Average Error: 53.0 → 0.3
Time: 9.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.03811431304857993:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.90048843936555456:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(2 \cdot x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.03811431304857993:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.90048843936555456:\\
\;\;\;\;\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(2 \cdot x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\\

\end{array}
double f(double x) {
        double r133701 = x;
        double r133702 = r133701 * r133701;
        double r133703 = 1.0;
        double r133704 = r133702 + r133703;
        double r133705 = sqrt(r133704);
        double r133706 = r133701 + r133705;
        double r133707 = log(r133706);
        return r133707;
}
↓
double f(double x) {
        double r133708 = x;
        double r133709 = -1.03811431304858;
        bool r133710 = r133708 <= r133709;
        double r133711 = 0.125;
        double r133712 = 3.0;
        double r133713 = pow(r133708, r133712);
        double r133714 = r133711 / r133713;
        double r133715 = 0.5;
        double r133716 = r133715 / r133708;
        double r133717 = 0.0625;
        double r133718 = 5.0;
        double r133719 = pow(r133708, r133718);
        double r133720 = r133717 / r133719;
        double r133721 = r133716 + r133720;
        double r133722 = r133714 - r133721;
        double r133723 = log(r133722);
        double r133724 = 0.9004884393655546;
        bool r133725 = r133708 <= r133724;
        double r133726 = 1.0;
        double r133727 = sqrt(r133726);
        double r133728 = log(r133727);
        double r133729 = r133708 / r133727;
        double r133730 = r133728 + r133729;
        double r133731 = 0.16666666666666666;
        double r133732 = pow(r133727, r133712);
        double r133733 = r133713 / r133732;
        double r133734 = r133731 * r133733;
        double r133735 = r133730 - r133734;
        double r133736 = 2.0;
        double r133737 = r133736 * r133708;
        double r133738 = r133737 + r133716;
        double r133739 = r133738 - r133714;
        double r133740 = log(r133739);
        double r133741 = r133725 ? r133735 : r133740;
        double r133742 = r133710 ? r133723 : r133741;
        return r133742;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.0
Target45.4
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 < -1.03811431304858

    1. Initial program 62.7

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

    2. Taylor expanded around -inf 0.3

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

    3. Simplified0.3

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

    if -1.03811431304858 < x < 0.9004884393655546

    1. Initial program 58.8

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

    2. Taylor expanded around 0 0.3

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

    if 0.9004884393655546 < x

    1. Initial program 32.6

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.03811431304857993:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.90048843936555456:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(2 \cdot x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))))