Average Error: 53.1 → 0.2
Time: 33.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.056937052753612:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9616249105092439:\\ \;\;\;\;\left(\frac{3}{40} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(x + x\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.056937052753612:\\
\;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.9616249105092439:\\
\;\;\;\;\left(\frac{3}{40} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\

\mathbf{else}:\\
\;\;\;\;\log \left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(x + x\right)\right)\right)\\

\end{array}
double f(double x) {
        double r6466876 = x;
        double r6466877 = r6466876 * r6466876;
        double r6466878 = 1.0;
        double r6466879 = r6466877 + r6466878;
        double r6466880 = sqrt(r6466879);
        double r6466881 = r6466876 + r6466880;
        double r6466882 = log(r6466881);
        return r6466882;
}
↓
double f(double x) {
        double r6466883 = x;
        double r6466884 = -1.056937052753612;
        bool r6466885 = r6466883 <= r6466884;
        double r6466886 = -0.0625;
        double r6466887 = r6466883 * r6466883;
        double r6466888 = r6466887 * r6466883;
        double r6466889 = r6466887 * r6466888;
        double r6466890 = r6466886 / r6466889;
        double r6466891 = 0.125;
        double r6466892 = r6466891 / r6466883;
        double r6466893 = r6466892 / r6466887;
        double r6466894 = 0.5;
        double r6466895 = r6466894 / r6466883;
        double r6466896 = r6466893 - r6466895;
        double r6466897 = r6466890 + r6466896;
        double r6466898 = log(r6466897);
        double r6466899 = 0.9616249105092439;
        bool r6466900 = r6466883 <= r6466899;
        double r6466901 = 0.075;
        double r6466902 = r6466901 * r6466889;
        double r6466903 = -0.16666666666666666;
        double r6466904 = r6466888 * r6466903;
        double r6466905 = r6466902 + r6466904;
        double r6466906 = r6466905 + r6466883;
        double r6466907 = r6466883 + r6466883;
        double r6466908 = r6466893 - r6466907;
        double r6466909 = r6466895 - r6466908;
        double r6466910 = log(r6466909);
        double r6466911 = r6466900 ? r6466906 : r6466910;
        double r6466912 = r6466885 ? r6466898 : r6466911;
        return r6466912;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.1
Target45.9
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.056937052753612

    1. Initial program 61.6

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

    2. Taylor expanded around -inf 0.2

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

    3. Simplified0.2

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

    if -1.056937052753612 < x < 0.9616249105092439

    1. Initial program 58.8

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

    if 0.9616249105092439 < x

    1. Initial program 33.0

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(x + x\right)\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.056937052753612:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9616249105092439:\\ \;\;\;\;\left(\frac{3}{40} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(x + x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(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)))))