Average Error: 53.4 → 0.2
Time: 14.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.020534956814445282091696753923315554857:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.8969537796209996471930026018526405096054:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.020534956814445282091696753923315554857:\\
\;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 0.8969537796209996471930026018526405096054:\\
\;\;\;\;\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(x + \left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\right)\\

\end{array}
double f(double x) {
        double r98568 = x;
        double r98569 = r98568 * r98568;
        double r98570 = 1.0;
        double r98571 = r98569 + r98570;
        double r98572 = sqrt(r98571);
        double r98573 = r98568 + r98572;
        double r98574 = log(r98573);
        return r98574;
}
↓
double f(double x) {
        double r98575 = x;
        double r98576 = -1.0205349568144453;
        bool r98577 = r98575 <= r98576;
        double r98578 = 0.125;
        double r98579 = 3.0;
        double r98580 = pow(r98575, r98579);
        double r98581 = r98578 / r98580;
        double r98582 = 0.5;
        double r98583 = r98582 / r98575;
        double r98584 = r98581 - r98583;
        double r98585 = 0.0625;
        double r98586 = 5.0;
        double r98587 = pow(r98575, r98586);
        double r98588 = r98585 / r98587;
        double r98589 = r98584 - r98588;
        double r98590 = log(r98589);
        double r98591 = 0.8969537796209996;
        bool r98592 = r98575 <= r98591;
        double r98593 = 1.0;
        double r98594 = sqrt(r98593);
        double r98595 = log(r98594);
        double r98596 = r98575 / r98594;
        double r98597 = r98595 + r98596;
        double r98598 = 0.16666666666666666;
        double r98599 = pow(r98594, r98579);
        double r98600 = r98580 / r98599;
        double r98601 = r98598 * r98600;
        double r98602 = r98597 - r98601;
        double r98603 = r98583 - r98581;
        double r98604 = r98575 + r98603;
        double r98605 = r98575 + r98604;
        double r98606 = log(r98605);
        double r98607 = r98592 ? r98602 : r98606;
        double r98608 = r98577 ? r98590 : r98607;
        return r98608;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.4
Target45.7
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0205349568144453

    1. Initial program 63.1

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

    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

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

    if -1.0205349568144453 < x < 0.8969537796209996

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.2

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

    if 0.8969537796209996 < x

    1. Initial program 32.4

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

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.020534956814445282091696753923315554857:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.8969537796209996471930026018526405096054:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(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)))))