Average Error: 52.2 → 0.2
Time: 1.0m
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0682549790295666:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{elif}\;x \le 0.9560728927963928:\\ \;\;\;\;\left(x - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{6}\right) + \frac{3}{40} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0682549790295666:\\
\;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\

\mathbf{elif}\;x \le 0.9560728927963928:\\
\;\;\;\;\left(x - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{6}\right) + \frac{3}{40} \cdot {x}^{5}\\

\mathbf{else}:\\
\;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\

\end{array}
double f(double x) {
        double r12419612 = x;
        double r12419613 = r12419612 * r12419612;
        double r12419614 = 1.0;
        double r12419615 = r12419613 + r12419614;
        double r12419616 = sqrt(r12419615);
        double r12419617 = r12419612 + r12419616;
        double r12419618 = log(r12419617);
        return r12419618;
}
↓
double f(double x) {
        double r12419619 = x;
        double r12419620 = -1.0682549790295666;
        bool r12419621 = r12419619 <= r12419620;
        double r12419622 = -0.5;
        double r12419623 = r12419622 / r12419619;
        double r12419624 = 0.0625;
        double r12419625 = 5.0;
        double r12419626 = pow(r12419619, r12419625);
        double r12419627 = r12419624 / r12419626;
        double r12419628 = 0.125;
        double r12419629 = r12419619 * r12419619;
        double r12419630 = r12419619 * r12419629;
        double r12419631 = r12419628 / r12419630;
        double r12419632 = r12419627 - r12419631;
        double r12419633 = r12419623 - r12419632;
        double r12419634 = log(r12419633);
        double r12419635 = 0.9560728927963928;
        bool r12419636 = r12419619 <= r12419635;
        double r12419637 = 0.16666666666666666;
        double r12419638 = r12419630 * r12419637;
        double r12419639 = r12419619 - r12419638;
        double r12419640 = 0.075;
        double r12419641 = r12419640 * r12419626;
        double r12419642 = r12419639 + r12419641;
        double r12419643 = 0.5;
        double r12419644 = r12419643 / r12419619;
        double r12419645 = r12419644 - r12419631;
        double r12419646 = r12419619 + r12419645;
        double r12419647 = r12419646 + r12419619;
        double r12419648 = log(r12419647);
        double r12419649 = r12419636 ? r12419642 : r12419648;
        double r12419650 = r12419621 ? r12419634 : r12419649;
        return r12419650;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.2
Target44.7
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0682549790295666

    1. Initial program 61.8

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

    2. Taylor expanded around -inf 0.2

      \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

    3. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x}\right)\right)}\]

    if -1.0682549790295666 < x < 0.9560728927963928

    1. Initial program 58.4

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

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{\left(x - \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{6}\right) + \frac{3}{40} \cdot {x}^{5}}\]

    if 0.9560728927963928 < x

    1. Initial program 29.9

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

    2. Taylor expanded around inf 0.2

      \[\leadsto \log \left(x + \color{blue}{\left(\left(x + \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)}\right)\]

    3. Simplified0.2

      \[\leadsto \log \left(x + \color{blue}{\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x}\right)\right)}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0682549790295666:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{elif}\;x \le 0.9560728927963928:\\ \;\;\;\;\left(x - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{6}\right) + \frac{3}{40} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))