Average Error: 53.1 → 0.3
Time: 15.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.9941877870754349810766825612518005073071:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.9042263775546853787901113719271961599588:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + x\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.9941877870754349810766825612518005073071:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.9042263775546853787901113719271961599588:\\
\;\;\;\;\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 + \left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + x\right)\right)\\

\end{array}
double f(double x) {
        double r99708 = x;
        double r99709 = r99708 * r99708;
        double r99710 = 1.0;
        double r99711 = r99709 + r99710;
        double r99712 = sqrt(r99711);
        double r99713 = r99708 + r99712;
        double r99714 = log(r99713);
        return r99714;
}
↓
double f(double x) {
        double r99715 = x;
        double r99716 = -0.994187787075435;
        bool r99717 = r99715 <= r99716;
        double r99718 = 0.125;
        double r99719 = 3.0;
        double r99720 = pow(r99715, r99719);
        double r99721 = r99718 / r99720;
        double r99722 = 0.5;
        double r99723 = r99722 / r99715;
        double r99724 = 0.0625;
        double r99725 = 5.0;
        double r99726 = pow(r99715, r99725);
        double r99727 = r99724 / r99726;
        double r99728 = r99723 + r99727;
        double r99729 = r99721 - r99728;
        double r99730 = log(r99729);
        double r99731 = 0.9042263775546854;
        bool r99732 = r99715 <= r99731;
        double r99733 = 1.0;
        double r99734 = sqrt(r99733);
        double r99735 = log(r99734);
        double r99736 = r99715 / r99734;
        double r99737 = r99735 + r99736;
        double r99738 = 0.16666666666666666;
        double r99739 = pow(r99734, r99719);
        double r99740 = r99720 / r99739;
        double r99741 = r99738 * r99740;
        double r99742 = r99737 - r99741;
        double r99743 = r99723 - r99721;
        double r99744 = r99743 + r99715;
        double r99745 = r99715 + r99744;
        double r99746 = log(r99745);
        double r99747 = r99732 ? r99742 : r99746;
        double r99748 = r99717 ? r99730 : r99747;
        return r99748;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.1
Target45.6
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 < -0.994187787075435

    1. Initial program 62.7

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

    2. Taylor expanded around -inf 0.3

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

    3. Simplified0.3

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

    if -0.994187787075435 < x < 0.9042263775546854

    1. Initial program 58.6

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

    2. Taylor expanded around 0 0.4

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

    if 0.9042263775546854 < x

    1. Initial program 33.0

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9941877870754349810766825612518005073071:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.9042263775546853787901113719271961599588:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(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)))))