Average Error: 52.4 → 0.2
Time: 16.0s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0467443314953524:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{elif}\;x \le 0.9512405390439127:\\ \;\;\;\;\left(x - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{6}\right) + \frac{3}{40} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0467443314953524:\\
\;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\

\mathbf{elif}\;x \le 0.9512405390439127:\\
\;\;\;\;\left(x - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{6}\right) + \frac{3}{40} \cdot {x}^{5}\\

\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 r4596482 = x;
        double r4596483 = r4596482 * r4596482;
        double r4596484 = 1.0;
        double r4596485 = r4596483 + r4596484;
        double r4596486 = sqrt(r4596485);
        double r4596487 = r4596482 + r4596486;
        double r4596488 = log(r4596487);
        return r4596488;
}
↓
double f(double x) {
        double r4596489 = x;
        double r4596490 = -1.0467443314953524;
        bool r4596491 = r4596489 <= r4596490;
        double r4596492 = -0.5;
        double r4596493 = r4596492 / r4596489;
        double r4596494 = 0.0625;
        double r4596495 = 5.0;
        double r4596496 = pow(r4596489, r4596495);
        double r4596497 = r4596494 / r4596496;
        double r4596498 = 0.125;
        double r4596499 = r4596489 * r4596489;
        double r4596500 = r4596489 * r4596499;
        double r4596501 = r4596498 / r4596500;
        double r4596502 = r4596497 - r4596501;
        double r4596503 = r4596493 - r4596502;
        double r4596504 = log(r4596503);
        double r4596505 = 0.9512405390439127;
        bool r4596506 = r4596489 <= r4596505;
        double r4596507 = 0.16666666666666666;
        double r4596508 = r4596500 * r4596507;
        double r4596509 = r4596489 - r4596508;
        double r4596510 = 0.075;
        double r4596511 = r4596510 * r4596496;
        double r4596512 = r4596509 + r4596511;
        double r4596513 = 0.5;
        double r4596514 = r4596513 / r4596489;
        double r4596515 = r4596514 - r4596501;
        double r4596516 = r4596489 + r4596515;
        double r4596517 = r4596516 + r4596489;
        double r4596518 = log(r4596517);
        double r4596519 = r4596506 ? r4596512 : r4596518;
        double r4596520 = r4596491 ? r4596504 : r4596519;
        return r4596520;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.4
Target44.7
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 < -1.0467443314953524

    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

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

    if -1.0467443314953524 < x < 0.9512405390439127

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 0.9512405390439127 < x

    1. Initial program 30.7

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0467443314953524:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{elif}\;x \le 0.9512405390439127:\\ \;\;\;\;\left(x - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{6}\right) + \frac{3}{40} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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)))))