Average Error: 52.3 → 0.2
Time: 45.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0413231664509972:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.9643127001303118:\\ \;\;\;\;{x}^{5} \cdot \frac{3}{40} + \left(x + x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(x + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0413231664509972:\\
\;\;\;\;\log \left(\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 0.9643127001303118:\\
\;\;\;\;{x}^{5} \cdot \frac{3}{40} + \left(x + x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\\

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

\end{array}
double f(double x) {
        double r20965527 = x;
        double r20965528 = r20965527 * r20965527;
        double r20965529 = 1.0;
        double r20965530 = r20965528 + r20965529;
        double r20965531 = sqrt(r20965530);
        double r20965532 = r20965527 + r20965531;
        double r20965533 = log(r20965532);
        return r20965533;
}
↓
double f(double x) {
        double r20965534 = x;
        double r20965535 = -1.0413231664509972;
        bool r20965536 = r20965534 <= r20965535;
        double r20965537 = 0.125;
        double r20965538 = r20965537 / r20965534;
        double r20965539 = r20965538 / r20965534;
        double r20965540 = r20965539 / r20965534;
        double r20965541 = -0.5;
        double r20965542 = r20965541 / r20965534;
        double r20965543 = r20965540 + r20965542;
        double r20965544 = 0.0625;
        double r20965545 = 5.0;
        double r20965546 = pow(r20965534, r20965545);
        double r20965547 = r20965544 / r20965546;
        double r20965548 = r20965543 - r20965547;
        double r20965549 = log(r20965548);
        double r20965550 = 0.9643127001303118;
        bool r20965551 = r20965534 <= r20965550;
        double r20965552 = 0.075;
        double r20965553 = r20965546 * r20965552;
        double r20965554 = r20965534 * r20965534;
        double r20965555 = -0.16666666666666666;
        double r20965556 = r20965554 * r20965555;
        double r20965557 = r20965534 * r20965556;
        double r20965558 = r20965534 + r20965557;
        double r20965559 = r20965553 + r20965558;
        double r20965560 = 0.5;
        double r20965561 = r20965560 / r20965534;
        double r20965562 = -0.125;
        double r20965563 = r20965534 * r20965554;
        double r20965564 = r20965562 / r20965563;
        double r20965565 = r20965534 + r20965564;
        double r20965566 = r20965561 + r20965565;
        double r20965567 = r20965534 + r20965566;
        double r20965568 = log(r20965567);
        double r20965569 = r20965551 ? r20965559 : r20965568;
        double r20965570 = r20965536 ? r20965549 : r20965569;
        return r20965570;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.3
Target44.6
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 < -1.0413231664509972

    1. Initial program 61.7

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

    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

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

    if -1.0413231664509972 < x < 0.9643127001303118

    1. Initial program 58.6

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

    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

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

    if 0.9643127001303118 < x

    1. Initial program 30.5

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

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0413231664509972:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.9643127001303118:\\ \;\;\;\;{x}^{5} \cdot \frac{3}{40} + \left(x + x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(x + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(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)))))