Average Error: 52.8 → 0.3
Time: 13.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0079740505275243:\\ \;\;\;\;\log \left(\sqrt{\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}}\right) + \log \left(\sqrt{\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}}\right)\\ \mathbf{elif}\;x \le 0.89474849272833001:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{6}, \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0079740505275243:\\
\;\;\;\;\log \left(\sqrt{\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}}\right) + \log \left(\sqrt{\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}}\right)\\

\mathbf{elif}\;x \le 0.89474849272833001:\\
\;\;\;\;\mathsf{fma}\left(\frac{-1}{6}, \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\

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

\end{array}
double f(double x) {
        double r217360 = x;
        double r217361 = r217360 * r217360;
        double r217362 = 1.0;
        double r217363 = r217361 + r217362;
        double r217364 = sqrt(r217363);
        double r217365 = r217360 + r217364;
        double r217366 = log(r217365);
        return r217366;
}
↓
double f(double x) {
        double r217367 = x;
        double r217368 = -1.0079740505275243;
        bool r217369 = r217367 <= r217368;
        double r217370 = 0.125;
        double r217371 = 3.0;
        double r217372 = pow(r217367, r217371);
        double r217373 = r217370 / r217372;
        double r217374 = 0.5;
        double r217375 = r217374 / r217367;
        double r217376 = r217373 - r217375;
        double r217377 = 0.0625;
        double r217378 = 5.0;
        double r217379 = pow(r217367, r217378);
        double r217380 = r217377 / r217379;
        double r217381 = r217376 - r217380;
        double r217382 = sqrt(r217381);
        double r217383 = log(r217382);
        double r217384 = r217383 + r217383;
        double r217385 = 0.89474849272833;
        bool r217386 = r217367 <= r217385;
        double r217387 = -0.16666666666666666;
        double r217388 = 1.0;
        double r217389 = sqrt(r217388);
        double r217390 = pow(r217389, r217371);
        double r217391 = r217372 / r217390;
        double r217392 = log(r217389);
        double r217393 = r217367 / r217389;
        double r217394 = r217392 + r217393;
        double r217395 = fma(r217387, r217391, r217394);
        double r217396 = 2.0;
        double r217397 = r217375 - r217373;
        double r217398 = fma(r217367, r217396, r217397);
        double r217399 = log(r217398);
        double r217400 = r217386 ? r217395 : r217399;
        double r217401 = r217369 ? r217384 : r217400;
        return r217401;
}

Error

Bits error versus x

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. Simplified62.8

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

    3. Taylor expanded around -inf 0.2

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

    4. Simplified0.2

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

    6. Applied add-sqr-sqrt0.2

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

    7. Applied log-prod0.2

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

    if -1.0079740505275243 < x < 0.89474849272833

    1. Initial program 58.5

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

    2. Simplified58.5

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

    3. Taylor expanded around 0 0.3

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

    4. Simplified0.3

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

    if 0.89474849272833 < x

    1. Initial program 31.1

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

    2. Simplified31.1

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

    3. Taylor expanded around inf 0.3

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

    4. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2020045 +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)))))