Average Error: 53.2 → 0.2
Time: 12.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.9993980925116870972502169934159610420465:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 9.582810861441525300194466119307890039636 \cdot 10^{-4}:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 \cdot \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.9993980925116870972502169934159610420465:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 9.582810861441525300194466119307890039636 \cdot 10^{-4}:\\
\;\;\;\;\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(1 \cdot \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\right)\\

\end{array}
double f(double x) {
        double r150301 = x;
        double r150302 = r150301 * r150301;
        double r150303 = 1.0;
        double r150304 = r150302 + r150303;
        double r150305 = sqrt(r150304);
        double r150306 = r150301 + r150305;
        double r150307 = log(r150306);
        return r150307;
}
↓
double f(double x) {
        double r150308 = x;
        double r150309 = -0.9993980925116871;
        bool r150310 = r150308 <= r150309;
        double r150311 = 0.125;
        double r150312 = 3.0;
        double r150313 = pow(r150308, r150312);
        double r150314 = r150311 / r150313;
        double r150315 = 0.5;
        double r150316 = r150315 / r150308;
        double r150317 = 0.0625;
        double r150318 = -r150317;
        double r150319 = 5.0;
        double r150320 = pow(r150308, r150319);
        double r150321 = r150318 / r150320;
        double r150322 = r150316 - r150321;
        double r150323 = r150314 - r150322;
        double r150324 = log(r150323);
        double r150325 = 0.0009582810861441525;
        bool r150326 = r150308 <= r150325;
        double r150327 = 1.0;
        double r150328 = sqrt(r150327);
        double r150329 = log(r150328);
        double r150330 = r150308 / r150328;
        double r150331 = r150329 + r150330;
        double r150332 = 0.16666666666666666;
        double r150333 = pow(r150328, r150312);
        double r150334 = r150313 / r150333;
        double r150335 = r150332 * r150334;
        double r150336 = r150331 - r150335;
        double r150337 = 1.0;
        double r150338 = hypot(r150308, r150328);
        double r150339 = r150338 + r150308;
        double r150340 = r150337 * r150339;
        double r150341 = log(r150340);
        double r150342 = r150326 ? r150336 : r150341;
        double r150343 = r150310 ? r150324 : r150342;
        return r150343;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.2
Target45.7
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 < -0.9993980925116871

    1. Initial program 62.9

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

    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 -0.9993980925116871 < x < 0.0009582810861441525

    1. Initial program 58.9

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

    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 0.0009582810861441525 < x

    1. Initial program 32.6

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
    2. Using strategy rm

    3. Applied *-un-lft-identity32.6

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

    4. Applied *-un-lft-identity32.6

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

    5. Applied distribute-lft-out32.6

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

    6. Simplified0.1

      \[\leadsto \log \left(1 \cdot \color{blue}{\left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9993980925116870972502169934159610420465:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 9.582810861441525300194466119307890039636 \cdot 10^{-4}:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 \cdot \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(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)))))