Hyperbolic arcsine

Average Error: 53.0 → 0.2

Time: 9.0s

Precision: binary64

Cost: 13444

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00105:\\ \;\;\;\;\left(1 - \log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right) + -1\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.00105)
   (+ (- 1.0 (log (- (hypot 1.0 x) x))) -1.0)
   (if (<= x 0.95)
     (+ x (* (* x x) (* x -0.16666666666666666)))
     (log (+ x (+ x (/ 0.5 x)))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.00105) {
		tmp = (1.0 - log((hypot(1.0, x) - x))) + -1.0;
	} else if (x <= 0.95) {
		tmp = x + ((x * x) * (x * -0.16666666666666666));
	} else {
		tmp = log((x + (x + (0.5 / 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 <= -0.00105) {
		tmp = (1.0 - Math.log((Math.hypot(1.0, x) - x))) + -1.0;
	} else if (x <= 0.95) {
		tmp = x + ((x * x) * (x * -0.16666666666666666));
	} else {
		tmp = Math.log((x + (x + (0.5 / x))));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -0.00105:
		tmp = (1.0 - math.log((math.hypot(1.0, x) - x))) + -1.0
	elif x <= 0.95:
		tmp = x + ((x * x) * (x * -0.16666666666666666))
	else:
		tmp = math.log((x + (x + (0.5 / 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 <= -0.00105)
		tmp = Float64(Float64(1.0 - log(Float64(hypot(1.0, x) - x))) + -1.0);
	elseif (x <= 0.95)
		tmp = Float64(x + Float64(Float64(x * x) * Float64(x * -0.16666666666666666)));
	else
		tmp = log(Float64(x + Float64(x + Float64(0.5 / 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 <= -0.00105)
		tmp = (1.0 - log((hypot(1.0, x) - x))) + -1.0;
	elseif (x <= 0.95)
		tmp = x + ((x * x) * (x * -0.16666666666666666));
	else
		tmp = log((x + (x + (0.5 / 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, -0.00105], N[(N[(1.0 - N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[x, 0.95], N[(x + N[(N[(x * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[(x + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	53.0
Target	45.3
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 < -0.00104999999999999994

   1. Initial program 62.3

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

   2. Applied egg-rr 0.1

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

   3. Simplified 0.1

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

      [Start] 0.1	\[ \log \left(\left(x \cdot \left(x - x\right) - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      *-commutative [=>] 0.1	\[ \log \left(\left(\color{blue}{\left(x - x\right) \cdot x} - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      +-inverses [=>] 0.1	\[ \log \left(\left(\color{blue}{0} \cdot x - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      mul0-lft [=>] 0.1	\[ \log \left(\left(\color{blue}{0} - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      metadata-eval [=>] 0.1	\[ \log \left(\color{blue}{-1} \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      associate-*r/ [=>] 0.1	\[ \log \color{blue}{\left(\frac{-1 \cdot 1}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
      metadata-eval [=>] 0.1	\[ \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      metadata-eval [<=] 0.1	\[ \log \left(\frac{\color{blue}{\frac{1}{-1}}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      associate-/r* [<=] 0.1	\[ \log \color{blue}{\left(\frac{1}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)} \]
      neg-mul-1 [<=] 0.1	\[ \log \left(\frac{1}{\color{blue}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right) \]
      neg-sub0 [=>] 0.1	\[ \log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right) \]
      associate--r- [=>] 0.1	\[ \log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right) \]
      neg-sub0 [<=] 0.1	\[ \log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right) \]
      mul-1-neg [<=] 0.1	\[ \log \left(\frac{1}{\color{blue}{-1 \cdot x} + \mathsf{hypot}\left(1, x\right)}\right) \]
      +-commutative [<=] 0.1	\[ \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + -1 \cdot x}}\right) \]
      mul-1-neg [=>] 0.1	\[ \log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) + \color{blue}{\left(-x\right)}}\right) \]
      sub-neg [<=] 0.1	\[ \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right) \]

   4. Applied egg-rr 63.3

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)} - 1} \]

   5. Applied egg-rr 0.1

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

   if -0.00104999999999999994 < x < 0.94999999999999996

   1. Initial program 58.8

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

   2. Taylor expanded in x around 0 0.1

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

   3. Applied egg-rr 0.3

      \[\leadsto \color{blue}{\left(\left(1 + -0.16666666666666666 \cdot {x}^{3}\right) - 1\right)} + x \]

   4. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)} + x \]

   if 0.94999999999999996 < x

   1. Initial program 31.8

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

   2. Taylor expanded in x around inf 0.4

      \[\leadsto \log \left(x + \color{blue}{\left(0.5 \cdot \frac{1}{x} + x\right)}\right) \]

   3. Simplified 0.4

      \[\leadsto \log \left(x + \color{blue}{\left(x + \frac{0.5}{x}\right)}\right) \]
      Proof

      [Start] 0.4	\[ \log \left(x + \left(0.5 \cdot \frac{1}{x} + x\right)\right) \]
      +-commutative [=>] 0.4	\[ \log \left(x + \color{blue}{\left(x + 0.5 \cdot \frac{1}{x}\right)}\right) \]
      associate-*r/ [=>] 0.4	\[ \log \left(x + \left(x + \color{blue}{\frac{0.5 \cdot 1}{x}}\right)\right) \]
      metadata-eval [=>] 0.4	\[ \log \left(x + \left(x + \frac{\color{blue}{0.5}}{x}\right)\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00105:\\ \;\;\;\;\left(1 - \log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right) + -1\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	13316

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0011:\\ \;\;\;\;\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	13252

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0011:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Alternative 4
Error	0.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;-\log \left(x \cdot -2 + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Alternative 5
Error	0.4
Cost	6856

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

Alternative 6
Error	15.4
Cost	6724

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

Alternative 7
Error	30.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023060 
(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)))))