Average Error: 53.3 → 0.2
Time: 14.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.004740488829904077050514388247393071651:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8999222701664713053304467393900267779827:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(2, x, \frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.004740488829904077050514388247393071651:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.8999222701664713053304467393900267779827:\\
\;\;\;\;\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(\mathsf{fma}\left(2, x, \frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\\

\end{array}
double f(double x) {
        double r94499 = x;
        double r94500 = r94499 * r94499;
        double r94501 = 1.0;
        double r94502 = r94500 + r94501;
        double r94503 = sqrt(r94502);
        double r94504 = r94499 + r94503;
        double r94505 = log(r94504);
        return r94505;
}
↓
double f(double x) {
        double r94506 = x;
        double r94507 = -1.004740488829904;
        bool r94508 = r94506 <= r94507;
        double r94509 = 0.125;
        double r94510 = 3.0;
        double r94511 = pow(r94506, r94510);
        double r94512 = r94509 / r94511;
        double r94513 = 0.0625;
        double r94514 = 5.0;
        double r94515 = pow(r94506, r94514);
        double r94516 = r94513 / r94515;
        double r94517 = 0.5;
        double r94518 = r94517 / r94506;
        double r94519 = r94516 + r94518;
        double r94520 = r94512 - r94519;
        double r94521 = log(r94520);
        double r94522 = 0.8999222701664713;
        bool r94523 = r94506 <= r94522;
        double r94524 = 1.0;
        double r94525 = sqrt(r94524);
        double r94526 = pow(r94525, r94510);
        double r94527 = r94511 / r94526;
        double r94528 = -0.16666666666666666;
        double r94529 = log(r94525);
        double r94530 = r94506 / r94525;
        double r94531 = r94529 + r94530;
        double r94532 = fma(r94527, r94528, r94531);
        double r94533 = 2.0;
        double r94534 = r94518 - r94512;
        double r94535 = fma(r94533, r94506, r94534);
        double r94536 = log(r94535);
        double r94537 = r94523 ? r94532 : r94536;
        double r94538 = r94508 ? r94521 : r94537;
        return r94538;
}

Error

Bits error versus x

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. Simplified62.5

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

    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 -1.004740488829904 < x < 0.8999222701664713

    1. Initial program 58.8

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

    2. Simplified58.8

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

    3. Taylor expanded around 0 0.2

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

    4. Simplified0.2

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

    if 0.8999222701664713 < x

    1. Initial program 32.8

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

    2. Simplified32.8

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

    3. Taylor expanded around inf 0.2

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

    4. 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 simplification0.2

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

Reproduce

herbie shell --seed 2019325 +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)))))