Average Error: 53.2 → 0.2
Time: 14.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.016091287793053909993545858014840632677:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.8917023207783242311208482533402275294065:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.016091287793053909993545858014840632677:\\
\;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 0.8917023207783242311208482533402275294065:\\
\;\;\;\;\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 r166463 = x;
        double r166464 = r166463 * r166463;
        double r166465 = 1.0;
        double r166466 = r166464 + r166465;
        double r166467 = sqrt(r166466);
        double r166468 = r166463 + r166467;
        double r166469 = log(r166468);
        return r166469;
}
↓
double f(double x) {
        double r166470 = x;
        double r166471 = -1.016091287793054;
        bool r166472 = r166470 <= r166471;
        double r166473 = 0.125;
        double r166474 = 3.0;
        double r166475 = pow(r166470, r166474);
        double r166476 = r166473 / r166475;
        double r166477 = 0.5;
        double r166478 = r166477 / r166470;
        double r166479 = r166476 - r166478;
        double r166480 = 0.0625;
        double r166481 = 5.0;
        double r166482 = pow(r166470, r166481);
        double r166483 = r166480 / r166482;
        double r166484 = r166479 - r166483;
        double r166485 = log(r166484);
        double r166486 = 0.8917023207783242;
        bool r166487 = r166470 <= r166486;
        double r166488 = 1.0;
        double r166489 = sqrt(r166488);
        double r166490 = log(r166489);
        double r166491 = r166470 / r166489;
        double r166492 = r166490 + r166491;
        double r166493 = 0.16666666666666666;
        double r166494 = pow(r166489, r166474);
        double r166495 = r166475 / r166494;
        double r166496 = r166493 * r166495;
        double r166497 = r166492 - r166496;
        double r166498 = r166478 - r166476;
        double r166499 = r166470 + r166498;
        double r166500 = r166470 + r166499;
        double r166501 = log(r166500);
        double r166502 = r166487 ? r166497 : r166501;
        double r166503 = r166472 ? r166485 : r166502;
        return r166503;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.2
Target45.4
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 < -1.016091287793054

    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

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

    if -1.016091287793054 < x < 0.8917023207783242

    1. Initial program 58.6

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

    2. Taylor expanded around 0 0.2

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

    if 0.8917023207783242 < x

    1. Initial program 32.5

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.016091287793053909993545858014840632677:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.8917023207783242311208482533402275294065:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :herbie-target
  (if (< x 0.0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))