Average Error: 53.0 → 0.1
Time: 17.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.03908666992687837:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 8.62376812508959946 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.03908666992687837:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 8.62376812508959946 \cdot 10^{-4}:\\
\;\;\;\;\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(x + \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\

\end{array}
double f(double x) {
        double r123550 = x;
        double r123551 = r123550 * r123550;
        double r123552 = 1.0;
        double r123553 = r123551 + r123552;
        double r123554 = sqrt(r123553);
        double r123555 = r123550 + r123554;
        double r123556 = log(r123555);
        return r123556;
}
↓
double f(double x) {
        double r123557 = x;
        double r123558 = -1.0390866699268784;
        bool r123559 = r123557 <= r123558;
        double r123560 = 0.125;
        double r123561 = 3.0;
        double r123562 = pow(r123557, r123561);
        double r123563 = r123560 / r123562;
        double r123564 = 0.5;
        double r123565 = r123564 / r123557;
        double r123566 = 0.0625;
        double r123567 = 5.0;
        double r123568 = pow(r123557, r123567);
        double r123569 = r123566 / r123568;
        double r123570 = r123565 + r123569;
        double r123571 = r123563 - r123570;
        double r123572 = log(r123571);
        double r123573 = 0.0008623768125089599;
        bool r123574 = r123557 <= r123573;
        double r123575 = 1.0;
        double r123576 = sqrt(r123575);
        double r123577 = pow(r123576, r123561);
        double r123578 = r123562 / r123577;
        double r123579 = -0.16666666666666666;
        double r123580 = log(r123576);
        double r123581 = r123557 / r123576;
        double r123582 = r123580 + r123581;
        double r123583 = fma(r123578, r123579, r123582);
        double r123584 = hypot(r123557, r123576);
        double r123585 = r123557 + r123584;
        double r123586 = log(r123585);
        double r123587 = r123574 ? r123583 : r123586;
        double r123588 = r123559 ? r123572 : r123587;
        return r123588;
}

Error

Bits error versus x

Target

Original53.0
Target45.5
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0390866699268784

    1. Initial program 62.8

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

    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 -1.0390866699268784 < x < 0.0008623768125089599

    1. Initial program 59.2

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    if 0.0008623768125089599 < x

    1. Initial program 31.7

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

    3. Applied add-sqr-sqrt31.7

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

    4. Applied hypot-def0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.03908666992687837:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 8.62376812508959946 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +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)))))