Hyperbolic arcsine

Average Error: 52.6 → 0.2

Time: 20.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0507751989105836:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9626554548619399:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{3}{40} - \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{6} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + x\right)\right)\right)\\ \end{array}\]

C

double f(double x) {
        double r5436282 = x;
        double r5436283 = r5436282 * r5436282;
        double r5436284 = 1.0;
        double r5436285 = r5436283 + r5436284;
        double r5436286 = sqrt(r5436285);
        double r5436287 = r5436282 + r5436286;
        double r5436288 = log(r5436287);
        return r5436288;
}

↓

double f(double x) {
        double r5436289 = x;
        double r5436290 = -1.0507751989105836;
        bool r5436291 = r5436289 <= r5436290;
        double r5436292 = -0.0625;
        double r5436293 = r5436289 * r5436289;
        double r5436294 = r5436293 * r5436289;
        double r5436295 = r5436293 * r5436294;
        double r5436296 = r5436292 / r5436295;
        double r5436297 = 0.125;
        double r5436298 = r5436297 / r5436294;
        double r5436299 = 0.5;
        double r5436300 = r5436299 / r5436289;
        double r5436301 = r5436298 - r5436300;
        double r5436302 = r5436296 + r5436301;
        double r5436303 = log(r5436302);
        double r5436304 = 0.9626554548619399;
        bool r5436305 = r5436289 <= r5436304;
        double r5436306 = 0.075;
        double r5436307 = r5436295 * r5436306;
        double r5436308 = 0.16666666666666666;
        double r5436309 = r5436294 * r5436308;
        double r5436310 = r5436309 - r5436289;
        double r5436311 = r5436307 - r5436310;
        double r5436312 = -0.125;
        double r5436313 = r5436312 / r5436294;
        double r5436314 = r5436313 + r5436289;
        double r5436315 = r5436300 + r5436314;
        double r5436316 = r5436289 + r5436315;
        double r5436317 = log(r5436316);
        double r5436318 = r5436305 ? r5436311 : r5436317;
        double r5436319 = r5436291 ? r5436303 : r5436318;
        return r5436319;
}

Error

Bits error versus x

Target

Original	52.6
Target	44.2
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 < -1.0507751989105836

   1. Initial program 61.9

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

   2. Taylor expanded around -inf 0.1

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

   3. Simplified 0.1

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

   if -1.0507751989105836 < x < 0.9626554548619399

   1. Initial program 58.5

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

   2. Taylor expanded around 0 0.3

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

   3. Simplified 0.3

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

   if 0.9626554548619399 < x

   1. Initial program 30.6

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

   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

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0507751989105836:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9626554548619399:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{3}{40} - \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{6} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(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)))))