Average Error: 52.9 → 0.2
Time: 14.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.008768630290992396325577828974928706884:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.8844350792771517033585837452847044914961:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(2 \cdot x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.008768630290992396325577828974928706884:\\
\;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 0.8844350792771517033585837452847044914961:\\
\;\;\;\;\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(\left(2 \cdot x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\\

\end{array}
double f(double x) {
        double r114504 = x;
        double r114505 = r114504 * r114504;
        double r114506 = 1.0;
        double r114507 = r114505 + r114506;
        double r114508 = sqrt(r114507);
        double r114509 = r114504 + r114508;
        double r114510 = log(r114509);
        return r114510;
}
↓
double f(double x) {
        double r114511 = x;
        double r114512 = -1.0087686302909924;
        bool r114513 = r114511 <= r114512;
        double r114514 = 0.125;
        double r114515 = 3.0;
        double r114516 = pow(r114511, r114515);
        double r114517 = r114514 / r114516;
        double r114518 = 0.5;
        double r114519 = r114518 / r114511;
        double r114520 = r114517 - r114519;
        double r114521 = 0.0625;
        double r114522 = 5.0;
        double r114523 = pow(r114511, r114522);
        double r114524 = r114521 / r114523;
        double r114525 = r114520 - r114524;
        double r114526 = log(r114525);
        double r114527 = 0.8844350792771517;
        bool r114528 = r114511 <= r114527;
        double r114529 = 1.0;
        double r114530 = sqrt(r114529);
        double r114531 = log(r114530);
        double r114532 = r114511 / r114530;
        double r114533 = r114531 + r114532;
        double r114534 = 0.16666666666666666;
        double r114535 = pow(r114530, r114515);
        double r114536 = r114516 / r114535;
        double r114537 = r114534 * r114536;
        double r114538 = r114533 - r114537;
        double r114539 = 2.0;
        double r114540 = r114539 * r114511;
        double r114541 = r114540 + r114519;
        double r114542 = r114541 - r114517;
        double r114543 = log(r114542);
        double r114544 = r114528 ? r114538 : r114543;
        double r114545 = r114513 ? r114526 : r114544;
        return r114545;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.9
Target45.3
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0087686302909924

    1. Initial program 62.7

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

    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

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

    if -1.0087686302909924 < x < 0.8844350792771517

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.3

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

    if 0.8844350792771517 < x

    1. Initial program 33.0

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

    2. Taylor expanded around inf 0.2

      \[\leadsto \log \color{blue}{\left(\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)}\]

    3. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\left(2 \cdot x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.008768630290992396325577828974928706884:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.8844350792771517033585837452847044914961:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(2 \cdot x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(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)))))