Average Error: 53.1 → 0.3
Time: 8.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.99895493447090078:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.90017583502977816:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.99895493447090078:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.90017583502977816:\\
\;\;\;\;\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(\frac{0.5}{x} + \left(x - \frac{0.125}{{x}^{3}}\right)\right)\right)\\

\end{array}
double f(double x) {
        double r125972 = x;
        double r125973 = r125972 * r125972;
        double r125974 = 1.0;
        double r125975 = r125973 + r125974;
        double r125976 = sqrt(r125975);
        double r125977 = r125972 + r125976;
        double r125978 = log(r125977);
        return r125978;
}
↓
double f(double x) {
        double r125979 = x;
        double r125980 = -0.9989549344709008;
        bool r125981 = r125979 <= r125980;
        double r125982 = 0.125;
        double r125983 = 3.0;
        double r125984 = pow(r125979, r125983);
        double r125985 = r125982 / r125984;
        double r125986 = 0.5;
        double r125987 = r125986 / r125979;
        double r125988 = 0.0625;
        double r125989 = 5.0;
        double r125990 = pow(r125979, r125989);
        double r125991 = r125988 / r125990;
        double r125992 = r125987 + r125991;
        double r125993 = r125985 - r125992;
        double r125994 = log(r125993);
        double r125995 = 0.9001758350297782;
        bool r125996 = r125979 <= r125995;
        double r125997 = 1.0;
        double r125998 = sqrt(r125997);
        double r125999 = log(r125998);
        double r126000 = r125979 / r125998;
        double r126001 = r125999 + r126000;
        double r126002 = 0.16666666666666666;
        double r126003 = pow(r125998, r125983);
        double r126004 = r125984 / r126003;
        double r126005 = r126002 * r126004;
        double r126006 = r126001 - r126005;
        double r126007 = r125979 - r125985;
        double r126008 = r125987 + r126007;
        double r126009 = r125979 + r126008;
        double r126010 = log(r126009);
        double r126011 = r125996 ? r126006 : r126010;
        double r126012 = r125981 ? r125994 : r126011;
        return r126012;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.1
Target45.5
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 < -0.9989549344709008

    1. Initial program 62.8

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

    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 -0.9989549344709008 < x < 0.9001758350297782

    1. Initial program 58.6

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

    2. Taylor expanded around 0 0.2

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

    if 0.9001758350297782 < x

    1. Initial program 32.1

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.99895493447090078:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.90017583502977816:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))))