Average Error: 53.5 → 0.2
Time: 6.3s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0059580948780908:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.90050708137317836:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} - \left(\frac{0.125}{{x}^{3}} - 2 \cdot x\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0059580948780908:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.90050708137317836:\\
\;\;\;\;\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(\frac{0.5}{x} - \left(\frac{0.125}{{x}^{3}} - 2 \cdot x\right)\right)\\

\end{array}
double f(double x) {
        double r161400 = x;
        double r161401 = r161400 * r161400;
        double r161402 = 1.0;
        double r161403 = r161401 + r161402;
        double r161404 = sqrt(r161403);
        double r161405 = r161400 + r161404;
        double r161406 = log(r161405);
        return r161406;
}
↓
double f(double x) {
        double r161407 = x;
        double r161408 = -1.0059580948780908;
        bool r161409 = r161407 <= r161408;
        double r161410 = 0.125;
        double r161411 = 3.0;
        double r161412 = pow(r161407, r161411);
        double r161413 = r161410 / r161412;
        double r161414 = 0.5;
        double r161415 = r161414 / r161407;
        double r161416 = 0.0625;
        double r161417 = -r161416;
        double r161418 = 5.0;
        double r161419 = pow(r161407, r161418);
        double r161420 = r161417 / r161419;
        double r161421 = r161415 - r161420;
        double r161422 = r161413 - r161421;
        double r161423 = log(r161422);
        double r161424 = 0.9005070813731784;
        bool r161425 = r161407 <= r161424;
        double r161426 = 1.0;
        double r161427 = sqrt(r161426);
        double r161428 = log(r161427);
        double r161429 = r161407 / r161427;
        double r161430 = r161428 + r161429;
        double r161431 = 0.16666666666666666;
        double r161432 = pow(r161427, r161411);
        double r161433 = r161412 / r161432;
        double r161434 = r161431 * r161433;
        double r161435 = r161430 - r161434;
        double r161436 = 2.0;
        double r161437 = r161436 * r161407;
        double r161438 = r161413 - r161437;
        double r161439 = r161415 - r161438;
        double r161440 = log(r161439);
        double r161441 = r161425 ? r161435 : r161440;
        double r161442 = r161409 ? r161423 : r161441;
        return r161442;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.5
Target45.8
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 < -1.0059580948780908

    1. Initial program 63.0

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

    2. Taylor expanded around -inf 0.2

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

    3. Simplified0.2

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

    if -1.0059580948780908 < x < 0.9005070813731784

    1. Initial program 58.8

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

    2. Taylor expanded around 0 0.2

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

    if 0.9005070813731784 < x

    1. Initial program 34.0

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

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0059580948780908:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.90050708137317836:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{0.5}{x} - \left(\frac{0.125}{{x}^{3}} - 2 \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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)))))