Average Error: 52.9 → 0.3
Time: 28.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.002637632358202246507516974816098809242:\\ \;\;\;\;\log \left(\left(\frac{0.125}{\left(x \cdot x\right) \cdot x} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.8920640501139610600844775945006404072046:\\ \;\;\;\;\left(\frac{x}{1} \cdot \left(\frac{x}{\sqrt{1}} \cdot x\right)\right) \cdot \frac{-1}{6} + \left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{0.5}{x} - \frac{0.125}{\left(x \cdot x\right) \cdot x}\right) + x\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.002637632358202246507516974816098809242:\\
\;\;\;\;\log \left(\left(\frac{0.125}{\left(x \cdot x\right) \cdot x} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 0.8920640501139610600844775945006404072046:\\
\;\;\;\;\left(\frac{x}{1} \cdot \left(\frac{x}{\sqrt{1}} \cdot x\right)\right) \cdot \frac{-1}{6} + \left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\

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

\end{array}
double f(double x) {
        double r7940290 = x;
        double r7940291 = r7940290 * r7940290;
        double r7940292 = 1.0;
        double r7940293 = r7940291 + r7940292;
        double r7940294 = sqrt(r7940293);
        double r7940295 = r7940290 + r7940294;
        double r7940296 = log(r7940295);
        return r7940296;
}
↓
double f(double x) {
        double r7940297 = x;
        double r7940298 = -1.0026376323582022;
        bool r7940299 = r7940297 <= r7940298;
        double r7940300 = 0.125;
        double r7940301 = r7940297 * r7940297;
        double r7940302 = r7940301 * r7940297;
        double r7940303 = r7940300 / r7940302;
        double r7940304 = 0.5;
        double r7940305 = r7940304 / r7940297;
        double r7940306 = r7940303 - r7940305;
        double r7940307 = 0.0625;
        double r7940308 = 5.0;
        double r7940309 = pow(r7940297, r7940308);
        double r7940310 = r7940307 / r7940309;
        double r7940311 = r7940306 - r7940310;
        double r7940312 = log(r7940311);
        double r7940313 = 0.8920640501139611;
        bool r7940314 = r7940297 <= r7940313;
        double r7940315 = 1.0;
        double r7940316 = r7940297 / r7940315;
        double r7940317 = sqrt(r7940315);
        double r7940318 = r7940297 / r7940317;
        double r7940319 = r7940318 * r7940297;
        double r7940320 = r7940316 * r7940319;
        double r7940321 = -0.16666666666666666;
        double r7940322 = r7940320 * r7940321;
        double r7940323 = log(r7940317);
        double r7940324 = r7940323 + r7940318;
        double r7940325 = r7940322 + r7940324;
        double r7940326 = r7940305 - r7940303;
        double r7940327 = r7940326 + r7940297;
        double r7940328 = r7940297 + r7940327;
        double r7940329 = log(r7940328);
        double r7940330 = r7940314 ? r7940325 : r7940329;
        double r7940331 = r7940299 ? r7940312 : r7940330;
        return r7940331;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.9
Target45.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0026376323582022

    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

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

    if -1.0026376323582022 < x < 0.8920640501139611

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.4

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

    3. Simplified0.4

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

    if 0.8920640501139611 < x

    1. Initial program 31.7

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.002637632358202246507516974816098809242:\\ \;\;\;\;\log \left(\left(\frac{0.125}{\left(x \cdot x\right) \cdot x} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.8920640501139610600844775945006404072046:\\ \;\;\;\;\left(\frac{x}{1} \cdot \left(\frac{x}{\sqrt{1}} \cdot x\right)\right) \cdot \frac{-1}{6} + \left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{0.5}{x} - \frac{0.125}{\left(x \cdot x\right) \cdot x}\right) + x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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)))))