Average Error: 52.8 → 0.3
Time: 11.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0079740505275243:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.89474849272833001:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + 2 \cdot x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0079740505275243:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.89474849272833001:\\
\;\;\;\;\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(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + 2 \cdot x\right)\\

\end{array}
double f(double x) {
        double r221329 = x;
        double r221330 = r221329 * r221329;
        double r221331 = 1.0;
        double r221332 = r221330 + r221331;
        double r221333 = sqrt(r221332);
        double r221334 = r221329 + r221333;
        double r221335 = log(r221334);
        return r221335;
}
↓
double f(double x) {
        double r221336 = x;
        double r221337 = -1.0079740505275243;
        bool r221338 = r221336 <= r221337;
        double r221339 = 0.125;
        double r221340 = 3.0;
        double r221341 = pow(r221336, r221340);
        double r221342 = r221339 / r221341;
        double r221343 = 0.5;
        double r221344 = r221343 / r221336;
        double r221345 = 0.0625;
        double r221346 = 5.0;
        double r221347 = pow(r221336, r221346);
        double r221348 = r221345 / r221347;
        double r221349 = r221344 + r221348;
        double r221350 = r221342 - r221349;
        double r221351 = log(r221350);
        double r221352 = 0.89474849272833;
        bool r221353 = r221336 <= r221352;
        double r221354 = 1.0;
        double r221355 = sqrt(r221354);
        double r221356 = log(r221355);
        double r221357 = r221336 / r221355;
        double r221358 = r221356 + r221357;
        double r221359 = 0.16666666666666666;
        double r221360 = pow(r221355, r221340);
        double r221361 = r221341 / r221360;
        double r221362 = r221359 * r221361;
        double r221363 = r221358 - r221362;
        double r221364 = r221344 - r221342;
        double r221365 = 2.0;
        double r221366 = r221365 * r221336;
        double r221367 = r221364 + r221366;
        double r221368 = log(r221367);
        double r221369 = r221353 ? r221363 : r221368;
        double r221370 = r221338 ? r221351 : r221369;
        return r221370;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.8
Target44.6
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 < -1.0079740505275243

    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

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

    if -1.0079740505275243 < x < 0.89474849272833

    1. Initial program 58.5

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

    2. Taylor expanded around 0 0.3

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

    if 0.89474849272833 < x

    1. Initial program 31.1

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

    2. Taylor expanded around inf 0.3

      \[\leadsto \log \color{blue}{\left(\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)}\]

    3. Simplified0.3

      \[\leadsto \log \color{blue}{\left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + 2 \cdot x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0079740505275243:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.89474849272833001:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + 2 \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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)))))