Average Error: 52.7 → 0.2
Time: 30.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.014477909773237040980120582389645278454:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.8909481582438814051272402139147743582726:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) + \frac{\left(x \cdot x\right) \cdot x}{1} \cdot \frac{\frac{-1}{6}}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{0.5}{x} - \frac{\frac{0.125}{x}}{x \cdot x}\right) + x\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.014477909773237040980120582389645278454:\\
\;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.8909481582438814051272402139147743582726:\\
\;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) + \frac{\left(x \cdot x\right) \cdot x}{1} \cdot \frac{\frac{-1}{6}}{\sqrt{1}}\\

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

\end{array}
double f(double x) {
        double r6867377 = x;
        double r6867378 = r6867377 * r6867377;
        double r6867379 = 1.0;
        double r6867380 = r6867378 + r6867379;
        double r6867381 = sqrt(r6867380);
        double r6867382 = r6867377 + r6867381;
        double r6867383 = log(r6867382);
        return r6867383;
}
↓
double f(double x) {
        double r6867384 = x;
        double r6867385 = -1.014477909773237;
        bool r6867386 = r6867384 <= r6867385;
        double r6867387 = 0.125;
        double r6867388 = r6867387 / r6867384;
        double r6867389 = r6867384 * r6867384;
        double r6867390 = r6867388 / r6867389;
        double r6867391 = 0.5;
        double r6867392 = r6867391 / r6867384;
        double r6867393 = 0.0625;
        double r6867394 = 5.0;
        double r6867395 = pow(r6867384, r6867394);
        double r6867396 = r6867393 / r6867395;
        double r6867397 = r6867392 + r6867396;
        double r6867398 = r6867390 - r6867397;
        double r6867399 = log(r6867398);
        double r6867400 = 0.8909481582438814;
        bool r6867401 = r6867384 <= r6867400;
        double r6867402 = 1.0;
        double r6867403 = sqrt(r6867402);
        double r6867404 = log(r6867403);
        double r6867405 = r6867384 / r6867403;
        double r6867406 = r6867404 + r6867405;
        double r6867407 = r6867389 * r6867384;
        double r6867408 = r6867407 / r6867402;
        double r6867409 = -0.16666666666666666;
        double r6867410 = r6867409 / r6867403;
        double r6867411 = r6867408 * r6867410;
        double r6867412 = r6867406 + r6867411;
        double r6867413 = r6867392 - r6867390;
        double r6867414 = r6867413 + r6867384;
        double r6867415 = r6867384 + r6867414;
        double r6867416 = log(r6867415);
        double r6867417 = r6867401 ? r6867412 : r6867416;
        double r6867418 = r6867386 ? r6867399 : r6867417;
        return r6867418;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.7
Target44.9
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 < -1.014477909773237

    1. Initial program 63.0

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

    2. Taylor expanded around -inf 0.2

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

    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 -1.014477909773237 < x < 0.8909481582438814

    1. Initial program 58.4

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 0.8909481582438814 < x

    1. Initial program 31.4

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.014477909773237040980120582389645278454:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.8909481582438814051272402139147743582726:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) + \frac{\left(x \cdot x\right) \cdot x}{1} \cdot \frac{\frac{-1}{6}}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{0.5}{x} - \frac{\frac{0.125}{x}}{x \cdot x}\right) + x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(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)))))