Average Error: 53.0 → 0.2
Time: 18.3s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.03908666992687837:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.89600402982816929:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.03908666992687837:\\
\;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 0.89600402982816929:\\
\;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\log \left(x + \left(x + \left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\right)\\

\end{array}
double f(double x) {
        double r113368 = x;
        double r113369 = r113368 * r113368;
        double r113370 = 1.0;
        double r113371 = r113369 + r113370;
        double r113372 = sqrt(r113371);
        double r113373 = r113368 + r113372;
        double r113374 = log(r113373);
        return r113374;
}
↓
double f(double x) {
        double r113375 = x;
        double r113376 = -1.0390866699268784;
        bool r113377 = r113375 <= r113376;
        double r113378 = 0.125;
        double r113379 = 3.0;
        double r113380 = pow(r113375, r113379);
        double r113381 = r113378 / r113380;
        double r113382 = 0.5;
        double r113383 = r113382 / r113375;
        double r113384 = r113381 - r113383;
        double r113385 = 0.0625;
        double r113386 = 5.0;
        double r113387 = pow(r113375, r113386);
        double r113388 = r113385 / r113387;
        double r113389 = r113384 - r113388;
        double r113390 = log(r113389);
        double r113391 = 0.8960040298281693;
        bool r113392 = r113375 <= r113391;
        double r113393 = 1.0;
        double r113394 = sqrt(r113393);
        double r113395 = log(r113394);
        double r113396 = r113375 / r113394;
        double r113397 = r113395 + r113396;
        double r113398 = 0.16666666666666666;
        double r113399 = pow(r113394, r113379);
        double r113400 = r113380 / r113399;
        double r113401 = r113398 * r113400;
        double r113402 = r113397 - r113401;
        double r113403 = r113383 - r113381;
        double r113404 = r113375 + r113403;
        double r113405 = r113375 + r113404;
        double r113406 = log(r113405);
        double r113407 = r113392 ? r113402 : r113406;
        double r113408 = r113377 ? r113390 : r113407;
        return r113408;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.0
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 < -1.0390866699268784

    1. Initial program 62.8

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

    2. Taylor expanded around -inf 0.2

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

    3. Simplified0.2

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

    if -1.0390866699268784 < x < 0.8960040298281693

    1. Initial program 59.0

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

    2. Taylor expanded around 0 0.2

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

    if 0.8960040298281693 < x

    1. Initial program 31.8

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.03908666992687837:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.89600402982816929:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(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)))))