Average Error: 53.0 → 0.3
Time: 19.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.021540982075884063107196197961457073689:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.8817659968088067401481566776055842638016:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{1 \cdot \sqrt{1}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(2, x, \frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.021540982075884063107196197961457073689:\\
\;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 0.8817659968088067401481566776055842638016:\\
\;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{1 \cdot \sqrt{1}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\mathsf{fma}\left(2, x, \frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\\

\end{array}
double f(double x) {
        double r138403 = x;
        double r138404 = r138403 * r138403;
        double r138405 = 1.0;
        double r138406 = r138404 + r138405;
        double r138407 = sqrt(r138406);
        double r138408 = r138403 + r138407;
        double r138409 = log(r138408);
        return r138409;
}
↓
double f(double x) {
        double r138410 = x;
        double r138411 = -1.021540982075884;
        bool r138412 = r138410 <= r138411;
        double r138413 = 0.125;
        double r138414 = 3.0;
        double r138415 = pow(r138410, r138414);
        double r138416 = r138413 / r138415;
        double r138417 = 0.5;
        double r138418 = r138417 / r138410;
        double r138419 = r138416 - r138418;
        double r138420 = 0.0625;
        double r138421 = 5.0;
        double r138422 = pow(r138410, r138421);
        double r138423 = r138420 / r138422;
        double r138424 = r138419 - r138423;
        double r138425 = log(r138424);
        double r138426 = 0.8817659968088067;
        bool r138427 = r138410 <= r138426;
        double r138428 = 1.0;
        double r138429 = sqrt(r138428);
        double r138430 = r138428 * r138429;
        double r138431 = r138415 / r138430;
        double r138432 = -0.16666666666666666;
        double r138433 = log(r138429);
        double r138434 = r138410 / r138429;
        double r138435 = r138433 + r138434;
        double r138436 = fma(r138431, r138432, r138435);
        double r138437 = 2.0;
        double r138438 = r138418 - r138416;
        double r138439 = fma(r138437, r138410, r138438);
        double r138440 = log(r138439);
        double r138441 = r138427 ? r138436 : r138440;
        double r138442 = r138412 ? r138425 : r138441;
        return r138442;
}

Error

Bits error versus x

Target

Original53.0
Target45.0
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 < -1.021540982075884

    1. Initial program 63.1

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

    2. Taylor expanded around -inf 0.2

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

    3. Simplified0.2

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

    if -1.021540982075884 < x < 0.8817659968088067

    1. Initial program 58.6

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 0.8817659968088067 < x

    1. Initial program 31.4

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

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.021540982075884063107196197961457073689:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.8817659968088067401481566776055842638016:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{1 \cdot \sqrt{1}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(2, x, \frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))