Average Error: 52.9 → 0.2
Time: 18.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0632494934709653:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.9505314742632508:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{6}, \left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{3}{40}, {x}^{5}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \mathsf{fma}\left(2, x, \frac{\frac{1}{2}}{x}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0632494934709653:\\
\;\;\;\;\log \left(\left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 0.9505314742632508:\\
\;\;\;\;\mathsf{fma}\left(\frac{-1}{6}, \left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{3}{40}, {x}^{5}, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \mathsf{fma}\left(2, x, \frac{\frac{1}{2}}{x}\right)\right)\\

\end{array}
double f(double x) {
        double r2041082 = x;
        double r2041083 = r2041082 * r2041082;
        double r2041084 = 1.0;
        double r2041085 = r2041083 + r2041084;
        double r2041086 = sqrt(r2041085);
        double r2041087 = r2041082 + r2041086;
        double r2041088 = log(r2041087);
        return r2041088;
}
↓
double f(double x) {
        double r2041089 = x;
        double r2041090 = -1.0632494934709653;
        bool r2041091 = r2041089 <= r2041090;
        double r2041092 = 0.125;
        double r2041093 = r2041089 * r2041089;
        double r2041094 = r2041092 / r2041093;
        double r2041095 = r2041094 / r2041089;
        double r2041096 = 0.5;
        double r2041097 = r2041096 / r2041089;
        double r2041098 = r2041095 - r2041097;
        double r2041099 = 0.0625;
        double r2041100 = 5.0;
        double r2041101 = pow(r2041089, r2041100);
        double r2041102 = r2041099 / r2041101;
        double r2041103 = r2041098 - r2041102;
        double r2041104 = log(r2041103);
        double r2041105 = 0.9505314742632508;
        bool r2041106 = r2041089 <= r2041105;
        double r2041107 = -0.16666666666666666;
        double r2041108 = r2041093 * r2041089;
        double r2041109 = 0.075;
        double r2041110 = fma(r2041109, r2041101, r2041089);
        double r2041111 = fma(r2041107, r2041108, r2041110);
        double r2041112 = -0.125;
        double r2041113 = r2041112 / r2041108;
        double r2041114 = 2.0;
        double r2041115 = fma(r2041114, r2041089, r2041097);
        double r2041116 = r2041113 + r2041115;
        double r2041117 = log(r2041116);
        double r2041118 = r2041106 ? r2041111 : r2041117;
        double r2041119 = r2041091 ? r2041104 : r2041118;
        return r2041119;
}

Error

Bits error versus x

Target

Original52.9
Target44.8
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0632494934709653

    1. Initial program 61.7

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

    2. Simplified60.9

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

    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

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

    if -1.0632494934709653 < x < 0.9505314742632508

    1. Initial program 58.7

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

    2. Simplified58.7

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

    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

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

    if 0.9505314742632508 < x

    1. Initial program 32.2

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

    2. Simplified0.0

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

    3. Taylor expanded around inf 0.3

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

    4. Simplified0.3

      \[\leadsto \log \color{blue}{\left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \mathsf{fma}\left(2, x, \frac{\frac{1}{2}}{x}\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0632494934709653:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.9505314742632508:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{6}, \left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{3}{40}, {x}^{5}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \mathsf{fma}\left(2, x, \frac{\frac{1}{2}}{x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 +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)))))