Hyperbolic arcsine

Average Error: 53.1 → 0.3

Time: 9.7s

Precision: binary64

Cost: 13316

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-5}:\\ \;\;\;\;\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1.02e-5)
   (log (/ 1.0 (- (hypot 1.0 x) x)))
   (if (<= x 0.8) x (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 <= -1.02e-5) {
		tmp = log((1.0 / (hypot(1.0, x) - x)));
	} else if (x <= 0.8) {
		tmp = x;
	} 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 <= -1.02e-5) {
		tmp = Math.log((1.0 / (Math.hypot(1.0, x) - x)));
	} else if (x <= 0.8) {
		tmp = x;
	} 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 <= -1.02e-5:
		tmp = math.log((1.0 / (math.hypot(1.0, x) - x)))
	elif x <= 0.8:
		tmp = x
	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 <= -1.02e-5)
		tmp = log(Float64(1.0 / Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.8)
		tmp = x;
	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 <= -1.02e-5)
		tmp = log((1.0 / (hypot(1.0, x) - x)));
	elseif (x <= 0.8)
		tmp = x;
	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, -1.02e-5], N[Log[N[(1.0 / N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.8], x, N[Log[N[(N[(x * 2.0), $MachinePrecision] + N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	53.1
Target	45.4
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if x < -1.0200000000000001e-5

   1. Initial program 61.8

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

   2. Simplified 61.8

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      Proof
      (log.f64 (+.f64 x (hypot.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 x x)))))): 3 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) 1))))): 0 points increase in error, 3 points decrease in error

   3. Applied egg-rr 61.8

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

   4. Simplified 0.2

      \[\leadsto \log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)} \]
      Proof
      (log.f64 (/.f64 1 (-.f64 (hypot.f64 1 x) x))): 0 points increase in error, 0 points decrease in error
      (log.f64 (/.f64 1 (Rewrite=> sub-neg_binary64 (+.f64 (hypot.f64 1 x) (neg.f64 x))))): 3 points increase in error, 11 points decrease in error
      (log.f64 (/.f64 1 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 x) (hypot.f64 1 x))))): 11 points increase in error, 3 points decrease in error
      (log.f64 (/.f64 1 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 x)) (hypot.f64 1 x)))): 4 points increase in error, 10 points decrease in error
      (log.f64 (/.f64 1 (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 x (hypot.f64 1 x)))))): 10 points increase in error, 4 points decrease in error
      (log.f64 (/.f64 1 (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 x (hypot.f64 1 x)))))): 0 points increase in error, 14 points decrease in error
      (log.f64 (/.f64 1 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 x (hypot.f64 1 x)))))): 14 points increase in error, 0 points decrease in error
      (log.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 1 -1) (-.f64 x (hypot.f64 1 x))))): 14 points increase in error, 0 points decrease in error
      (log.f64 (/.f64 (Rewrite=> metadata-eval -1) (-.f64 x (hypot.f64 1 x)))): 0 points increase in error, 14 points decrease in error
      (log.f64 (/.f64 (Rewrite<= metadata-eval (-.f64 0 1)) (-.f64 x (hypot.f64 1 x)))): 14 points increase in error, 0 points decrease in error
      (log.f64 (/.f64 (-.f64 (Rewrite<= +-inverses_binary64 (-.f64 (*.f64 x x) (*.f64 x x))) 1) (-.f64 x (hypot.f64 1 x)))): 0 points increase in error, 14 points decrease in error
      (log.f64 (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 (*.f64 x x) (+.f64 (*.f64 x x) 1))) (-.f64 x (hypot.f64 1 x)))): 4 points increase in error, 10 points decrease in error
      (log.f64 (/.f64 (-.f64 (*.f64 x x) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 x x)))) (-.f64 x (hypot.f64 1 x)))): 10 points increase in error, 4 points decrease in error
      (log.f64 (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (*.f64 x x) (-.f64 x (hypot.f64 1 x))) (/.f64 (+.f64 1 (*.f64 x x)) (-.f64 x (hypot.f64 1 x)))))): 0 points increase in error, 14 points decrease in error

   if -1.0200000000000001e-5 < x < 0.80000000000000004

   1. Initial program 59.0

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

   2. Simplified 59.0

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      Proof
      (log.f64 (+.f64 x (hypot.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 x x)))))): 3 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) 1))))): 0 points increase in error, 3 points decrease in error

   3. Taylor expanded in x around 0 0.3

      \[\leadsto \color{blue}{x} \]

   if 0.80000000000000004 < x

   1. Initial program 32.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      Proof
      (log.f64 (+.f64 x (hypot.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 x x)))))): 3 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) 1))))): 0 points increase in error, 3 points decrease in error

   3. Taylor expanded in x around inf 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-5}:\\ \;\;\;\;\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	13316

\[\begin{array}{l} \mathbf{if}\;x \leq -0.061:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	13252

\[\begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-6}:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \]

Alternative 4
Error	0.3
Cost	7240

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

Alternative 5
Error	0.5
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	0.6
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\left(x \cdot \left(x + 2\right)\right) \cdot \frac{1}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 7
Error	15.0
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 0.75:\\ \;\;\;\;\frac{x \cdot 2}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 8
Error	25.7
Cost	6596

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-14}:\\ \;\;\;\;\frac{x \cdot 2}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x\right)\\ \end{array} \]

Alternative 9
Error	28.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq 10^{-8}:\\ \;\;\;\;\frac{x \cdot 2}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \frac{1}{x + 2}\\ \end{array} \]

Alternative 10
Error	29.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\frac{x \cdot 2}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	30.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022341 
(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)))))