Average Error: 53.3 → 0.2
Time: 13.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.004740488829904077050514388247393071651:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \le 0.8999222701664713053304467393900267779827:\\ \;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{-1}{6} \cdot \frac{x \cdot x}{1} + 1\right) \cdot \frac{x}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + 2 \cdot x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.004740488829904077050514388247393071651:\\
\;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\

\mathbf{elif}\;x \le 0.8999222701664713053304467393900267779827:\\
\;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{-1}{6} \cdot \frac{x \cdot x}{1} + 1\right) \cdot \frac{x}{\sqrt{1}}\\

\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 r145435 = x;
        double r145436 = r145435 * r145435;
        double r145437 = 1.0;
        double r145438 = r145436 + r145437;
        double r145439 = sqrt(r145438);
        double r145440 = r145435 + r145439;
        double r145441 = log(r145440);
        return r145441;
}
↓
double f(double x) {
        double r145442 = x;
        double r145443 = -1.004740488829904;
        bool r145444 = r145442 <= r145443;
        double r145445 = 0.125;
        double r145446 = 3.0;
        double r145447 = pow(r145442, r145446);
        double r145448 = r145445 / r145447;
        double r145449 = 0.0625;
        double r145450 = 5.0;
        double r145451 = pow(r145442, r145450);
        double r145452 = r145449 / r145451;
        double r145453 = r145448 - r145452;
        double r145454 = 0.5;
        double r145455 = r145454 / r145442;
        double r145456 = r145453 - r145455;
        double r145457 = log(r145456);
        double r145458 = 0.8999222701664713;
        bool r145459 = r145442 <= r145458;
        double r145460 = 1.0;
        double r145461 = sqrt(r145460);
        double r145462 = log(r145461);
        double r145463 = -0.16666666666666666;
        double r145464 = r145442 * r145442;
        double r145465 = r145464 / r145460;
        double r145466 = r145463 * r145465;
        double r145467 = 1.0;
        double r145468 = r145466 + r145467;
        double r145469 = r145442 / r145461;
        double r145470 = r145468 * r145469;
        double r145471 = r145462 + r145470;
        double r145472 = r145455 - r145448;
        double r145473 = 2.0;
        double r145474 = r145473 * r145442;
        double r145475 = r145472 + r145474;
        double r145476 = log(r145475);
        double r145477 = r145459 ? r145471 : r145476;
        double r145478 = r145444 ? r145457 : r145477;
        return r145478;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.3
Target45.6
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 < -1.004740488829904

    1. Initial program 62.5

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

    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

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

    if -1.004740488829904 < x < 0.8999222701664713

    1. Initial program 58.8

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

    if 0.8999222701664713 < x

    1. Initial program 32.8

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.004740488829904077050514388247393071651:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \le 0.8999222701664713053304467393900267779827:\\ \;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{-1}{6} \cdot \frac{x \cdot x}{1} + 1\right) \cdot \frac{x}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + 2 \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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)))))