Average Error: 52.6 → 0.2
Time: 13.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.076228701950242:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} + \frac{\frac{-1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9496967329184121:\\ \;\;\;\;\left(x + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}\right) + {x}^{5} \cdot \frac{3}{40}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.076228701950242:\\
\;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} + \frac{\frac{-1}{2}}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.9496967329184121:\\
\;\;\;\;\left(x + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}\right) + {x}^{5} \cdot \frac{3}{40}\\

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

\end{array}
double f(double x) {
        double r2314557 = x;
        double r2314558 = r2314557 * r2314557;
        double r2314559 = 1.0;
        double r2314560 = r2314558 + r2314559;
        double r2314561 = sqrt(r2314560);
        double r2314562 = r2314557 + r2314561;
        double r2314563 = log(r2314562);
        return r2314563;
}
↓
double f(double x) {
        double r2314564 = x;
        double r2314565 = -1.076228701950242;
        bool r2314566 = r2314564 <= r2314565;
        double r2314567 = -0.0625;
        double r2314568 = 5.0;
        double r2314569 = pow(r2314564, r2314568);
        double r2314570 = r2314567 / r2314569;
        double r2314571 = 0.125;
        double r2314572 = r2314564 * r2314564;
        double r2314573 = r2314564 * r2314572;
        double r2314574 = r2314571 / r2314573;
        double r2314575 = -0.5;
        double r2314576 = r2314575 / r2314564;
        double r2314577 = r2314574 + r2314576;
        double r2314578 = r2314570 + r2314577;
        double r2314579 = log(r2314578);
        double r2314580 = 0.9496967329184121;
        bool r2314581 = r2314564 <= r2314580;
        double r2314582 = -0.16666666666666666;
        double r2314583 = r2314573 * r2314582;
        double r2314584 = r2314564 + r2314583;
        double r2314585 = 0.075;
        double r2314586 = r2314569 * r2314585;
        double r2314587 = r2314584 + r2314586;
        double r2314588 = 0.5;
        double r2314589 = r2314588 / r2314564;
        double r2314590 = r2314589 - r2314574;
        double r2314591 = r2314564 + r2314590;
        double r2314592 = r2314591 + r2314564;
        double r2314593 = log(r2314592);
        double r2314594 = r2314581 ? r2314587 : r2314593;
        double r2314595 = r2314566 ? r2314579 : r2314594;
        return r2314595;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.6
Target44.6
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 < -1.076228701950242

    1. Initial program 61.8

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

    2. Taylor expanded around -inf 0.1

      \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

    3. Simplified0.1

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

    if -1.076228701950242 < x < 0.9496967329184121

    1. Initial program 58.5

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

    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

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

    if 0.9496967329184121 < x

    1. Initial program 31.2

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.076228701950242:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} + \frac{\frac{-1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9496967329184121:\\ \;\;\;\;\left(x + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}\right) + {x}^{5} \cdot \frac{3}{40}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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)))))