Average Error: 52.8 → 0.2
Time: 18.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0869167841195677:\\ \;\;\;\;\log \left(\left(\frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.9419391586635008:\\ \;\;\;\;\left(\frac{3}{40} \cdot {x}^{5} - \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) \cdot x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x}\right) + x\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0869167841195677:\\
\;\;\;\;\log \left(\left(\frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 0.9419391586635008:\\
\;\;\;\;\left(\frac{3}{40} \cdot {x}^{5} - \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) \cdot x\right) + x\\

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

\end{array}
double f(double x) {
        double r4811265 = x;
        double r4811266 = r4811265 * r4811265;
        double r4811267 = 1.0;
        double r4811268 = r4811266 + r4811267;
        double r4811269 = sqrt(r4811268);
        double r4811270 = r4811265 + r4811269;
        double r4811271 = log(r4811270);
        return r4811271;
}
↓
double f(double x) {
        double r4811272 = x;
        double r4811273 = -1.0869167841195677;
        bool r4811274 = r4811272 <= r4811273;
        double r4811275 = 0.125;
        double r4811276 = r4811272 * r4811272;
        double r4811277 = r4811276 * r4811272;
        double r4811278 = r4811275 / r4811277;
        double r4811279 = 0.5;
        double r4811280 = r4811279 / r4811272;
        double r4811281 = r4811278 - r4811280;
        double r4811282 = 0.0625;
        double r4811283 = 5.0;
        double r4811284 = pow(r4811272, r4811283);
        double r4811285 = r4811282 / r4811284;
        double r4811286 = r4811281 - r4811285;
        double r4811287 = log(r4811286);
        double r4811288 = 0.9419391586635008;
        bool r4811289 = r4811272 <= r4811288;
        double r4811290 = 0.075;
        double r4811291 = r4811290 * r4811284;
        double r4811292 = 0.16666666666666666;
        double r4811293 = r4811276 * r4811292;
        double r4811294 = r4811293 * r4811272;
        double r4811295 = r4811291 - r4811294;
        double r4811296 = r4811295 + r4811272;
        double r4811297 = r4811280 - r4811278;
        double r4811298 = r4811297 + r4811272;
        double r4811299 = r4811272 + r4811298;
        double r4811300 = log(r4811299);
        double r4811301 = r4811289 ? r4811296 : r4811300;
        double r4811302 = r4811274 ? r4811287 : r4811301;
        return r4811302;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.8
Target45.2
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 < -1.0869167841195677

    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

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

    if -1.0869167841195677 < x < 0.9419391586635008

    1. Initial program 58.6

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

    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

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

    if 0.9419391586635008 < x

    1. Initial program 31.8

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0869167841195677:\\ \;\;\;\;\log \left(\left(\frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.9419391586635008:\\ \;\;\;\;\left(\frac{3}{40} \cdot {x}^{5} - \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) \cdot x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x}\right) + x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(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)))))