Average Error: 53.4 → 0.2
Time: 13.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.9950068448859530345629309522337280213833:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 9.563381925855174566994398865915627538925 \cdot 10^{-4}:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{\mathsf{hypot}\left(x, \sqrt{1}\right) + x}\right) + \log \left(\sqrt{\mathsf{hypot}\left(x, \sqrt{1}\right) + x}\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.9950068448859530345629309522337280213833:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 9.563381925855174566994398865915627538925 \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(\sqrt{\mathsf{hypot}\left(x, \sqrt{1}\right) + x}\right) + \log \left(\sqrt{\mathsf{hypot}\left(x, \sqrt{1}\right) + x}\right)\\

\end{array}
double f(double x) {
        double r143183 = x;
        double r143184 = r143183 * r143183;
        double r143185 = 1.0;
        double r143186 = r143184 + r143185;
        double r143187 = sqrt(r143186);
        double r143188 = r143183 + r143187;
        double r143189 = log(r143188);
        return r143189;
}
↓
double f(double x) {
        double r143190 = x;
        double r143191 = -0.995006844885953;
        bool r143192 = r143190 <= r143191;
        double r143193 = 0.125;
        double r143194 = 3.0;
        double r143195 = pow(r143190, r143194);
        double r143196 = r143193 / r143195;
        double r143197 = 0.5;
        double r143198 = r143197 / r143190;
        double r143199 = 0.0625;
        double r143200 = -r143199;
        double r143201 = 5.0;
        double r143202 = pow(r143190, r143201);
        double r143203 = r143200 / r143202;
        double r143204 = r143198 - r143203;
        double r143205 = r143196 - r143204;
        double r143206 = log(r143205);
        double r143207 = 0.0009563381925855175;
        bool r143208 = r143190 <= r143207;
        double r143209 = 1.0;
        double r143210 = sqrt(r143209);
        double r143211 = log(r143210);
        double r143212 = r143190 / r143210;
        double r143213 = r143211 + r143212;
        double r143214 = 0.16666666666666666;
        double r143215 = pow(r143210, r143194);
        double r143216 = r143195 / r143215;
        double r143217 = r143214 * r143216;
        double r143218 = r143213 - r143217;
        double r143219 = hypot(r143190, r143210);
        double r143220 = r143219 + r143190;
        double r143221 = sqrt(r143220);
        double r143222 = log(r143221);
        double r143223 = r143222 + r143222;
        double r143224 = r143208 ? r143218 : r143223;
        double r143225 = r143192 ? r143206 : r143224;
        return r143225;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.4
Target45.4
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 < -0.995006844885953

    1. Initial program 63.1

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

    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 -0.995006844885953 < x < 0.0009563381925855175

    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 0.0009563381925855175 < x

    1. Initial program 32.4

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

    3. Applied add-log-exp32.4

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

    4. Simplified0.1

      \[\leadsto \log \color{blue}{\left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt0.1

      \[\leadsto \log \color{blue}{\left(\sqrt{\mathsf{hypot}\left(x, \sqrt{1}\right) + x} \cdot \sqrt{\mathsf{hypot}\left(x, \sqrt{1}\right) + x}\right)}\]

    7. Applied log-prod0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9950068448859530345629309522337280213833:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 9.563381925855174566994398865915627538925 \cdot 10^{-4}:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{\mathsf{hypot}\left(x, \sqrt{1}\right) + x}\right) + \log \left(\sqrt{\mathsf{hypot}\left(x, \sqrt{1}\right) + x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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)))))