Hyperbolic arcsine

Average Error: 53.3 → 0.2

Time: 4.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2603231646087443:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.0010462575660624225:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1.2603231646087443)
   (log (/ -0.5 x))
   (if (<= x 0.0010462575660624225)
     (fma (pow x 3.0) -0.16666666666666666 x)
     (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 <= -1.2603231646087443) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.0010462575660624225) {
		tmp = fma(pow(x, 3.0), -0.16666666666666666, x);
	} else {
		tmp = log((x + 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 <= -1.2603231646087443)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.0010462575660624225)
		tmp = fma((x ^ 3.0), -0.16666666666666666, x);
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return 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, -1.2603231646087443], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.0010462575660624225], N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Error

Bits error versus x

Target

Original	53.3
Target	45.5
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 < -1.26032316460874427

   1. Initial program 63.0

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

   2. Simplified 63.0

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

   3. Taylor expanded in x around -inf 0.5

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

   if -1.26032316460874427 < x < 0.0010462575660624225

   1. Initial program 58.9

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

   2. Simplified 58.9

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

   3. Taylor expanded in x around 0 0.1

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

   4. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \]

   if 0.0010462575660624225 < x

   1. Initial program 32.8

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

   2. Simplified 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2603231646087443:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.0010462575660624225:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022148 
(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)))))