Average Error: 53.1 → 0.1
Time: 5.3s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.9971603401393445:\\ \;\;\;\;\left(\frac{0.09375}{{x}^{4}} + \left(\log \left(\frac{-1}{x}\right) + \log 0.5\right)\right) - \left(\frac{0.25}{x \cdot x} + \frac{0.052083333333333336}{{x}^{6}}\right)\\ \mathbf{elif}\;x \leq 0.023563724942838992:\\ \;\;\;\;\mathsf{fma}\left(0.075, {x}^{5}, x\right) - \mathsf{fma}\left(0.16666666666666666, {x}^{3}, 0.044642857142857144 \cdot {x}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.9971603401393445:\\
\;\;\;\;\left(\frac{0.09375}{{x}^{4}} + \left(\log \left(\frac{-1}{x}\right) + \log 0.5\right)\right) - \left(\frac{0.25}{x \cdot x} + \frac{0.052083333333333336}{{x}^{6}}\right)\\

\mathbf{elif}\;x \leq 0.023563724942838992:\\
\;\;\;\;\mathsf{fma}\left(0.075, {x}^{5}, x\right) - \mathsf{fma}\left(0.16666666666666666, {x}^{3}, 0.044642857142857144 \cdot {x}^{7}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\


\end{array}

(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.9971603401393445)
   (-
    (+ (/ 0.09375 (pow x 4.0)) (+ (log (/ -1.0 x)) (log 0.5)))
    (+ (/ 0.25 (* x x)) (/ 0.052083333333333336 (pow x 6.0))))
   (if (<= x 0.023563724942838992)
     (-
      (fma 0.075 (pow x 5.0) x)
      (fma
       0.16666666666666666
       (pow x 3.0)
       (* 0.044642857142857144 (pow x 7.0))))
     (log (+ x (hypot 1.0 x))))))
double code(double x) {
	return log(x + sqrt((x * x) + 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -0.9971603401393445) {
		tmp = ((0.09375 / pow(x, 4.0)) + (log(-1.0 / x) + log(0.5))) - ((0.25 / (x * x)) + (0.052083333333333336 / pow(x, 6.0)));
	} else if (x <= 0.023563724942838992) {
		tmp = fma(0.075, pow(x, 5.0), x) - fma(0.16666666666666666, pow(x, 3.0), (0.044642857142857144 * pow(x, 7.0)));
	} else {
		tmp = log(x + hypot(1.0, x));
	}
	return tmp;
}

Error

Bits error versus x

Target

Original53.1
Target45.0
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.99716034013934451

    1. Initial program 63.2

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

    2. Simplified63.2

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

    3. Taylor expanded in x around -inf 0.3

      \[\leadsto \color{blue}{\left(0.09375 \cdot \frac{1}{{x}^{4}} + \left(\log \left(\frac{-1}{x}\right) + \log 0.5\right)\right) - \left(0.052083333333333336 \cdot \frac{1}{{x}^{6}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

    4. Simplified0.3

      \[\leadsto \color{blue}{\left(\frac{0.09375}{{x}^{4}} + \left(\log \left(\frac{-1}{x}\right) + \log 0.5\right)\right) - \left(\frac{0.25}{x \cdot x} + \frac{0.052083333333333336}{{x}^{6}}\right)} \]

    if -0.99716034013934451 < x < 0.0235637249428389921

    1. Initial program 58.6

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

    2. Simplified58.6

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

    3. Taylor expanded in x around 0 0.1

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

    4. Simplified0.1

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

    if 0.0235637249428389921 < x

    1. Initial program 32.3

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

    2. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.9971603401393445:\\ \;\;\;\;\left(\frac{0.09375}{{x}^{4}} + \left(\log \left(\frac{-1}{x}\right) + \log 0.5\right)\right) - \left(\frac{0.25}{x \cdot x} + \frac{0.052083333333333336}{{x}^{6}}\right)\\ \mathbf{elif}\;x \leq 0.023563724942838992:\\ \;\;\;\;\mathsf{fma}\left(0.075, {x}^{5}, x\right) - \mathsf{fma}\left(0.16666666666666666, {x}^{3}, 0.044642857142857144 \cdot {x}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(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)))))