Average Error: 53.2 → 0.2
Time: 16.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.026591513399979893605973302328493446112:\\ \;\;\;\;\log \left(\frac{0.125}{\left(x \cdot x\right) \cdot x} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8907247814494133608675952018529642373323:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{\frac{1}{6}}{1} \cdot \frac{\left(x \cdot x\right) \cdot x}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{\left(x \cdot x\right) \cdot x}\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.026591513399979893605973302328493446112:\\
\;\;\;\;\log \left(\frac{0.125}{\left(x \cdot x\right) \cdot x} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.8907247814494133608675952018529642373323:\\
\;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{\frac{1}{6}}{1} \cdot \frac{\left(x \cdot x\right) \cdot x}{\sqrt{1}}\\

\mathbf{else}:\\
\;\;\;\;\log \left(\left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{\left(x \cdot x\right) \cdot x}\right) + x\right)\\

\end{array}
double f(double x) {
        double r8289152 = x;
        double r8289153 = r8289152 * r8289152;
        double r8289154 = 1.0;
        double r8289155 = r8289153 + r8289154;
        double r8289156 = sqrt(r8289155);
        double r8289157 = r8289152 + r8289156;
        double r8289158 = log(r8289157);
        return r8289158;
}
↓
double f(double x) {
        double r8289159 = x;
        double r8289160 = -1.02659151339998;
        bool r8289161 = r8289159 <= r8289160;
        double r8289162 = 0.125;
        double r8289163 = r8289159 * r8289159;
        double r8289164 = r8289163 * r8289159;
        double r8289165 = r8289162 / r8289164;
        double r8289166 = 0.0625;
        double r8289167 = 5.0;
        double r8289168 = pow(r8289159, r8289167);
        double r8289169 = r8289166 / r8289168;
        double r8289170 = 0.5;
        double r8289171 = r8289170 / r8289159;
        double r8289172 = r8289169 + r8289171;
        double r8289173 = r8289165 - r8289172;
        double r8289174 = log(r8289173);
        double r8289175 = 0.8907247814494134;
        bool r8289176 = r8289159 <= r8289175;
        double r8289177 = 1.0;
        double r8289178 = sqrt(r8289177);
        double r8289179 = log(r8289178);
        double r8289180 = r8289159 / r8289178;
        double r8289181 = r8289179 + r8289180;
        double r8289182 = 0.16666666666666666;
        double r8289183 = r8289182 / r8289177;
        double r8289184 = r8289164 / r8289178;
        double r8289185 = r8289183 * r8289184;
        double r8289186 = r8289181 - r8289185;
        double r8289187 = r8289159 + r8289171;
        double r8289188 = r8289187 - r8289165;
        double r8289189 = r8289188 + r8289159;
        double r8289190 = log(r8289189);
        double r8289191 = r8289176 ? r8289186 : r8289190;
        double r8289192 = r8289161 ? r8289174 : r8289191;
        return r8289192;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.2
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 < -1.02659151339998

    1. Initial program 63.0

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

    2. Taylor expanded around -inf 0.2

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

    3. Simplified0.2

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

    if -1.02659151339998 < x < 0.8907247814494134

    1. Initial program 58.6

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 0.8907247814494134 < x

    1. Initial program 32.1

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

      \[\leadsto \log \left(x + \color{blue}{\left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{x \cdot \left(x \cdot x\right)}\right)}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.026591513399979893605973302328493446112:\\ \;\;\;\;\log \left(\frac{0.125}{\left(x \cdot x\right) \cdot x} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8907247814494133608675952018529642373323:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{\frac{1}{6}}{1} \cdot \frac{\left(x \cdot x\right) \cdot x}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{\left(x \cdot x\right) \cdot x}\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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)))))