Average Error: 52.6 → 0.2
Time: 15.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0612400279589977:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} + \frac{\frac{-1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9581083600858236:\\ \;\;\;\;\left(x + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}\right) + {x}^{5} \cdot \frac{3}{40}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0612400279589977:\\
\;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} + \frac{\frac{-1}{2}}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.9581083600858236:\\
\;\;\;\;\left(x + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}\right) + {x}^{5} \cdot \frac{3}{40}\\

\mathbf{else}:\\
\;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\

\end{array}
double f(double x) {
        double r3346152 = x;
        double r3346153 = r3346152 * r3346152;
        double r3346154 = 1.0;
        double r3346155 = r3346153 + r3346154;
        double r3346156 = sqrt(r3346155);
        double r3346157 = r3346152 + r3346156;
        double r3346158 = log(r3346157);
        return r3346158;
}
↓
double f(double x) {
        double r3346159 = x;
        double r3346160 = -1.0612400279589977;
        bool r3346161 = r3346159 <= r3346160;
        double r3346162 = -0.0625;
        double r3346163 = 5.0;
        double r3346164 = pow(r3346159, r3346163);
        double r3346165 = r3346162 / r3346164;
        double r3346166 = 0.125;
        double r3346167 = r3346159 * r3346159;
        double r3346168 = r3346159 * r3346167;
        double r3346169 = r3346166 / r3346168;
        double r3346170 = -0.5;
        double r3346171 = r3346170 / r3346159;
        double r3346172 = r3346169 + r3346171;
        double r3346173 = r3346165 + r3346172;
        double r3346174 = log(r3346173);
        double r3346175 = 0.9581083600858236;
        bool r3346176 = r3346159 <= r3346175;
        double r3346177 = -0.16666666666666666;
        double r3346178 = r3346168 * r3346177;
        double r3346179 = r3346159 + r3346178;
        double r3346180 = 0.075;
        double r3346181 = r3346164 * r3346180;
        double r3346182 = r3346179 + r3346181;
        double r3346183 = 0.5;
        double r3346184 = r3346183 / r3346159;
        double r3346185 = r3346184 - r3346169;
        double r3346186 = r3346159 + r3346185;
        double r3346187 = r3346186 + r3346159;
        double r3346188 = log(r3346187);
        double r3346189 = r3346176 ? r3346182 : r3346188;
        double r3346190 = r3346161 ? r3346174 : r3346189;
        return r3346190;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.6
Target44.7
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0612400279589977

    1. Initial program 61.7

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

    2. Taylor expanded around -inf 0.2

      \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

    3. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{-1}{2}}{x} + \frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x}\right)\right)}\]

    if -1.0612400279589977 < x < 0.9581083600858236

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]

    3. Simplified0.1

      \[\leadsto \color{blue}{\left(x + \frac{-1}{6} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \frac{3}{40} \cdot {x}^{5}}\]

    if 0.9581083600858236 < x

    1. Initial program 31.3

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

    2. Taylor expanded around inf 0.4

      \[\leadsto \log \left(x + \color{blue}{\left(\left(x + \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)}\right)\]

    3. Simplified0.4

      \[\leadsto \log \left(x + \color{blue}{\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x}\right)\right)}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0612400279589977:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} + \frac{\frac{-1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9581083600858236:\\ \;\;\;\;\left(x + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}\right) + {x}^{5} \cdot \frac{3}{40}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))