Average Error: 52.0 → 0.2
Time: 18.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0651328544879792:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} - \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x}\right)\right)\\ \mathbf{elif}\;x \le 0.9510515308963582:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{3}{40} + \left(x - \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{6}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{\frac{1}{2}}{x} + \left(\frac{\frac{\frac{-1}{8}}{x \cdot x}}{x} + x\right)\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0651328544879792:\\
\;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} - \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x}\right)\right)\\

\mathbf{elif}\;x \le 0.9510515308963582:\\
\;\;\;\;\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{3}{40} + \left(x - \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{6}\right)\\

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

\end{array}
double f(double x) {
        double r5965111 = x;
        double r5965112 = r5965111 * r5965111;
        double r5965113 = 1.0;
        double r5965114 = r5965112 + r5965113;
        double r5965115 = sqrt(r5965114);
        double r5965116 = r5965111 + r5965115;
        double r5965117 = log(r5965116);
        return r5965117;
}
↓
double f(double x) {
        double r5965118 = x;
        double r5965119 = -1.0651328544879792;
        bool r5965120 = r5965118 <= r5965119;
        double r5965121 = -0.0625;
        double r5965122 = r5965118 * r5965118;
        double r5965123 = r5965122 * r5965118;
        double r5965124 = r5965122 * r5965123;
        double r5965125 = r5965121 / r5965124;
        double r5965126 = 0.5;
        double r5965127 = r5965126 / r5965118;
        double r5965128 = 0.125;
        double r5965129 = r5965128 / r5965123;
        double r5965130 = r5965127 - r5965129;
        double r5965131 = r5965125 - r5965130;
        double r5965132 = log(r5965131);
        double r5965133 = 0.9510515308963582;
        bool r5965134 = r5965118 <= r5965133;
        double r5965135 = 0.075;
        double r5965136 = r5965124 * r5965135;
        double r5965137 = 0.16666666666666666;
        double r5965138 = r5965123 * r5965137;
        double r5965139 = r5965118 - r5965138;
        double r5965140 = r5965136 + r5965139;
        double r5965141 = -0.125;
        double r5965142 = r5965141 / r5965122;
        double r5965143 = r5965142 / r5965118;
        double r5965144 = r5965143 + r5965118;
        double r5965145 = r5965127 + r5965144;
        double r5965146 = r5965145 + r5965118;
        double r5965147 = log(r5965146);
        double r5965148 = r5965134 ? r5965140 : r5965147;
        double r5965149 = r5965120 ? r5965132 : r5965148;
        return r5965149;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.0
Target44.3
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 < -1.0651328544879792

    1. Initial program 61.6

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

    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

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

    if -1.0651328544879792 < x < 0.9510515308963582

    1. Initial program 58.5

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

    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

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

    if 0.9510515308963582 < x

    1. Initial program 30.9

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0651328544879792:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} - \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x}\right)\right)\\ \mathbf{elif}\;x \le 0.9510515308963582:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{3}{40} + \left(x - \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{6}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{\frac{1}{2}}{x} + \left(\frac{\frac{\frac{-1}{8}}{x \cdot x}}{x} + x\right)\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(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)))))