Average Error: 53.3 → 0.3
Time: 6.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.02867483106351787:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.89493471582712192:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.02867483106351787:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.89493471582712192:\\
\;\;\;\;\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(x + \left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\

\end{array}
double f(double x) {
        double r273002 = x;
        double r273003 = r273002 * r273002;
        double r273004 = 1.0;
        double r273005 = r273003 + r273004;
        double r273006 = sqrt(r273005);
        double r273007 = r273002 + r273006;
        double r273008 = log(r273007);
        return r273008;
}
↓
double f(double x) {
        double r273009 = x;
        double r273010 = -1.0286748310635179;
        bool r273011 = r273009 <= r273010;
        double r273012 = 0.125;
        double r273013 = 3.0;
        double r273014 = pow(r273009, r273013);
        double r273015 = r273012 / r273014;
        double r273016 = 0.5;
        double r273017 = r273016 / r273009;
        double r273018 = 0.0625;
        double r273019 = -r273018;
        double r273020 = 5.0;
        double r273021 = pow(r273009, r273020);
        double r273022 = r273019 / r273021;
        double r273023 = r273017 - r273022;
        double r273024 = r273015 - r273023;
        double r273025 = log(r273024);
        double r273026 = 0.8949347158271219;
        bool r273027 = r273009 <= r273026;
        double r273028 = 1.0;
        double r273029 = sqrt(r273028);
        double r273030 = log(r273029);
        double r273031 = r273009 / r273029;
        double r273032 = r273030 + r273031;
        double r273033 = 0.16666666666666666;
        double r273034 = pow(r273029, r273013);
        double r273035 = r273014 / r273034;
        double r273036 = r273033 * r273035;
        double r273037 = r273032 - r273036;
        double r273038 = r273009 + r273017;
        double r273039 = r273038 - r273015;
        double r273040 = r273009 + r273039;
        double r273041 = log(r273040);
        double r273042 = r273027 ? r273037 : r273041;
        double r273043 = r273011 ? r273025 : r273042;
        return r273043;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.3
Target45.6
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 < -1.0286748310635179

    1. Initial program 63.0

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

    2. Taylor expanded around -inf 0.1

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

    3. Simplified0.1

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

    if -1.0286748310635179 < x < 0.8949347158271219

    1. Initial program 58.6

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

    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 0.8949347158271219 < x

    1. Initial program 32.5

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.02867483106351787:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.89493471582712192:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(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)))))