Average Error: 52.9 → 0.1
Time: 15.0s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.03845383551809634:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.00121273027943644379:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{1} \cdot \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.03845383551809634:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.00121273027943644379:\\
\;\;\;\;\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 r215743 = x;
        double r215744 = r215743 * r215743;
        double r215745 = 1.0;
        double r215746 = r215744 + r215745;
        double r215747 = sqrt(r215746);
        double r215748 = r215743 + r215747;
        double r215749 = log(r215748);
        return r215749;
}
↓
double f(double x) {
        double r215750 = x;
        double r215751 = -1.0384538355180963;
        bool r215752 = r215750 <= r215751;
        double r215753 = 0.125;
        double r215754 = 3.0;
        double r215755 = pow(r215750, r215754);
        double r215756 = r215753 / r215755;
        double r215757 = 0.5;
        double r215758 = r215757 / r215750;
        double r215759 = 0.0625;
        double r215760 = -r215759;
        double r215761 = 5.0;
        double r215762 = pow(r215750, r215761);
        double r215763 = r215760 / r215762;
        double r215764 = r215758 - r215763;
        double r215765 = r215756 - r215764;
        double r215766 = log(r215765);
        double r215767 = 0.0012127302794364438;
        bool r215768 = r215750 <= r215767;
        double r215769 = 1.0;
        double r215770 = sqrt(r215769);
        double r215771 = log(r215770);
        double r215772 = r215750 / r215770;
        double r215773 = r215771 + r215772;
        double r215774 = 0.16666666666666666;
        double r215775 = pow(r215770, r215754);
        double r215776 = r215755 / r215775;
        double r215777 = r215774 * r215776;
        double r215778 = r215773 - r215777;
        double r215779 = 1.0;
        double r215780 = sqrt(r215779);
        double r215781 = hypot(r215750, r215770);
        double r215782 = r215780 * r215781;
        double r215783 = r215750 + r215782;
        double r215784 = log(r215783);
        double r215785 = r215768 ? r215778 : r215784;
        double r215786 = r215752 ? r215766 : r215785;
        return r215786;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.9
Target44.8
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 < -1.0384538355180963

    1. Initial program 63.0

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

    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 -1.0384538355180963 < x < 0.0012127302794364438

    1. Initial program 58.8

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

    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 0.0012127302794364438 < x

    1. Initial program 32.3

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

    3. Applied *-un-lft-identity32.3

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

    4. Applied sqrt-prod32.3

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.03845383551809634:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.00121273027943644379:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{1} \cdot \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))))