Average Error: 53.1 → 0.3
Time: 18.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.000827500383000945305411732988432049751:\\ \;\;\;\;\log \left(0.125 \cdot \frac{1}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.8895218929182532319188680958177428692579:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{-0.125}{{x}^{3}} + \frac{0.5}{x}\right) + 2 \cdot x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.000827500383000945305411732988432049751:\\
\;\;\;\;\log \left(0.125 \cdot \frac{1}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.8895218929182532319188680958177428692579:\\
\;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\log \left(\left(\frac{-0.125}{{x}^{3}} + \frac{0.5}{x}\right) + 2 \cdot x\right)\\

\end{array}
double f(double x) {
        double r163108 = x;
        double r163109 = r163108 * r163108;
        double r163110 = 1.0;
        double r163111 = r163109 + r163110;
        double r163112 = sqrt(r163111);
        double r163113 = r163108 + r163112;
        double r163114 = log(r163113);
        return r163114;
}
↓
double f(double x) {
        double r163115 = x;
        double r163116 = -1.000827500383001;
        bool r163117 = r163115 <= r163116;
        double r163118 = 0.125;
        double r163119 = 1.0;
        double r163120 = 3.0;
        double r163121 = pow(r163115, r163120);
        double r163122 = r163119 / r163121;
        double r163123 = r163118 * r163122;
        double r163124 = 0.5;
        double r163125 = r163124 / r163115;
        double r163126 = 0.0625;
        double r163127 = -r163126;
        double r163128 = 5.0;
        double r163129 = pow(r163115, r163128);
        double r163130 = r163127 / r163129;
        double r163131 = r163125 - r163130;
        double r163132 = r163123 - r163131;
        double r163133 = log(r163132);
        double r163134 = 0.8895218929182532;
        bool r163135 = r163115 <= r163134;
        double r163136 = 1.0;
        double r163137 = sqrt(r163136);
        double r163138 = log(r163137);
        double r163139 = r163115 / r163137;
        double r163140 = r163138 + r163139;
        double r163141 = 0.16666666666666666;
        double r163142 = pow(r163137, r163120);
        double r163143 = r163121 / r163142;
        double r163144 = r163141 * r163143;
        double r163145 = r163140 - r163144;
        double r163146 = -r163118;
        double r163147 = r163146 / r163121;
        double r163148 = r163147 + r163125;
        double r163149 = 2.0;
        double r163150 = r163149 * r163115;
        double r163151 = r163148 + r163150;
        double r163152 = log(r163151);
        double r163153 = r163135 ? r163145 : r163152;
        double r163154 = r163117 ? r163133 : r163153;
        return r163154;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.1
Target45.7
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.000827500383001

    1. Initial program 62.9

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

    2. Taylor expanded around -inf 0.2

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

    3. Simplified0.2

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

    if -1.000827500383001 < x < 0.8895218929182532

    1. Initial program 58.5

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

    2. Taylor expanded around 0 0.4

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

    if 0.8895218929182532 < x

    1. Initial program 32.6

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

    2. Taylor expanded around inf 0.3

      \[\leadsto \log \color{blue}{\left(\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)}\]

    3. Simplified0.3

      \[\leadsto \log \color{blue}{\left(\left(\frac{-0.125}{{x}^{3}} + \frac{0.5}{x}\right) + 2 \cdot x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.000827500383000945305411732988432049751:\\ \;\;\;\;\log \left(0.125 \cdot \frac{1}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.8895218929182532319188680958177428692579:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{-0.125}{{x}^{3}} + \frac{0.5}{x}\right) + 2 \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :herbie-target
  (if (< x 0.0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

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