Average Error: 52.5 → 0.1
Time: 13.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0715992056992538:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9586135763949255:\\ \;\;\;\;\left(\frac{-1}{6} \cdot \left(\left(x \cdot x\right) \cdot x\right) + \frac{3}{40} \cdot {x}^{5}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{\frac{1}{2}}{x} + \frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x}\right) + x\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0715992056992538:\\
\;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{2}}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.9586135763949255:\\
\;\;\;\;\left(\frac{-1}{6} \cdot \left(\left(x \cdot x\right) \cdot x\right) + \frac{3}{40} \cdot {x}^{5}\right) + x\\

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

\end{array}
double f(double x) {
        double r2955135 = x;
        double r2955136 = r2955135 * r2955135;
        double r2955137 = 1.0;
        double r2955138 = r2955136 + r2955137;
        double r2955139 = sqrt(r2955138);
        double r2955140 = r2955135 + r2955139;
        double r2955141 = log(r2955140);
        return r2955141;
}
↓
double f(double x) {
        double r2955142 = x;
        double r2955143 = -1.0715992056992538;
        bool r2955144 = r2955142 <= r2955143;
        double r2955145 = -0.0625;
        double r2955146 = 5.0;
        double r2955147 = pow(r2955142, r2955146);
        double r2955148 = r2955145 / r2955147;
        double r2955149 = 0.125;
        double r2955150 = r2955142 * r2955142;
        double r2955151 = r2955149 / r2955150;
        double r2955152 = r2955151 / r2955142;
        double r2955153 = 0.5;
        double r2955154 = r2955153 / r2955142;
        double r2955155 = r2955152 - r2955154;
        double r2955156 = r2955148 + r2955155;
        double r2955157 = log(r2955156);
        double r2955158 = 0.9586135763949255;
        bool r2955159 = r2955142 <= r2955158;
        double r2955160 = -0.16666666666666666;
        double r2955161 = r2955150 * r2955142;
        double r2955162 = r2955160 * r2955161;
        double r2955163 = 0.075;
        double r2955164 = r2955163 * r2955147;
        double r2955165 = r2955162 + r2955164;
        double r2955166 = r2955165 + r2955142;
        double r2955167 = -0.125;
        double r2955168 = r2955167 / r2955161;
        double r2955169 = r2955154 + r2955168;
        double r2955170 = r2955169 + r2955142;
        double r2955171 = r2955142 + r2955170;
        double r2955172 = log(r2955171);
        double r2955173 = r2955159 ? r2955166 : r2955172;
        double r2955174 = r2955144 ? r2955157 : r2955173;
        return r2955174;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.5
Target44.8
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0715992056992538

    1. Initial program 61.8

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

    2. Taylor expanded around -inf 0.1

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

    3. Simplified0.1

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

    if -1.0715992056992538 < x < 0.9586135763949255

    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

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

    if 0.9586135763949255 < x

    1. Initial program 30.8

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0715992056992538:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9586135763949255:\\ \;\;\;\;\left(\frac{-1}{6} \cdot \left(\left(x \cdot x\right) \cdot x\right) + \frac{3}{40} \cdot {x}^{5}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{\frac{1}{2}}{x} + \frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x}\right) + x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(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)))))