Average Error: 52.4 → 0.1
Time: 15.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0467443314953524:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} + \left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{16}}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.00749865384513097:\\ \;\;\;\;\mathsf{fma}\left({x}^{5}, \frac{3}{40}, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{-1}{6}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(1, x\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0467443314953524:\\
\;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} + \left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{16}}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.00749865384513097:\\
\;\;\;\;\mathsf{fma}\left({x}^{5}, \frac{3}{40}, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{-1}{6}, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\mathsf{hypot}\left(1, x\right) + x\right)\\

\end{array}
double f(double x) {
        double r6111304 = x;
        double r6111305 = r6111304 * r6111304;
        double r6111306 = 1.0;
        double r6111307 = r6111305 + r6111306;
        double r6111308 = sqrt(r6111307);
        double r6111309 = r6111304 + r6111308;
        double r6111310 = log(r6111309);
        return r6111310;
}
↓
double f(double x) {
        double r6111311 = x;
        double r6111312 = -1.0467443314953524;
        bool r6111313 = r6111311 <= r6111312;
        double r6111314 = -0.5;
        double r6111315 = r6111314 / r6111311;
        double r6111316 = 0.125;
        double r6111317 = r6111311 * r6111311;
        double r6111318 = r6111316 / r6111317;
        double r6111319 = r6111318 / r6111311;
        double r6111320 = 0.0625;
        double r6111321 = 5.0;
        double r6111322 = pow(r6111311, r6111321);
        double r6111323 = r6111320 / r6111322;
        double r6111324 = r6111319 - r6111323;
        double r6111325 = r6111315 + r6111324;
        double r6111326 = log(r6111325);
        double r6111327 = 0.00749865384513097;
        bool r6111328 = r6111311 <= r6111327;
        double r6111329 = 0.075;
        double r6111330 = r6111311 * r6111317;
        double r6111331 = -0.16666666666666666;
        double r6111332 = fma(r6111330, r6111331, r6111311);
        double r6111333 = fma(r6111322, r6111329, r6111332);
        double r6111334 = 1.0;
        double r6111335 = hypot(r6111334, r6111311);
        double r6111336 = r6111335 + r6111311;
        double r6111337 = log(r6111336);
        double r6111338 = r6111328 ? r6111333 : r6111337;
        double r6111339 = r6111313 ? r6111326 : r6111338;
        return r6111339;
}

Error

Bits error versus x

Target

Original52.4
Target44.7
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0467443314953524

    1. Initial program 61.8

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

    2. Simplified61.0

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

    3. Taylor expanded around -inf 0.1

      \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

    4. Simplified0.1

      \[\leadsto \log \color{blue}{\left(\frac{\frac{-1}{2}}{x} + \left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{16}}{{x}^{5}}\right)\right)}\]

    if -1.0467443314953524 < x < 0.00749865384513097

    1. Initial program 58.9

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

    2. Simplified58.9

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

    3. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]

    4. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{5}, \frac{3}{40}, \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{-1}{6}, x\right)\right)}\]

    if 0.00749865384513097 < x

    1. Initial program 30.5

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

    2. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0467443314953524:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} + \left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{16}}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.00749865384513097:\\ \;\;\;\;\mathsf{fma}\left({x}^{5}, \frac{3}{40}, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{-1}{6}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(1, x\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))