Hyperbolic arcsine

Average Error: 52.0 → 0.2

Time: 23.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0884602506497811:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x} - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \le 0.9577042263772265:\\ \;\;\;\;\left(x + \frac{3}{40} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) + \left(x \cdot x\right) \cdot \left(\frac{-1}{6} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{\frac{1}{2}}{x} - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right) + x\right)\right)\\ \end{array}\]

C

double f(double x) {
        double r5238282 = x;
        double r5238283 = r5238282 * r5238282;
        double r5238284 = 1.0;
        double r5238285 = r5238283 + r5238284;
        double r5238286 = sqrt(r5238285);
        double r5238287 = r5238282 + r5238286;
        double r5238288 = log(r5238287);
        return r5238288;
}

↓

double f(double x) {
        double r5238289 = x;
        double r5238290 = -1.0884602506497811;
        bool r5238291 = r5238289 <= r5238290;
        double r5238292 = -0.5;
        double r5238293 = r5238292 / r5238289;
        double r5238294 = 0.0625;
        double r5238295 = r5238289 * r5238289;
        double r5238296 = r5238295 * r5238295;
        double r5238297 = r5238296 * r5238289;
        double r5238298 = r5238294 / r5238297;
        double r5238299 = 0.125;
        double r5238300 = r5238299 / r5238289;
        double r5238301 = r5238300 / r5238295;
        double r5238302 = r5238298 - r5238301;
        double r5238303 = r5238293 - r5238302;
        double r5238304 = log(r5238303);
        double r5238305 = 0.9577042263772265;
        bool r5238306 = r5238289 <= r5238305;
        double r5238307 = 0.075;
        double r5238308 = r5238307 * r5238297;
        double r5238309 = r5238289 + r5238308;
        double r5238310 = -0.16666666666666666;
        double r5238311 = r5238310 * r5238289;
        double r5238312 = r5238295 * r5238311;
        double r5238313 = r5238309 + r5238312;
        double r5238314 = 0.5;
        double r5238315 = r5238314 / r5238289;
        double r5238316 = r5238315 - r5238301;
        double r5238317 = r5238316 + r5238289;
        double r5238318 = r5238289 + r5238317;
        double r5238319 = log(r5238318);
        double r5238320 = r5238306 ? r5238313 : r5238319;
        double r5238321 = r5238291 ? r5238304 : r5238320;
        return r5238321;
}

Error

Bits error versus x

Target

Original	52.0
Target	44.3
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -1.0884602506497811

   1. Initial program 61.5

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

   2. Taylor expanded around -inf 0.3

      \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

   3. Simplified 0.3

      \[\leadsto \log \color{blue}{\left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)}\]

   if -1.0884602506497811 < x < 0.9577042263772265

   1. Initial program 58.5

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

   2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]

   3. Simplified 0.2

      \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\frac{-1}{6} \cdot x\right) \cdot \left(x \cdot x\right)}\]

   if 0.9577042263772265 < x

   1. Initial program 29.9

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

   2. Taylor expanded around inf 0.3

      \[\leadsto \log \left(x + \color{blue}{\left(\left(x + \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)}\right)\]

   3. Simplified 0.3

      \[\leadsto \log \left(x + \color{blue}{\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0884602506497811:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x} - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \le 0.9577042263772265:\\ \;\;\;\;\left(x + \frac{3}{40} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) + \left(x \cdot x\right) \cdot \left(\frac{-1}{6} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{\frac{1}{2}}{x} - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right) + x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))