Average Error: 53.0 → 0.3
Time: 15.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.006460284952229500277098850347101688385:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8931378377170595683764986461028456687927:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(2, x, \frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.006460284952229500277098850347101688385:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.8931378377170595683764986461028456687927:\\
\;\;\;\;\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(\mathsf{fma}\left(2, x, \frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\\

\end{array}
double f(double x) {
        double r99158 = x;
        double r99159 = r99158 * r99158;
        double r99160 = 1.0;
        double r99161 = r99159 + r99160;
        double r99162 = sqrt(r99161);
        double r99163 = r99158 + r99162;
        double r99164 = log(r99163);
        return r99164;
}
↓
double f(double x) {
        double r99165 = x;
        double r99166 = -1.0064602849522295;
        bool r99167 = r99165 <= r99166;
        double r99168 = 0.125;
        double r99169 = 3.0;
        double r99170 = pow(r99165, r99169);
        double r99171 = r99168 / r99170;
        double r99172 = 0.0625;
        double r99173 = 5.0;
        double r99174 = pow(r99165, r99173);
        double r99175 = r99172 / r99174;
        double r99176 = 0.5;
        double r99177 = r99176 / r99165;
        double r99178 = r99175 + r99177;
        double r99179 = r99171 - r99178;
        double r99180 = log(r99179);
        double r99181 = 0.8931378377170596;
        bool r99182 = r99165 <= r99181;
        double r99183 = 1.0;
        double r99184 = sqrt(r99183);
        double r99185 = pow(r99184, r99169);
        double r99186 = r99170 / r99185;
        double r99187 = -0.16666666666666666;
        double r99188 = log(r99184);
        double r99189 = r99165 / r99184;
        double r99190 = r99188 + r99189;
        double r99191 = fma(r99186, r99187, r99190);
        double r99192 = 2.0;
        double r99193 = r99177 - r99171;
        double r99194 = fma(r99192, r99165, r99193);
        double r99195 = log(r99194);
        double r99196 = r99182 ? r99191 : r99195;
        double r99197 = r99167 ? r99180 : r99196;
        return r99197;
}

Error

Bits error versus x

Target

Original53.0
Target45.0
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0064602849522295

    1. Initial program 62.9

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

    2. Simplified62.9

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

    3. Taylor expanded around -inf 0.1

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

    4. Simplified0.1

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

    if -1.0064602849522295 < x < 0.8931378377170596

    1. Initial program 58.7

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

    2. Simplified58.7

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

    3. Taylor expanded around 0 0.4

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

    4. Simplified0.4

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

    if 0.8931378377170596 < x

    1. Initial program 31.9

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

    2. Simplified31.9

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

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.006460284952229500277098850347101688385:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8931378377170595683764986461028456687927:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(2, x, \frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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)))))