Average Error: 52.8 → 0.2
Time: 20.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.002275654089841649962977498944383114576:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.8884086436842496548038639048172626644373:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{{x}^{3}}{1}}{\sqrt{1}}, \frac{-1}{6}, \frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.002275654089841649962977498944383114576:\\
\;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 0.8884086436842496548038639048172626644373:\\
\;\;\;\;\mathsf{fma}\left(\frac{\frac{{x}^{3}}{1}}{\sqrt{1}}, \frac{-1}{6}, \frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\\

\end{array}
double f(double x) {
        double r189971 = x;
        double r189972 = r189971 * r189971;
        double r189973 = 1.0;
        double r189974 = r189972 + r189973;
        double r189975 = sqrt(r189974);
        double r189976 = r189971 + r189975;
        double r189977 = log(r189976);
        return r189977;
}
↓
double f(double x) {
        double r189978 = x;
        double r189979 = -1.0022756540898416;
        bool r189980 = r189978 <= r189979;
        double r189981 = 0.125;
        double r189982 = 3.0;
        double r189983 = pow(r189978, r189982);
        double r189984 = r189981 / r189983;
        double r189985 = 0.5;
        double r189986 = r189985 / r189978;
        double r189987 = r189984 - r189986;
        double r189988 = 0.0625;
        double r189989 = 5.0;
        double r189990 = pow(r189978, r189989);
        double r189991 = r189988 / r189990;
        double r189992 = r189987 - r189991;
        double r189993 = log(r189992);
        double r189994 = 0.8884086436842497;
        bool r189995 = r189978 <= r189994;
        double r189996 = 1.0;
        double r189997 = r189983 / r189996;
        double r189998 = sqrt(r189996);
        double r189999 = r189997 / r189998;
        double r190000 = -0.16666666666666666;
        double r190001 = r189978 / r189998;
        double r190002 = log(r189998);
        double r190003 = r190001 + r190002;
        double r190004 = fma(r189999, r190000, r190003);
        double r190005 = 2.0;
        double r190006 = fma(r189978, r190005, r189986);
        double r190007 = r190006 - r189984;
        double r190008 = log(r190007);
        double r190009 = r189995 ? r190004 : r190008;
        double r190010 = r189980 ? r189993 : r190009;
        return r190010;
}

Error

Bits error versus x

Target

Original52.8
Target45.0
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0022756540898416

    1. Initial program 63.1

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

    2. Simplified63.1

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

    3. Taylor expanded around -inf 0.1

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

    4. Simplified0.1

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

    if -1.0022756540898416 < x < 0.8884086436842497

    1. Initial program 58.7

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

    2. Simplified58.7

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

    3. Taylor expanded around 0 0.3

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

    4. Simplified0.3

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

    if 0.8884086436842497 < x

    1. Initial program 31.4

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

    2. Simplified31.4

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

    3. Taylor expanded around inf 0.3

      \[\leadsto \log \color{blue}{\left(\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)}\]

    4. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.002275654089841649962977498944383114576:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.8884086436842496548038639048172626644373:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{{x}^{3}}{1}}{\sqrt{1}}, \frac{-1}{6}, \frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +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)))))