Average Error: 52.8 → 0.2
Time: 20.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.9983730585657322187387308076722547411919:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8840407169458701641673314952640794217587:\\ \;\;\;\;\frac{x}{\sqrt{1}} + \left(\log \left(\sqrt{1}\right) - \frac{\frac{1}{6}}{\frac{\sqrt{1}}{\frac{\left(x \cdot x\right) \cdot x}{1}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{\frac{0.125}{x}}{x \cdot x}\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.9983730585657322187387308076722547411919:\\
\;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.8840407169458701641673314952640794217587:\\
\;\;\;\;\frac{x}{\sqrt{1}} + \left(\log \left(\sqrt{1}\right) - \frac{\frac{1}{6}}{\frac{\sqrt{1}}{\frac{\left(x \cdot x\right) \cdot x}{1}}}\right)\\

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

\end{array}
double f(double x) {
        double r5332539 = x;
        double r5332540 = r5332539 * r5332539;
        double r5332541 = 1.0;
        double r5332542 = r5332540 + r5332541;
        double r5332543 = sqrt(r5332542);
        double r5332544 = r5332539 + r5332543;
        double r5332545 = log(r5332544);
        return r5332545;
}
↓
double f(double x) {
        double r5332546 = x;
        double r5332547 = -0.9983730585657322;
        bool r5332548 = r5332546 <= r5332547;
        double r5332549 = 0.125;
        double r5332550 = r5332549 / r5332546;
        double r5332551 = r5332546 * r5332546;
        double r5332552 = r5332550 / r5332551;
        double r5332553 = 0.0625;
        double r5332554 = 5.0;
        double r5332555 = pow(r5332546, r5332554);
        double r5332556 = r5332553 / r5332555;
        double r5332557 = 0.5;
        double r5332558 = r5332557 / r5332546;
        double r5332559 = r5332556 + r5332558;
        double r5332560 = r5332552 - r5332559;
        double r5332561 = log(r5332560);
        double r5332562 = 0.8840407169458702;
        bool r5332563 = r5332546 <= r5332562;
        double r5332564 = 1.0;
        double r5332565 = sqrt(r5332564);
        double r5332566 = r5332546 / r5332565;
        double r5332567 = log(r5332565);
        double r5332568 = 0.16666666666666666;
        double r5332569 = r5332551 * r5332546;
        double r5332570 = r5332569 / r5332564;
        double r5332571 = r5332565 / r5332570;
        double r5332572 = r5332568 / r5332571;
        double r5332573 = r5332567 - r5332572;
        double r5332574 = r5332566 + r5332573;
        double r5332575 = r5332546 - r5332552;
        double r5332576 = r5332558 + r5332575;
        double r5332577 = r5332546 + r5332576;
        double r5332578 = log(r5332577);
        double r5332579 = r5332563 ? r5332574 : r5332578;
        double r5332580 = r5332548 ? r5332561 : r5332579;
        return r5332580;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.8
Target45.1
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 < -0.9983730585657322

    1. Initial program 63.0

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

    2. Taylor expanded around -inf 0.1

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

    3. Simplified0.1

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

    if -0.9983730585657322 < x < 0.8840407169458702

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 0.8840407169458702 < x

    1. Initial program 30.3

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9983730585657322187387308076722547411919:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8840407169458701641673314952640794217587:\\ \;\;\;\;\frac{x}{\sqrt{1}} + \left(\log \left(\sqrt{1}\right) - \frac{\frac{1}{6}}{\frac{\sqrt{1}}{\frac{\left(x \cdot x\right) \cdot x}{1}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{\frac{0.125}{x}}{x \cdot x}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))