Average Error: 53.0 → 0.2
Time: 5.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.02086207945877905:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.894743900678192428:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} - \left(\frac{0.125}{{x}^{3}} - 2 \cdot x\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.02086207945877905:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.894743900678192428:\\
\;\;\;\;\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(\frac{0.5}{x} - \left(\frac{0.125}{{x}^{3}} - 2 \cdot x\right)\right)\\

\end{array}
double f(double x) {
        double r120703 = x;
        double r120704 = r120703 * r120703;
        double r120705 = 1.0;
        double r120706 = r120704 + r120705;
        double r120707 = sqrt(r120706);
        double r120708 = r120703 + r120707;
        double r120709 = log(r120708);
        return r120709;
}
↓
double f(double x) {
        double r120710 = x;
        double r120711 = -1.020862079458779;
        bool r120712 = r120710 <= r120711;
        double r120713 = 0.125;
        double r120714 = 3.0;
        double r120715 = pow(r120710, r120714);
        double r120716 = r120713 / r120715;
        double r120717 = 0.5;
        double r120718 = r120717 / r120710;
        double r120719 = 0.0625;
        double r120720 = -r120719;
        double r120721 = 5.0;
        double r120722 = pow(r120710, r120721);
        double r120723 = r120720 / r120722;
        double r120724 = r120718 - r120723;
        double r120725 = r120716 - r120724;
        double r120726 = log(r120725);
        double r120727 = 0.8947439006781924;
        bool r120728 = r120710 <= r120727;
        double r120729 = 1.0;
        double r120730 = sqrt(r120729);
        double r120731 = log(r120730);
        double r120732 = r120710 / r120730;
        double r120733 = r120731 + r120732;
        double r120734 = 0.16666666666666666;
        double r120735 = pow(r120730, r120714);
        double r120736 = r120715 / r120735;
        double r120737 = r120734 * r120736;
        double r120738 = r120733 - r120737;
        double r120739 = 2.0;
        double r120740 = r120739 * r120710;
        double r120741 = r120716 - r120740;
        double r120742 = r120718 - r120741;
        double r120743 = log(r120742);
        double r120744 = r120728 ? r120738 : r120743;
        double r120745 = r120712 ? r120726 : r120744;
        return r120745;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.0
Target45.1
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 < -1.020862079458779

    1. Initial program 63.1

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

    2. Taylor expanded around -inf 0.1

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

    3. Simplified0.1

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

    if -1.020862079458779 < x < 0.8947439006781924

    1. Initial program 58.8

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

    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 0.8947439006781924 < x

    1. Initial program 32.4

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

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.02086207945877905:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.894743900678192428:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} - \left(\frac{0.125}{{x}^{3}} - 2 \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(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)))))