Average Error: 52.9 → 0.1
Time: 12.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.01388879764755901:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 7.87465144273405394 \cdot 10^{-4}:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.01388879764755901:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 7.87465144273405394 \cdot 10^{-4}:\\
\;\;\;\;\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(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\\

\end{array}
double f(double x) {
        double r226408 = x;
        double r226409 = r226408 * r226408;
        double r226410 = 1.0;
        double r226411 = r226409 + r226410;
        double r226412 = sqrt(r226411);
        double r226413 = r226408 + r226412;
        double r226414 = log(r226413);
        return r226414;
}
↓
double f(double x) {
        double r226415 = x;
        double r226416 = -1.013888797647559;
        bool r226417 = r226415 <= r226416;
        double r226418 = 0.125;
        double r226419 = 3.0;
        double r226420 = pow(r226415, r226419);
        double r226421 = r226418 / r226420;
        double r226422 = 0.5;
        double r226423 = r226422 / r226415;
        double r226424 = 0.0625;
        double r226425 = -r226424;
        double r226426 = 5.0;
        double r226427 = pow(r226415, r226426);
        double r226428 = r226425 / r226427;
        double r226429 = r226423 - r226428;
        double r226430 = r226421 - r226429;
        double r226431 = log(r226430);
        double r226432 = 0.0007874651442734054;
        bool r226433 = r226415 <= r226432;
        double r226434 = 1.0;
        double r226435 = sqrt(r226434);
        double r226436 = log(r226435);
        double r226437 = r226415 / r226435;
        double r226438 = r226436 + r226437;
        double r226439 = 0.16666666666666666;
        double r226440 = pow(r226435, r226419);
        double r226441 = r226420 / r226440;
        double r226442 = r226439 * r226441;
        double r226443 = r226438 - r226442;
        double r226444 = hypot(r226415, r226435);
        double r226445 = r226444 + r226415;
        double r226446 = log(r226445);
        double r226447 = r226433 ? r226443 : r226446;
        double r226448 = r226417 ? r226431 : r226447;
        return r226448;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.9
Target44.8
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.013888797647559

    1. Initial program 62.8

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

    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 -1.013888797647559 < x < 0.0007874651442734054

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.1

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

    if 0.0007874651442734054 < x

    1. Initial program 31.1

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
    2. Using strategy rm

    3. Applied add-log-exp31.1

      \[\leadsto \color{blue}{\log \left(e^{\log \left(x + \sqrt{x \cdot x + 1}\right)}\right)}\]

    4. Simplified0.1

      \[\leadsto \log \color{blue}{\left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.01388879764755901:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 7.87465144273405394 \cdot 10^{-4}:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(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)))))