Average Error: 52.9 → 0.3
Time: 11.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.02004933856096858:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.88957403075873587:\\ \;\;\;\;\left(\frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} + \left(2 \cdot x - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.02004933856096858:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.88957403075873587:\\
\;\;\;\;\left(\frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\

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

\end{array}
double f(double x) {
        double r140717 = x;
        double r140718 = r140717 * r140717;
        double r140719 = 1.0;
        double r140720 = r140718 + r140719;
        double r140721 = sqrt(r140720);
        double r140722 = r140717 + r140721;
        double r140723 = log(r140722);
        return r140723;
}
↓
double f(double x) {
        double r140724 = x;
        double r140725 = -1.0200493385609686;
        bool r140726 = r140724 <= r140725;
        double r140727 = 0.125;
        double r140728 = 3.0;
        double r140729 = pow(r140724, r140728);
        double r140730 = r140727 / r140729;
        double r140731 = 0.5;
        double r140732 = r140731 / r140724;
        double r140733 = 0.0625;
        double r140734 = 5.0;
        double r140735 = pow(r140724, r140734);
        double r140736 = r140733 / r140735;
        double r140737 = r140732 + r140736;
        double r140738 = r140730 - r140737;
        double r140739 = log(r140738);
        double r140740 = 0.8895740307587359;
        bool r140741 = r140724 <= r140740;
        double r140742 = 1.0;
        double r140743 = sqrt(r140742);
        double r140744 = r140724 / r140743;
        double r140745 = log(r140743);
        double r140746 = r140744 + r140745;
        double r140747 = 0.16666666666666666;
        double r140748 = pow(r140743, r140728);
        double r140749 = r140729 / r140748;
        double r140750 = r140747 * r140749;
        double r140751 = r140746 - r140750;
        double r140752 = 2.0;
        double r140753 = r140752 * r140724;
        double r140754 = r140753 - r140730;
        double r140755 = r140732 + r140754;
        double r140756 = log(r140755);
        double r140757 = r140741 ? r140751 : r140756;
        double r140758 = r140726 ? r140739 : r140757;
        return r140758;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.9
Target45.4
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0200493385609686

    1. Initial program 62.7

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

    2. Taylor expanded around -inf 0.2

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

    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 -1.0200493385609686 < x < 0.8895740307587359

    1. Initial program 58.6

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

    2. Taylor expanded around 0 0.3

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

    if 0.8895740307587359 < x

    1. Initial program 31.1

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.02004933856096858:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.88957403075873587:\\ \;\;\;\;\left(\frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} + \left(2 \cdot x - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x)
  :name "Hyperbolic arcsine"

  :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)))))