x) - x)))
	elif x <= 0.0225:
		tmp = (-0.16666666666666666 * math.pow(x, 3.0)) + ((0.075 * math.pow(x, 5.0)) + (x + (-0.044642857142857144 * math.pow(x, 7.0))))
	else:
		tmp = math.log((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 <= -0.023)
		tmp = log(Float64(1.0 / Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.0225)
		tmp = Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + Float64(Float64(0.075 * (x ^ 5.0)) + Float64(x + Float64(-0.044642857142857144 * (x ^ 7.0)))));
	else
		tmp = log(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 <= -0.023)
		tmp = log((1.0 / (hypot(1.0, x) - x)));
	elseif (x <= 0.0225)
		tmp = (-0.16666666666666666 * (x ^ 3.0)) + ((0.075 * (x ^ 5.0)) + (x + (-0.044642857142857144 * (x ^ 7.0))));
	else
		tmp = log((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, -0.023], N[Log[N[(1.0 / N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.0225], N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(-0.044642857142857144 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	83.12%
Target	70.99%
Herbie	0.1%

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

Derivation

1. Split input into 3 regimes

2. if x < -0.023

   1. Initial program 97.86

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

   2. Applied egg-rr 0.12

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

   3. Simplified 0.12

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

      [Start] 0.12	\[ \log \left(\left(x \cdot \left(x - x\right) - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      *-commutative [=>] 0.12	\[ \log \left(\left(\color{blue}{\left(x - x\right) \cdot x} - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      +-inverses [=>] 0.12	\[ \log \left(\left(\color{blue}{0} \cdot x - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      mul0-lft [=>] 0.12	\[ \log \left(\left(\color{blue}{0} - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      metadata-eval [=>] 0.12	\[ \log \left(\color{blue}{-1} \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      associate-*r/ [=>] 0.12	\[ \log \color{blue}{\left(\frac{-1 \cdot 1}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
      metadata-eval [=>] 0.12	\[ \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      metadata-eval [<=] 0.12	\[ \log \left(\frac{\color{blue}{\frac{1}{-1}}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      associate-/r* [<=] 0.12	\[ \log \color{blue}{\left(\frac{1}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)} \]
      neg-mul-1 [<=] 0.12	\[ \log \left(\frac{1}{\color{blue}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right) \]
      neg-sub0 [=>] 0.12	\[ \log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right) \]
      associate--r- [=>] 0.12	\[ \log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right) \]
      neg-sub0 [<=] 0.12	\[ \log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right) \]
      mul-1-neg [<=] 0.12	\[ \log \left(\frac{1}{\color{blue}{-1 \cdot x} + \mathsf{hypot}\left(1, x\right)}\right) \]
      +-commutative [<=] 0.12	\[ \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + -1 \cdot x}}\right) \]
      mul-1-neg [=>] 0.12	\[ \log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) + \color{blue}{\left(-x\right)}}\right) \]
      sub-neg [<=] 0.12	\[ \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right) \]

   if -0.023 < x < 0.022499999999999999

   1. Initial program 91.93

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

   2. Taylor expanded in x around 0 0.01

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right)} \]

   if 0.022499999999999999 < x

   1. Initial program 50.32

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

   2. Simplified 0.27

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

      [Start] 50.32	\[ \log \left(x + \sqrt{x \cdot x + 1}\right) \]
      +-commutative [=>] 50.32	\[ \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]
      hypot-1-def [=>] 0.27	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.023:\\ \;\;\;\;\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)\\ \mathbf{elif}\;x \leq 0.0225:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.11%
Cost	13768

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0064:\\ \;\;\;\;\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)\\ \mathbf{elif}\;x \leq 0.0072:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 2
Error	0.43%
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00092:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 3
Error	0.12%
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00088:\\ \;\;\;\;\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)\\ \mathbf{elif}\;x \leq 0.00092:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 4
Error	0.6%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Alternative 5
Error	0.68%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 6
Error	0.96%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 7
Error	24.66%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 8
Error	41.34%
Cost	6596

\[\begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x\right)\\ \end{array} \]

Alternative 9
Error	47.76%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(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)))))