Hyperbolic arcsine

Average Accuracy: 17.3% → 99.8%

Time: 6.7s

Precision: binary64

Cost: 13316

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00045:\\ \;\;\;\;\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.00045)
   (log (/ 1.0 (- (hypot 1.0 x) x)))
   (if (<= x 0.95)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (+ (* x 2.0) (* 0.5 (/ 1.0 x)))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.00045) {
		tmp = log((1.0 / (hypot(1.0, x) - x)));
	} else if (x <= 0.95) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log(((x * 2.0) + (0.5 * (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 <= -0.00045) {
		tmp = Math.log((1.0 / (Math.hypot(1.0, x) - x)));
	} else if (x <= 0.95) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log(((x * 2.0) + (0.5 * (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 <= -0.00045:
		tmp = math.log((1.0 / (math.hypot(1.0, x) - x)))
	elif x <= 0.95:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log(((x * 2.0) + (0.5 * (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 <= -0.00045)
		tmp = log(Float64(1.0 / Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.95)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(Float64(x * 2.0) + Float64(0.5 * Float64(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 <= -0.00045)
		tmp = log((1.0 / (hypot(1.0, x) - x)));
	elseif (x <= 0.95)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log(((x * 2.0) + (0.5 * (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, -0.00045], N[Log[N[(1.0 / N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.95], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(x * 2.0), $MachinePrecision] + N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	17.3%
Target	29.4%
Herbie	99.8%

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

Derivation

1. Split input into 3 regimes

2. if x < -4.4999999999999999e-4

   1. Initial program 2.1%

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

   2. Simplified 2.1%

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

      [Start] 2.1	\[ \log \left(x + \sqrt{x \cdot x + 1}\right) \]
      +-commutative [=>] 2.1	\[ \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]
      hypot-1-def [=>] 2.1	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Applied egg-rr 2.1%

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

   4. Simplified 99.9%

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

      [Start] 2.1	\[ \log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      div-sub [<=] 3.0	\[ \log \color{blue}{\left(\frac{x \cdot x - \left(1 + x \cdot x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
      +-commutative [=>] 3.0	\[ \log \left(\frac{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      associate--r+ [=>] 50.2	\[ \log \left(\frac{\color{blue}{\left(x \cdot x - x \cdot x\right) - 1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      +-inverses [=>] 99.9	\[ \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      metadata-eval [=>] 99.9	\[ \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      metadata-eval [<=] 99.9	\[ \log \left(\frac{\color{blue}{\frac{1}{-1}}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      associate-/r* [<=] 99.9	\[ \log \color{blue}{\left(\frac{1}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)} \]
      neg-mul-1 [<=] 99.9	\[ \log \left(\frac{1}{\color{blue}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right) \]
      neg-sub0 [=>] 99.9	\[ \log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right) \]
      associate--r- [=>] 99.9	\[ \log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right) \]
      neg-sub0 [<=] 99.9	\[ \log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right) \]
      +-commutative [<=] 99.9	\[ \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + \left(-x\right)}}\right) \]
      sub-neg [<=] 99.9	\[ \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right) \]

   if -4.4999999999999999e-4 < x < 0.94999999999999996

   1. Initial program 8.2%

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

   2. Simplified 8.3%

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

      [Start] 8.2	\[ \log \left(x + \sqrt{x \cdot x + 1}\right) \]
      +-commutative [=>] 8.2	\[ \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]
      hypot-1-def [=>] 8.3	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Taylor expanded in x around 0 99.8%

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

   if 0.94999999999999996 < x

   1. Initial program 49.8%

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

   2. Simplified 99.8%

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

      [Start] 49.8	\[ \log \left(x + \sqrt{x \cdot x + 1}\right) \]
      +-commutative [=>] 49.8	\[ \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]
      hypot-1-def [=>] 99.8	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Taylor expanded in x around inf 99.5%

      \[\leadsto \log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00045:\\ \;\;\;\;\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0009:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \]

Alternative 2
Accuracy	99.6%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;-\log \left(x \cdot -2 + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 4
Accuracy	99.5%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;-\log \left(x \cdot -2 + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 5
Accuracy	99.2%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 6
Accuracy	75.7%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 1.26:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 7
Accuracy	51.8%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023122 
(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)))))