Average Error: 52.7 → 0.1
Time: 20.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0840635159626253:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} + \left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{16}}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.007778482819478155:\\ \;\;\;\;\mathsf{fma}\left({x}^{5}, \frac{3}{40}, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{-1}{6}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(1, x\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0840635159626253:\\
\;\;\;\;\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 0.007778482819478155:\\
\;\;\;\;\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 r6142971 = x;
        double r6142972 = r6142971 * r6142971;
        double r6142973 = 1.0;
        double r6142974 = r6142972 + r6142973;
        double r6142975 = sqrt(r6142974);
        double r6142976 = r6142971 + r6142975;
        double r6142977 = log(r6142976);
        return r6142977;
}
↓
double f(double x) {
        double r6142978 = x;
        double r6142979 = -1.0840635159626253;
        bool r6142980 = r6142978 <= r6142979;
        double r6142981 = -0.5;
        double r6142982 = r6142981 / r6142978;
        double r6142983 = 0.125;
        double r6142984 = r6142978 * r6142978;
        double r6142985 = r6142983 / r6142984;
        double r6142986 = r6142985 / r6142978;
        double r6142987 = 0.0625;
        double r6142988 = 5.0;
        double r6142989 = pow(r6142978, r6142988);
        double r6142990 = r6142987 / r6142989;
        double r6142991 = r6142986 - r6142990;
        double r6142992 = r6142982 + r6142991;
        double r6142993 = log(r6142992);
        double r6142994 = 0.007778482819478155;
        bool r6142995 = r6142978 <= r6142994;
        double r6142996 = 0.075;
        double r6142997 = r6142978 * r6142984;
        double r6142998 = -0.16666666666666666;
        double r6142999 = fma(r6142997, r6142998, r6142978);
        double r6143000 = fma(r6142989, r6142996, r6142999);
        double r6143001 = 1.0;
        double r6143002 = hypot(r6143001, r6142978);
        double r6143003 = r6143002 + r6142978;
        double r6143004 = log(r6143003);
        double r6143005 = r6142995 ? r6143000 : r6143004;
        double r6143006 = r6142980 ? r6142993 : r6143005;
        return r6143006;
}

Error

Bits error versus x

Target

Original52.7
Target44.8
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 < -1.0840635159626253

    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 0.2

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

    4. Simplified0.2

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

    if -1.0840635159626253 < x < 0.007778482819478155

    1. Initial program 58.6

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

    2. Simplified58.6

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

    3. Taylor expanded around 0 0.2

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

    4. Simplified0.2

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

    if 0.007778482819478155 < x

    1. Initial program 32.1

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0840635159626253:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} + \left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{16}}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.007778482819478155:\\ \;\;\;\;\mathsf{fma}\left({x}^{5}, \frac{3}{40}, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{-1}{6}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(1, x\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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)))))