Average Error: 52.8 → 0.2
Time: 15.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.002275654089841649962977498944383114576:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8884086436842496548038639048172626644373:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{\frac{1}{6}}{1} \cdot \frac{{x}^{3}}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.002275654089841649962977498944383114576:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.8884086436842496548038639048172626644373:\\
\;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{\frac{1}{6}}{1} \cdot \frac{{x}^{3}}{\sqrt{1}}\\

\mathbf{else}:\\
\;\;\;\;\log \left(\left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right) + x\right)\\

\end{array}
double f(double x) {
        double r150180 = x;
        double r150181 = r150180 * r150180;
        double r150182 = 1.0;
        double r150183 = r150181 + r150182;
        double r150184 = sqrt(r150183);
        double r150185 = r150180 + r150184;
        double r150186 = log(r150185);
        return r150186;
}
↓
double f(double x) {
        double r150187 = x;
        double r150188 = -1.0022756540898416;
        bool r150189 = r150187 <= r150188;
        double r150190 = 0.125;
        double r150191 = 3.0;
        double r150192 = pow(r150187, r150191);
        double r150193 = r150190 / r150192;
        double r150194 = 0.0625;
        double r150195 = 5.0;
        double r150196 = pow(r150187, r150195);
        double r150197 = r150194 / r150196;
        double r150198 = 0.5;
        double r150199 = r150198 / r150187;
        double r150200 = r150197 + r150199;
        double r150201 = r150193 - r150200;
        double r150202 = log(r150201);
        double r150203 = 0.8884086436842497;
        bool r150204 = r150187 <= r150203;
        double r150205 = 1.0;
        double r150206 = sqrt(r150205);
        double r150207 = log(r150206);
        double r150208 = r150187 / r150206;
        double r150209 = r150207 + r150208;
        double r150210 = 0.16666666666666666;
        double r150211 = r150210 / r150205;
        double r150212 = r150192 / r150206;
        double r150213 = r150211 * r150212;
        double r150214 = r150209 - r150213;
        double r150215 = r150187 + r150199;
        double r150216 = r150215 - r150193;
        double r150217 = r150216 + r150187;
        double r150218 = log(r150217);
        double r150219 = r150204 ? r150214 : r150218;
        double r150220 = r150189 ? r150202 : r150219;
        return r150220;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.8
Target45.0
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 < -1.0022756540898416

    1. Initial program 63.1

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

    2. Taylor expanded around -inf 0.1

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

    3. Simplified0.1

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

    if -1.0022756540898416 < x < 0.8884086436842497

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 0.8884086436842497 < x

    1. Initial program 31.4

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.002275654089841649962977498944383114576:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8884086436842496548038639048172626644373:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{\frac{1}{6}}{1} \cdot \frac{{x}^{3}}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))