Average Error: 52.8 → 0.2
Time: 16.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.9983730585657322187387308076722547411919:\\ \;\;\;\;\log \left(\frac{0.125}{\left(x \cdot x\right) \cdot x} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8840407169458701641673314952640794217587:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{\frac{1}{6}}{1} \cdot \frac{\left(x \cdot x\right) \cdot x}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{\left(x \cdot x\right) \cdot x}\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.9983730585657322187387308076722547411919:\\
\;\;\;\;\log \left(\frac{0.125}{\left(x \cdot x\right) \cdot x} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\

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

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

\end{array}
double f(double x) {
        double r5561650 = x;
        double r5561651 = r5561650 * r5561650;
        double r5561652 = 1.0;
        double r5561653 = r5561651 + r5561652;
        double r5561654 = sqrt(r5561653);
        double r5561655 = r5561650 + r5561654;
        double r5561656 = log(r5561655);
        return r5561656;
}
↓
double f(double x) {
        double r5561657 = x;
        double r5561658 = -0.9983730585657322;
        bool r5561659 = r5561657 <= r5561658;
        double r5561660 = 0.125;
        double r5561661 = r5561657 * r5561657;
        double r5561662 = r5561661 * r5561657;
        double r5561663 = r5561660 / r5561662;
        double r5561664 = 0.0625;
        double r5561665 = 5.0;
        double r5561666 = pow(r5561657, r5561665);
        double r5561667 = r5561664 / r5561666;
        double r5561668 = 0.5;
        double r5561669 = r5561668 / r5561657;
        double r5561670 = r5561667 + r5561669;
        double r5561671 = r5561663 - r5561670;
        double r5561672 = log(r5561671);
        double r5561673 = 0.8840407169458702;
        bool r5561674 = r5561657 <= r5561673;
        double r5561675 = 1.0;
        double r5561676 = sqrt(r5561675);
        double r5561677 = log(r5561676);
        double r5561678 = r5561657 / r5561676;
        double r5561679 = r5561677 + r5561678;
        double r5561680 = 0.16666666666666666;
        double r5561681 = r5561680 / r5561675;
        double r5561682 = r5561662 / r5561676;
        double r5561683 = r5561681 * r5561682;
        double r5561684 = r5561679 - r5561683;
        double r5561685 = r5561657 + r5561669;
        double r5561686 = r5561685 - r5561663;
        double r5561687 = r5561686 + r5561657;
        double r5561688 = log(r5561687);
        double r5561689 = r5561674 ? r5561684 : r5561688;
        double r5561690 = r5561659 ? r5561672 : r5561689;
        return r5561690;
}

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

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

    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

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

    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

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

  4. Final simplification0.2

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

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)))))