Average Error: 53.3 → 0.2
Time: 18.3s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.009816372300169629028232520795427262783:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8987700841367661785952236641605850309134:\\ \;\;\;\;\frac{x}{\sqrt{1}} + \left(\log \left(\sqrt{1}\right) - \frac{\frac{1}{6}}{\frac{\sqrt{1}}{\frac{\left(x \cdot x\right) \cdot x}{1}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{\frac{0.125}{x}}{x \cdot x}\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.009816372300169629028232520795427262783:\\
\;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.8987700841367661785952236641605850309134:\\
\;\;\;\;\frac{x}{\sqrt{1}} + \left(\log \left(\sqrt{1}\right) - \frac{\frac{1}{6}}{\frac{\sqrt{1}}{\frac{\left(x \cdot x\right) \cdot x}{1}}}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{\frac{0.125}{x}}{x \cdot x}\right)\right)\right)\\

\end{array}
double f(double x) {
        double r6930187 = x;
        double r6930188 = r6930187 * r6930187;
        double r6930189 = 1.0;
        double r6930190 = r6930188 + r6930189;
        double r6930191 = sqrt(r6930190);
        double r6930192 = r6930187 + r6930191;
        double r6930193 = log(r6930192);
        return r6930193;
}
↓
double f(double x) {
        double r6930194 = x;
        double r6930195 = -1.0098163723001696;
        bool r6930196 = r6930194 <= r6930195;
        double r6930197 = 0.125;
        double r6930198 = r6930197 / r6930194;
        double r6930199 = r6930194 * r6930194;
        double r6930200 = r6930198 / r6930199;
        double r6930201 = 0.0625;
        double r6930202 = 5.0;
        double r6930203 = pow(r6930194, r6930202);
        double r6930204 = r6930201 / r6930203;
        double r6930205 = 0.5;
        double r6930206 = r6930205 / r6930194;
        double r6930207 = r6930204 + r6930206;
        double r6930208 = r6930200 - r6930207;
        double r6930209 = log(r6930208);
        double r6930210 = 0.8987700841367662;
        bool r6930211 = r6930194 <= r6930210;
        double r6930212 = 1.0;
        double r6930213 = sqrt(r6930212);
        double r6930214 = r6930194 / r6930213;
        double r6930215 = log(r6930213);
        double r6930216 = 0.16666666666666666;
        double r6930217 = r6930199 * r6930194;
        double r6930218 = r6930217 / r6930212;
        double r6930219 = r6930213 / r6930218;
        double r6930220 = r6930216 / r6930219;
        double r6930221 = r6930215 - r6930220;
        double r6930222 = r6930214 + r6930221;
        double r6930223 = r6930194 - r6930200;
        double r6930224 = r6930206 + r6930223;
        double r6930225 = r6930194 + r6930224;
        double r6930226 = log(r6930225);
        double r6930227 = r6930211 ? r6930222 : r6930226;
        double r6930228 = r6930196 ? r6930209 : r6930227;
        return r6930228;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.3
Target45.5
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0098163723001696

    1. Initial program 62.8

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

    2. Taylor expanded around -inf 0.2

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right)}\]

    3. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)}\]

    if -1.0098163723001696 < x < 0.8987700841367662

    1. Initial program 58.8

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

    if 0.8987700841367662 < x

    1. Initial program 32.7

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.009816372300169629028232520795427262783:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8987700841367661785952236641605850309134:\\ \;\;\;\;\frac{x}{\sqrt{1}} + \left(\log \left(\sqrt{1}\right) - \frac{\frac{1}{6}}{\frac{\sqrt{1}}{\frac{\left(x \cdot x\right) \cdot x}{1}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{\frac{0.125}{x}}{x \cdot x}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))