Average Error: 52.8 → 0.2
Time: 19.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.9983730585657322187387308076722547411919:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8840407169458701641673314952640794217587:\\ \;\;\;\;\frac{x}{\sqrt{1}} + \left(\log \left(\sqrt{1}\right) - \frac{\frac{1}{6}}{\frac{\sqrt{1}}{\frac{\left(x \cdot x\right) \cdot x}{1}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{\frac{0.125}{x}}{x \cdot x}\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.9983730585657322187387308076722547411919:\\
\;\;\;\;\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 0.8840407169458701641673314952640794217587:\\
\;\;\;\;\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 r7142251 = x;
        double r7142252 = r7142251 * r7142251;
        double r7142253 = 1.0;
        double r7142254 = r7142252 + r7142253;
        double r7142255 = sqrt(r7142254);
        double r7142256 = r7142251 + r7142255;
        double r7142257 = log(r7142256);
        return r7142257;
}
↓
double f(double x) {
        double r7142258 = x;
        double r7142259 = -0.9983730585657322;
        bool r7142260 = r7142258 <= r7142259;
        double r7142261 = 0.125;
        double r7142262 = r7142261 / r7142258;
        double r7142263 = r7142258 * r7142258;
        double r7142264 = r7142262 / r7142263;
        double r7142265 = 0.0625;
        double r7142266 = 5.0;
        double r7142267 = pow(r7142258, r7142266);
        double r7142268 = r7142265 / r7142267;
        double r7142269 = 0.5;
        double r7142270 = r7142269 / r7142258;
        double r7142271 = r7142268 + r7142270;
        double r7142272 = r7142264 - r7142271;
        double r7142273 = log(r7142272);
        double r7142274 = 0.8840407169458702;
        bool r7142275 = r7142258 <= r7142274;
        double r7142276 = 1.0;
        double r7142277 = sqrt(r7142276);
        double r7142278 = r7142258 / r7142277;
        double r7142279 = log(r7142277);
        double r7142280 = 0.16666666666666666;
        double r7142281 = r7142263 * r7142258;
        double r7142282 = r7142281 / r7142276;
        double r7142283 = r7142277 / r7142282;
        double r7142284 = r7142280 / r7142283;
        double r7142285 = r7142279 - r7142284;
        double r7142286 = r7142278 + r7142285;
        double r7142287 = r7142258 - r7142264;
        double r7142288 = r7142270 + r7142287;
        double r7142289 = r7142258 + r7142288;
        double r7142290 = log(r7142289);
        double r7142291 = r7142275 ? r7142286 : r7142290;
        double r7142292 = r7142260 ? r7142273 : r7142291;
        return r7142292;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.8
Target45.1
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 < -0.9983730585657322

    1. Initial program 63.0

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

    2. Taylor expanded around -inf 0.1

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

    3. Simplified0.1

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

    if -0.9983730585657322 < x < 0.8840407169458702

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 0.8840407169458702 < x

    1. Initial program 30.3

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9983730585657322187387308076722547411919:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8840407169458701641673314952640794217587:\\ \;\;\;\;\frac{x}{\sqrt{1}} + \left(\log \left(\sqrt{1}\right) - \frac{\frac{1}{6}}{\frac{\sqrt{1}}{\frac{\left(x \cdot x\right) \cdot x}{1}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{\frac{0.125}{x}}{x \cdot x}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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)))))