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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0593607199305546:\\ \;\;\;\;\log \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(\frac{\frac{1}{2}}{x} + \frac{\frac{1}{16}}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.004359746085169371:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot \frac{-1}{6}\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\frac{3}{40}, \left({x}^{5}\right), x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{\mathsf{hypot}\left(1, x\right) + x}\right) + \log \left(\sqrt{\mathsf{hypot}\left(1, x\right) + x}\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.0593607199305546:\\
\;\;\;\;\log \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(\frac{\frac{1}{2}}{x} + \frac{\frac{1}{16}}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.004359746085169371:\\
\;\;\;\;\mathsf{fma}\left(\left(x \cdot \frac{-1}{6}\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\frac{3}{40}, \left({x}^{5}\right), x\right)\right)\right)\\

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

\end{array}

double f(double x) {
        double r27031884 = x;
        double r27031885 = r27031884 * r27031884;
        double r27031886 = 1.0;
        double r27031887 = r27031885 + r27031886;
        double r27031888 = sqrt(r27031887);
        double r27031889 = r27031884 + r27031888;
        double r27031890 = log(r27031889);
        return r27031890;
}

↓

double f(double x) {
        double r27031891 = x;
        double r27031892 = -1.0593607199305546;
        bool r27031893 = r27031891 <= r27031892;
        double r27031894 = 0.125;
        double r27031895 = r27031894 / r27031891;
        double r27031896 = r27031891 * r27031891;
        double r27031897 = r27031895 / r27031896;
        double r27031898 = 0.5;
        double r27031899 = r27031898 / r27031891;
        double r27031900 = 0.0625;
        double r27031901 = 5.0;
        double r27031902 = pow(r27031891, r27031901);
        double r27031903 = r27031900 / r27031902;
        double r27031904 = r27031899 + r27031903;
        double r27031905 = r27031897 - r27031904;
        double r27031906 = log(r27031905);
        double r27031907 = 0.004359746085169371;
        bool r27031908 = r27031891 <= r27031907;
        double r27031909 = -0.16666666666666666;
        double r27031910 = r27031891 * r27031909;
        double r27031911 = 0.075;
        double r27031912 = fma(r27031911, r27031902, r27031891);
        double r27031913 = fma(r27031910, r27031896, r27031912);
        double r27031914 = 1.0;
        double r27031915 = hypot(r27031914, r27031891);
        double r27031916 = r27031915 + r27031891;
        double r27031917 = sqrt(r27031916);
        double r27031918 = log(r27031917);
        double r27031919 = r27031918 + r27031918;
        double r27031920 = r27031908 ? r27031913 : r27031919;
        double r27031921 = r27031893 ? r27031906 : r27031920;
        return r27031921;
}

Original	52.0
Target	44.4
Herbie	0.2

Derivation

Split input into 3 regimes
if x < -1.0593607199305546
1. Initial program 61.6
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Simplified60.8
  \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}\]
3. Taylor expanded around -inf 0.3
  \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]
4. Simplified0.3
  \[\leadsto \log \color{blue}{\left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(\frac{\frac{1}{16}}{{x}^{5}} + \frac{\frac{1}{2}}{x}\right)\right)}\]
if -1.0593607199305546 < x < 0.004359746085169371
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 0.1
  \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]
4. Simplified0.1
  \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot \frac{-1}{6}\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\frac{3}{40}, \left({x}^{5}\right), x\right)\right)\right)}\]
if 0.004359746085169371 < x
1. Initial program 30.3
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Simplified0.1
  \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}\]
3. Using strategy rm
4. Applied add-sqr-sqrt0.1
  \[\leadsto \log \color{blue}{\left(\sqrt{x + \mathsf{hypot}\left(1, x\right)} \cdot \sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)}\]
5. Applied log-prod0.1
  \[\leadsto \color{blue}{\log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right) + \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)}\]
Recombined 3 regimes into one program.
Final simplification0.2
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0593607199305546:\\ \;\;\;\;\log \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(\frac{\frac{1}{2}}{x} + \frac{\frac{1}{16}}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.004359746085169371:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot \frac{-1}{6}\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\frac{3}{40}, \left({x}^{5}\right), x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{\mathsf{hypot}\left(1, x\right) + x}\right) + \log \left(\sqrt{\mathsf{hypot}\left(1, x\right) + x}\right)\\ \end{array}\]

Error

Target

Derivation

`if x < -1.0593607199305546`

`if -1.0593607199305546 < x < 0.004359746085169371`

`if 0.004359746085169371 < x`

Reproduce