Average Error: 53.1 → 0.3
Time: 15.3s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.004281043358186797931352884916123002768:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8904752561288424850260980747407302260399:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + x\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.004281043358186797931352884916123002768:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.8904752561288424850260980747407302260399:\\
\;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + x\right) + x\right)\\

\end{array}
double f(double x) {
        double r166410 = x;
        double r166411 = r166410 * r166410;
        double r166412 = 1.0;
        double r166413 = r166411 + r166412;
        double r166414 = sqrt(r166413);
        double r166415 = r166410 + r166414;
        double r166416 = log(r166415);
        return r166416;
}
↓
double f(double x) {
        double r166417 = x;
        double r166418 = -1.0042810433581868;
        bool r166419 = r166417 <= r166418;
        double r166420 = 0.125;
        double r166421 = 3.0;
        double r166422 = pow(r166417, r166421);
        double r166423 = r166420 / r166422;
        double r166424 = 0.0625;
        double r166425 = 5.0;
        double r166426 = pow(r166417, r166425);
        double r166427 = r166424 / r166426;
        double r166428 = 0.5;
        double r166429 = r166428 / r166417;
        double r166430 = r166427 + r166429;
        double r166431 = r166423 - r166430;
        double r166432 = log(r166431);
        double r166433 = 0.8904752561288425;
        bool r166434 = r166417 <= r166433;
        double r166435 = 1.0;
        double r166436 = sqrt(r166435);
        double r166437 = pow(r166436, r166421);
        double r166438 = r166422 / r166437;
        double r166439 = -0.16666666666666666;
        double r166440 = log(r166436);
        double r166441 = r166417 / r166436;
        double r166442 = r166440 + r166441;
        double r166443 = fma(r166438, r166439, r166442);
        double r166444 = r166429 - r166423;
        double r166445 = r166444 + r166417;
        double r166446 = r166445 + r166417;
        double r166447 = log(r166446);
        double r166448 = r166434 ? r166443 : r166447;
        double r166449 = r166419 ? r166432 : r166448;
        return r166449;
}

Error

Bits error versus x

Target

Original53.1
Target45.4
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 < -1.0042810433581868

    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

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

    if -1.0042810433581868 < x < 0.8904752561288425

    1. Initial program 58.6

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

    2. Simplified58.6

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

    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

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

    if 0.8904752561288425 < x

    1. Initial program 32.2

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

    2. Simplified32.2

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

    3. Taylor expanded around inf 0.2

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

    4. Simplified0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.004281043358186797931352884916123002768:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8904752561288424850260980747407302260399:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + x\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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)))))