Average Error: 53.3 → 0.1
Time: 12.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.010680203662621456928150109888520091772:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.001041760398045713442369275547605411702534:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{1} \cdot \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.010680203662621456928150109888520091772:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.001041760398045713442369275547605411702534:\\
\;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\

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

\end{array}
double f(double x) {
        double r224497 = x;
        double r224498 = r224497 * r224497;
        double r224499 = 1.0;
        double r224500 = r224498 + r224499;
        double r224501 = sqrt(r224500);
        double r224502 = r224497 + r224501;
        double r224503 = log(r224502);
        return r224503;
}
↓
double f(double x) {
        double r224504 = x;
        double r224505 = -1.0106802036626215;
        bool r224506 = r224504 <= r224505;
        double r224507 = 0.125;
        double r224508 = 3.0;
        double r224509 = pow(r224504, r224508);
        double r224510 = r224507 / r224509;
        double r224511 = 0.5;
        double r224512 = r224511 / r224504;
        double r224513 = 0.0625;
        double r224514 = -r224513;
        double r224515 = 5.0;
        double r224516 = pow(r224504, r224515);
        double r224517 = r224514 / r224516;
        double r224518 = r224512 - r224517;
        double r224519 = r224510 - r224518;
        double r224520 = log(r224519);
        double r224521 = 0.0010417603980457134;
        bool r224522 = r224504 <= r224521;
        double r224523 = 1.0;
        double r224524 = sqrt(r224523);
        double r224525 = log(r224524);
        double r224526 = r224504 / r224524;
        double r224527 = r224525 + r224526;
        double r224528 = 0.16666666666666666;
        double r224529 = pow(r224524, r224508);
        double r224530 = r224509 / r224529;
        double r224531 = r224528 * r224530;
        double r224532 = r224527 - r224531;
        double r224533 = 1.0;
        double r224534 = sqrt(r224533);
        double r224535 = hypot(r224504, r224524);
        double r224536 = r224534 * r224535;
        double r224537 = r224504 + r224536;
        double r224538 = log(r224537);
        double r224539 = r224522 ? r224532 : r224538;
        double r224540 = r224506 ? r224520 : r224539;
        return r224540;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.3
Target45.4
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0106802036626215

    1. Initial program 62.9

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

    2. Taylor expanded around -inf 0.2

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)}\]

    3. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)}\]

    if -1.0106802036626215 < x < 0.0010417603980457134

    1. Initial program 59.1

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

    2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}}\]

    if 0.0010417603980457134 < x

    1. Initial program 31.9

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
    2. Using strategy rm

    3. Applied *-un-lft-identity31.9

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

    4. Applied sqrt-prod31.9

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

    5. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.010680203662621456928150109888520091772:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.001041760398045713442369275547605411702534:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{1} \cdot \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :herbie-target
  (if (< x 0.0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))