Average Error: 53.0 → 0.3
Time: 7.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.006460284952229500277098850347101688385:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.8931378377170595683764986461028456687927:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.006460284952229500277098850347101688385:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.8931378377170595683764986461028456687927:\\
\;\;\;\;\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(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\

\end{array}
double f(double x) {
        double r185827 = x;
        double r185828 = r185827 * r185827;
        double r185829 = 1.0;
        double r185830 = r185828 + r185829;
        double r185831 = sqrt(r185830);
        double r185832 = r185827 + r185831;
        double r185833 = log(r185832);
        return r185833;
}
↓
double f(double x) {
        double r185834 = x;
        double r185835 = -1.0064602849522295;
        bool r185836 = r185834 <= r185835;
        double r185837 = 0.125;
        double r185838 = 3.0;
        double r185839 = pow(r185834, r185838);
        double r185840 = r185837 / r185839;
        double r185841 = 0.5;
        double r185842 = r185841 / r185834;
        double r185843 = 0.0625;
        double r185844 = -r185843;
        double r185845 = 5.0;
        double r185846 = pow(r185834, r185845);
        double r185847 = r185844 / r185846;
        double r185848 = r185842 - r185847;
        double r185849 = r185840 - r185848;
        double r185850 = log(r185849);
        double r185851 = 0.8931378377170596;
        bool r185852 = r185834 <= r185851;
        double r185853 = 1.0;
        double r185854 = sqrt(r185853);
        double r185855 = log(r185854);
        double r185856 = r185834 / r185854;
        double r185857 = r185855 + r185856;
        double r185858 = 0.16666666666666666;
        double r185859 = pow(r185854, r185838);
        double r185860 = r185839 / r185859;
        double r185861 = r185858 * r185860;
        double r185862 = r185857 - r185861;
        double r185863 = r185834 + r185842;
        double r185864 = r185863 - r185840;
        double r185865 = r185834 + r185864;
        double r185866 = log(r185865);
        double r185867 = r185852 ? r185862 : r185866;
        double r185868 = r185836 ? r185850 : r185867;
        return r185868;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.0
Target45.0
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 < -1.0064602849522295

    1. Initial program 62.9

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

    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 -1.0064602849522295 < x < 0.8931378377170596

    1. Initial program 58.7

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

    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 0.8931378377170596 < x

    1. Initial program 31.9

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.006460284952229500277098850347101688385:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.8931378377170595683764986461028456687927:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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)))))