Average Error: 52.9 → 0.3
Time: 8.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.03845383551809634:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.89194585549241023:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} - \left(\frac{0.125}{{x}^{3}} - 2 \cdot x\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.03845383551809634:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.89194585549241023:\\
\;\;\;\;\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 r264909 = x;
        double r264910 = r264909 * r264909;
        double r264911 = 1.0;
        double r264912 = r264910 + r264911;
        double r264913 = sqrt(r264912);
        double r264914 = r264909 + r264913;
        double r264915 = log(r264914);
        return r264915;
}
↓
double f(double x) {
        double r264916 = x;
        double r264917 = -1.0384538355180963;
        bool r264918 = r264916 <= r264917;
        double r264919 = 0.125;
        double r264920 = 3.0;
        double r264921 = pow(r264916, r264920);
        double r264922 = r264919 / r264921;
        double r264923 = 0.5;
        double r264924 = r264923 / r264916;
        double r264925 = 0.0625;
        double r264926 = -r264925;
        double r264927 = 5.0;
        double r264928 = pow(r264916, r264927);
        double r264929 = r264926 / r264928;
        double r264930 = r264924 - r264929;
        double r264931 = r264922 - r264930;
        double r264932 = log(r264931);
        double r264933 = 0.8919458554924102;
        bool r264934 = r264916 <= r264933;
        double r264935 = 1.0;
        double r264936 = sqrt(r264935);
        double r264937 = log(r264936);
        double r264938 = r264916 / r264936;
        double r264939 = r264937 + r264938;
        double r264940 = 0.16666666666666666;
        double r264941 = pow(r264936, r264920);
        double r264942 = r264921 / r264941;
        double r264943 = r264940 * r264942;
        double r264944 = r264939 - r264943;
        double r264945 = 2.0;
        double r264946 = r264945 * r264916;
        double r264947 = r264922 - r264946;
        double r264948 = r264924 - r264947;
        double r264949 = log(r264948);
        double r264950 = r264934 ? r264944 : r264949;
        double r264951 = r264918 ? r264932 : r264950;
        return r264951;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.9
Target44.8
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0384538355180963

    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 < 0.8919458554924102

    1. Initial program 58.6

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

    2. Taylor expanded around 0 0.3

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

    if 0.8919458554924102 < x

    1. Initial program 32.5

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

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.03845383551809634:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.89194585549241023:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} - \left(\frac{0.125}{{x}^{3}} - 2 \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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)))))