Average Error: 52.3 → 0.1
Time: 1.7m
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0413231664509972:\\ \;\;\;\;\log \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(\frac{\frac{1}{2}}{x} + \frac{\frac{1}{16}}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.00732269910368124:\\ \;\;\;\;(\left(x \cdot \frac{-1}{6}\right) \cdot \left(x \cdot x\right) + \left((\frac{3}{40} \cdot \left({x}^{5}\right) + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{\sqrt{1^2 + x^2}^*} \cdot \sqrt{\sqrt{1^2 + x^2}^*}\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0413231664509972:\\
\;\;\;\;\log \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(\frac{\frac{1}{2}}{x} + \frac{\frac{1}{16}}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.00732269910368124:\\
\;\;\;\;(\left(x \cdot \frac{-1}{6}\right) \cdot \left(x \cdot x\right) + \left((\frac{3}{40} \cdot \left({x}^{5}\right) + x)_*\right))_*\\

\mathbf{else}:\\
\;\;\;\;\log \left(x + \sqrt{\sqrt{1^2 + x^2}^*} \cdot \sqrt{\sqrt{1^2 + x^2}^*}\right)\\

\end{array}
double f(double x) {
        double r7264892 = x;
        double r7264893 = r7264892 * r7264892;
        double r7264894 = 1.0;
        double r7264895 = r7264893 + r7264894;
        double r7264896 = sqrt(r7264895);
        double r7264897 = r7264892 + r7264896;
        double r7264898 = log(r7264897);
        return r7264898;
}
↓
double f(double x) {
        double r7264899 = x;
        double r7264900 = -1.0413231664509972;
        bool r7264901 = r7264899 <= r7264900;
        double r7264902 = 0.125;
        double r7264903 = r7264902 / r7264899;
        double r7264904 = r7264899 * r7264899;
        double r7264905 = r7264903 / r7264904;
        double r7264906 = 0.5;
        double r7264907 = r7264906 / r7264899;
        double r7264908 = 0.0625;
        double r7264909 = 5.0;
        double r7264910 = pow(r7264899, r7264909);
        double r7264911 = r7264908 / r7264910;
        double r7264912 = r7264907 + r7264911;
        double r7264913 = r7264905 - r7264912;
        double r7264914 = log(r7264913);
        double r7264915 = 0.00732269910368124;
        bool r7264916 = r7264899 <= r7264915;
        double r7264917 = -0.16666666666666666;
        double r7264918 = r7264899 * r7264917;
        double r7264919 = 0.075;
        double r7264920 = fma(r7264919, r7264910, r7264899);
        double r7264921 = fma(r7264918, r7264904, r7264920);
        double r7264922 = 1.0;
        double r7264923 = hypot(r7264922, r7264899);
        double r7264924 = sqrt(r7264923);
        double r7264925 = r7264924 * r7264924;
        double r7264926 = r7264899 + r7264925;
        double r7264927 = log(r7264926);
        double r7264928 = r7264916 ? r7264921 : r7264927;
        double r7264929 = r7264901 ? r7264914 : r7264928;
        return r7264929;
}

Error

Bits error versus x

Target

Original52.3
Target44.6
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0413231664509972

    1. Initial program 61.7

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

    2. Simplified60.9

      \[\leadsto \color{blue}{\log \left(x + \sqrt{1^2 + x^2}^*\right)}\]

    3. Taylor expanded around -inf 0.2

      \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

    4. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(\frac{\frac{1}{16}}{{x}^{5}} + \frac{\frac{1}{2}}{x}\right)\right)}\]

    if -1.0413231664509972 < x < 0.00732269910368124

    1. Initial program 58.8

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

    2. Simplified58.8

      \[\leadsto \color{blue}{\log \left(x + \sqrt{1^2 + x^2}^*\right)}\]

    3. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]

    4. Simplified0.1

      \[\leadsto \color{blue}{(\left(x \cdot \frac{-1}{6}\right) \cdot \left(x \cdot x\right) + \left((\frac{3}{40} \cdot \left({x}^{5}\right) + x)_*\right))_*}\]

    if 0.00732269910368124 < x

    1. Initial program 30.2

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

    2. Simplified0.1

      \[\leadsto \color{blue}{\log \left(x + \sqrt{1^2 + x^2}^*\right)}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt0.1

      \[\leadsto \log \left(x + \color{blue}{\sqrt{\sqrt{1^2 + x^2}^*} \cdot \sqrt{\sqrt{1^2 + x^2}^*}}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0413231664509972:\\ \;\;\;\;\log \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(\frac{\frac{1}{2}}{x} + \frac{\frac{1}{16}}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.00732269910368124:\\ \;\;\;\;(\left(x \cdot \frac{-1}{6}\right) \cdot \left(x \cdot x\right) + \left((\frac{3}{40} \cdot \left({x}^{5}\right) + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{\sqrt{1^2 + x^2}^*} \cdot \sqrt{\sqrt{1^2 + x^2}^*}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))