Average Error: 53.0 → 0.3
Time: 13.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.03811431304857993:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.90048843936555456:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{6}, \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.03811431304857993:\\
\;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 0.90048843936555456:\\
\;\;\;\;\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 r198477 = x;
        double r198478 = r198477 * r198477;
        double r198479 = 1.0;
        double r198480 = r198478 + r198479;
        double r198481 = sqrt(r198480);
        double r198482 = r198477 + r198481;
        double r198483 = log(r198482);
        return r198483;
}
↓
double f(double x) {
        double r198484 = x;
        double r198485 = -1.03811431304858;
        bool r198486 = r198484 <= r198485;
        double r198487 = 0.125;
        double r198488 = 3.0;
        double r198489 = pow(r198484, r198488);
        double r198490 = r198487 / r198489;
        double r198491 = 0.5;
        double r198492 = r198491 / r198484;
        double r198493 = r198490 - r198492;
        double r198494 = 0.0625;
        double r198495 = 5.0;
        double r198496 = pow(r198484, r198495);
        double r198497 = r198494 / r198496;
        double r198498 = r198493 - r198497;
        double r198499 = log(r198498);
        double r198500 = 0.9004884393655546;
        bool r198501 = r198484 <= r198500;
        double r198502 = -0.16666666666666666;
        double r198503 = 1.0;
        double r198504 = sqrt(r198503);
        double r198505 = pow(r198504, r198488);
        double r198506 = r198489 / r198505;
        double r198507 = log(r198504);
        double r198508 = r198484 / r198504;
        double r198509 = r198507 + r198508;
        double r198510 = fma(r198502, r198506, r198509);
        double r198511 = 2.0;
        double r198512 = r198492 - r198490;
        double r198513 = fma(r198484, r198511, r198512);
        double r198514 = log(r198513);
        double r198515 = r198501 ? r198510 : r198514;
        double r198516 = r198486 ? r198499 : r198515;
        return r198516;
}

Error

Bits error versus x

Target

Original53.0
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 < -1.03811431304858

    1. Initial program 62.7

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

    2. Simplified62.7

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

    3. Taylor expanded around -inf 0.3

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

    4. Simplified0.3

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

    if -1.03811431304858 < x < 0.9004884393655546

    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 0.3

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

    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 0.9004884393655546 < x

    1. Initial program 32.6

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

    2. Simplified32.6

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

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.03811431304857993:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.90048843936555456:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{6}, \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))))