Average Error: 52.9 → 0.3
Time: 12.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.02004933856096858:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \le 0.88957403075873587:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(2, x, \frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.02004933856096858:\\
\;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\

\mathbf{elif}\;x \le 0.88957403075873587:\\
\;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\mathsf{fma}\left(2, x, \frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\\

\end{array}
double f(double x) {
        double r100253 = x;
        double r100254 = r100253 * r100253;
        double r100255 = 1.0;
        double r100256 = r100254 + r100255;
        double r100257 = sqrt(r100256);
        double r100258 = r100253 + r100257;
        double r100259 = log(r100258);
        return r100259;
}
↓
double f(double x) {
        double r100260 = x;
        double r100261 = -1.0200493385609686;
        bool r100262 = r100260 <= r100261;
        double r100263 = 0.125;
        double r100264 = 3.0;
        double r100265 = pow(r100260, r100264);
        double r100266 = r100263 / r100265;
        double r100267 = 0.0625;
        double r100268 = 5.0;
        double r100269 = pow(r100260, r100268);
        double r100270 = r100267 / r100269;
        double r100271 = r100266 - r100270;
        double r100272 = 0.5;
        double r100273 = r100272 / r100260;
        double r100274 = r100271 - r100273;
        double r100275 = log(r100274);
        double r100276 = 0.8895740307587359;
        bool r100277 = r100260 <= r100276;
        double r100278 = 1.0;
        double r100279 = sqrt(r100278);
        double r100280 = pow(r100279, r100264);
        double r100281 = r100265 / r100280;
        double r100282 = -0.16666666666666666;
        double r100283 = r100260 / r100279;
        double r100284 = log(r100279);
        double r100285 = r100283 + r100284;
        double r100286 = fma(r100281, r100282, r100285);
        double r100287 = 2.0;
        double r100288 = r100273 - r100266;
        double r100289 = fma(r100287, r100260, r100288);
        double r100290 = log(r100289);
        double r100291 = r100277 ? r100286 : r100290;
        double r100292 = r100262 ? r100275 : r100291;
        return r100292;
}

Error

Bits error versus x

Target

Original52.9
Target45.4
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0200493385609686

    1. Initial program 62.7

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

    2. Simplified62.7

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

    3. Taylor expanded around -inf 0.2

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

    4. Simplified0.2

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

    if -1.0200493385609686 < x < 0.8895740307587359

    1. Initial program 58.6

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

    2. Simplified58.6

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

    3. Taylor expanded around 0 0.3

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

    4. Simplified0.3

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

    if 0.8895740307587359 < x

    1. Initial program 31.1

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

    2. Simplified31.1

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

    3. Taylor expanded around inf 0.3

      \[\leadsto \log \color{blue}{\left(\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)}\]

    4. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.02004933856096858:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \le 0.88957403075873587:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(2, x, \frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(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)))))