Average Error: 53.1 → 0.3
Time: 14.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.02966525420296184378798898251261562109:\\ \;\;\;\;\log \left(\frac{-0.0625}{{x}^{5}} + \frac{1}{x} \cdot \left(\frac{0.125}{{x}^{2}} - 0.5\right)\right)\\ \mathbf{elif}\;x \le 0.8937069043444494376160491810878738760948:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(2 \cdot x - 0.125 \cdot \frac{1}{{x}^{3}}\right) + \frac{0.5}{x}\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.02966525420296184378798898251261562109:\\
\;\;\;\;\log \left(\frac{-0.0625}{{x}^{5}} + \frac{1}{x} \cdot \left(\frac{0.125}{{x}^{2}} - 0.5\right)\right)\\

\mathbf{elif}\;x \le 0.8937069043444494376160491810878738760948:\\
\;\;\;\;\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 - 0.125 \cdot \frac{1}{{x}^{3}}\right) + \frac{0.5}{x}\right)\\

\end{array}
double f(double x) {
        double r137585 = x;
        double r137586 = r137585 * r137585;
        double r137587 = 1.0;
        double r137588 = r137586 + r137587;
        double r137589 = sqrt(r137588);
        double r137590 = r137585 + r137589;
        double r137591 = log(r137590);
        return r137591;
}
↓
double f(double x) {
        double r137592 = x;
        double r137593 = -1.0296652542029618;
        bool r137594 = r137592 <= r137593;
        double r137595 = 0.0625;
        double r137596 = -r137595;
        double r137597 = 5.0;
        double r137598 = pow(r137592, r137597);
        double r137599 = r137596 / r137598;
        double r137600 = 1.0;
        double r137601 = r137600 / r137592;
        double r137602 = 0.125;
        double r137603 = 2.0;
        double r137604 = pow(r137592, r137603);
        double r137605 = r137602 / r137604;
        double r137606 = 0.5;
        double r137607 = r137605 - r137606;
        double r137608 = r137601 * r137607;
        double r137609 = r137599 + r137608;
        double r137610 = log(r137609);
        double r137611 = 0.8937069043444494;
        bool r137612 = r137592 <= r137611;
        double r137613 = 1.0;
        double r137614 = sqrt(r137613);
        double r137615 = log(r137614);
        double r137616 = r137592 / r137614;
        double r137617 = r137615 + r137616;
        double r137618 = 0.16666666666666666;
        double r137619 = 3.0;
        double r137620 = pow(r137592, r137619);
        double r137621 = pow(r137614, r137619);
        double r137622 = r137620 / r137621;
        double r137623 = r137618 * r137622;
        double r137624 = r137617 - r137623;
        double r137625 = r137603 * r137592;
        double r137626 = r137600 / r137620;
        double r137627 = r137602 * r137626;
        double r137628 = r137625 - r137627;
        double r137629 = r137606 / r137592;
        double r137630 = r137628 + r137629;
        double r137631 = log(r137630);
        double r137632 = r137612 ? r137624 : r137631;
        double r137633 = r137594 ? r137610 : r137632;
        return r137633;
}

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 < -1.0296652542029618

    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

      \[\leadsto \log \color{blue}{\left(\frac{-0.0625}{{x}^{5}} + \frac{1}{x} \cdot \left(\frac{0.125}{{x}^{2}} - 0.5\right)\right)}\]

    if -1.0296652542029618 < x < 0.8937069043444494

    1. Initial program 58.3

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

    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 0.8937069043444494 < x

    1. Initial program 33.4

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

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.02966525420296184378798898251261562109:\\ \;\;\;\;\log \left(\frac{-0.0625}{{x}^{5}} + \frac{1}{x} \cdot \left(\frac{0.125}{{x}^{2}} - 0.5\right)\right)\\ \mathbf{elif}\;x \le 0.8937069043444494376160491810878738760948:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(2 \cdot x - 0.125 \cdot \frac{1}{{x}^{3}}\right) + \frac{0.5}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(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)))))