Average Error: 53.2 → 0.1
Time: 12.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.016091287793053909993545858014840632677:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 9.192273826232662953250596160614804830402 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.016091287793053909993545858014840632677:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 9.192273826232662953250596160614804830402 \cdot 10^{-4}:\\
\;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\

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

\end{array}
double f(double x) {
        double r151991 = x;
        double r151992 = r151991 * r151991;
        double r151993 = 1.0;
        double r151994 = r151992 + r151993;
        double r151995 = sqrt(r151994);
        double r151996 = r151991 + r151995;
        double r151997 = log(r151996);
        return r151997;
}
↓
double f(double x) {
        double r151998 = x;
        double r151999 = -1.016091287793054;
        bool r152000 = r151998 <= r151999;
        double r152001 = 0.125;
        double r152002 = 3.0;
        double r152003 = pow(r151998, r152002);
        double r152004 = r152001 / r152003;
        double r152005 = 0.5;
        double r152006 = r152005 / r151998;
        double r152007 = 0.0625;
        double r152008 = 5.0;
        double r152009 = pow(r151998, r152008);
        double r152010 = r152007 / r152009;
        double r152011 = r152006 + r152010;
        double r152012 = r152004 - r152011;
        double r152013 = log(r152012);
        double r152014 = 0.0009192273826232663;
        bool r152015 = r151998 <= r152014;
        double r152016 = 1.0;
        double r152017 = sqrt(r152016);
        double r152018 = pow(r152017, r152002);
        double r152019 = r152003 / r152018;
        double r152020 = -0.16666666666666666;
        double r152021 = log(r152017);
        double r152022 = r151998 / r152017;
        double r152023 = r152021 + r152022;
        double r152024 = fma(r152019, r152020, r152023);
        double r152025 = hypot(r151998, r152017);
        double r152026 = r151998 + r152025;
        double r152027 = log(r152026);
        double r152028 = r152015 ? r152024 : r152027;
        double r152029 = r152000 ? r152013 : r152028;
        return r152029;
}

Error

Bits error versus x

Target

Original53.2
Target45.4
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.016091287793054

    1. Initial program 63.0

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

    2. Taylor expanded around -inf 0.2

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

    3. Simplified0.2

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

    if -1.016091287793054 < x < 0.0009192273826232663

    1. Initial program 58.8

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    if 0.0009192273826232663 < x

    1. Initial program 32.3

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt32.3

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

    4. Applied hypot-def0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.016091287793053909993545858014840632677:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 9.192273826232662953250596160614804830402 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +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)))))