Average Error: 53.1 → 0.3
Time: 10.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.99895493447090078:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.90017583502977816:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + x\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.99895493447090078:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.90017583502977816:\\
\;\;\;\;\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(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + x\right)\right)\\

\end{array}
double f(double x) {
        double r151506 = x;
        double r151507 = r151506 * r151506;
        double r151508 = 1.0;
        double r151509 = r151507 + r151508;
        double r151510 = sqrt(r151509);
        double r151511 = r151506 + r151510;
        double r151512 = log(r151511);
        return r151512;
}
↓
double f(double x) {
        double r151513 = x;
        double r151514 = -0.9989549344709008;
        bool r151515 = r151513 <= r151514;
        double r151516 = 0.125;
        double r151517 = 3.0;
        double r151518 = pow(r151513, r151517);
        double r151519 = r151516 / r151518;
        double r151520 = 0.5;
        double r151521 = r151520 / r151513;
        double r151522 = 0.0625;
        double r151523 = 5.0;
        double r151524 = pow(r151513, r151523);
        double r151525 = r151522 / r151524;
        double r151526 = r151521 + r151525;
        double r151527 = r151519 - r151526;
        double r151528 = log(r151527);
        double r151529 = 0.9001758350297782;
        bool r151530 = r151513 <= r151529;
        double r151531 = 1.0;
        double r151532 = sqrt(r151531);
        double r151533 = log(r151532);
        double r151534 = r151513 / r151532;
        double r151535 = r151533 + r151534;
        double r151536 = 0.16666666666666666;
        double r151537 = pow(r151532, r151517);
        double r151538 = r151518 / r151537;
        double r151539 = r151536 * r151538;
        double r151540 = r151535 - r151539;
        double r151541 = r151521 - r151519;
        double r151542 = r151541 + r151513;
        double r151543 = r151513 + r151542;
        double r151544 = log(r151543);
        double r151545 = r151530 ? r151540 : r151544;
        double r151546 = r151515 ? r151528 : r151545;
        return r151546;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.1
Target45.5
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.9989549344709008

    1. Initial program 62.8

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

    2. Taylor expanded around -inf 0.3

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

    3. Simplified0.3

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

    if -0.9989549344709008 < x < 0.9001758350297782

    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 0.9001758350297782 < x

    1. Initial program 32.1

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.99895493447090078:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.90017583502977816:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))))