Average Error: 52.9 → 0.2
Time: 13.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0632494934709653:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} + \frac{\frac{-1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9505314742632508:\\ \;\;\;\;\left(x + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}\right) + {x}^{5} \cdot \frac{3}{40}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0632494934709653:\\
\;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} + \frac{\frac{-1}{2}}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.9505314742632508:\\
\;\;\;\;\left(x + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}\right) + {x}^{5} \cdot \frac{3}{40}\\

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

\end{array}
double f(double x) {
        double r2830150 = x;
        double r2830151 = r2830150 * r2830150;
        double r2830152 = 1.0;
        double r2830153 = r2830151 + r2830152;
        double r2830154 = sqrt(r2830153);
        double r2830155 = r2830150 + r2830154;
        double r2830156 = log(r2830155);
        return r2830156;
}
↓
double f(double x) {
        double r2830157 = x;
        double r2830158 = -1.0632494934709653;
        bool r2830159 = r2830157 <= r2830158;
        double r2830160 = -0.0625;
        double r2830161 = 5.0;
        double r2830162 = pow(r2830157, r2830161);
        double r2830163 = r2830160 / r2830162;
        double r2830164 = 0.125;
        double r2830165 = r2830157 * r2830157;
        double r2830166 = r2830157 * r2830165;
        double r2830167 = r2830164 / r2830166;
        double r2830168 = -0.5;
        double r2830169 = r2830168 / r2830157;
        double r2830170 = r2830167 + r2830169;
        double r2830171 = r2830163 + r2830170;
        double r2830172 = log(r2830171);
        double r2830173 = 0.9505314742632508;
        bool r2830174 = r2830157 <= r2830173;
        double r2830175 = -0.16666666666666666;
        double r2830176 = r2830166 * r2830175;
        double r2830177 = r2830157 + r2830176;
        double r2830178 = 0.075;
        double r2830179 = r2830162 * r2830178;
        double r2830180 = r2830177 + r2830179;
        double r2830181 = 0.5;
        double r2830182 = r2830181 / r2830157;
        double r2830183 = r2830182 - r2830167;
        double r2830184 = r2830157 + r2830183;
        double r2830185 = r2830184 + r2830157;
        double r2830186 = log(r2830185);
        double r2830187 = r2830174 ? r2830180 : r2830186;
        double r2830188 = r2830159 ? r2830172 : r2830187;
        return r2830188;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.9
Target44.8
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0632494934709653

    1. Initial program 61.7

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

    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

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

    if -1.0632494934709653 < x < 0.9505314742632508

    1. Initial program 58.7

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

    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

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

    if 0.9505314742632508 < x

    1. Initial program 32.2

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0632494934709653:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} + \frac{\frac{-1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9505314742632508:\\ \;\;\;\;\left(x + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}\right) + {x}^{5} \cdot \frac{3}{40}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 
(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)))))