Asymptote C

Average Error: 28.6 → 0.2

Time: 1.2m

Precision: 64

Math

\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.008231828666988:\\ \;\;\;\;\frac{-3}{x} + \left(\frac{-3}{x \cdot \left(x \cdot x\right)} + \frac{-1}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 13687.362331495488:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \sqrt{1 + x}\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \sqrt{1 + x}\right) + \left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \sqrt{1 + x}\right) \cdot \sqrt{1 + x}\right)\right) \cdot \left(\left(\left(1 - x\right) + x \cdot x\right) \cdot \left(\frac{x}{\sqrt{1 + x}} \cdot \left(x \cdot x - 1\right)\right)\right) - \left(1 + x\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{3} + {\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \sqrt{1 + x}\right)}^{3}\right)}{\left(\left(\left(x - 1\right) \cdot \sqrt{1 + x}\right) \cdot \left(\left(1 - x\right) + x \cdot x\right)\right) \cdot \left(\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \sqrt{1 + x}\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \sqrt{1 + x}\right) + \left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \sqrt{1 + x}\right) \cdot \sqrt{1 + x}\right)\right) \cdot \left(1 + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \left(\frac{-3}{x \cdot \left(x \cdot x\right)} + \frac{-1}{x \cdot x}\right)\\ \end{array}\]

C

double f(double x) {
        double r7814278 = x;
        double r7814279 = 1.0;
        double r7814280 = r7814278 + r7814279;
        double r7814281 = r7814278 / r7814280;
        double r7814282 = r7814278 - r7814279;
        double r7814283 = r7814280 / r7814282;
        double r7814284 = r7814281 - r7814283;
        return r7814284;
}

↓

double f(double x) {
        double r7814285 = x;
        double r7814286 = -1.008231828666988;
        bool r7814287 = r7814285 <= r7814286;
        double r7814288 = -3.0;
        double r7814289 = r7814288 / r7814285;
        double r7814290 = r7814285 * r7814285;
        double r7814291 = r7814285 * r7814290;
        double r7814292 = r7814288 / r7814291;
        double r7814293 = -1.0;
        double r7814294 = r7814293 / r7814290;
        double r7814295 = r7814292 + r7814294;
        double r7814296 = r7814289 + r7814295;
        double r7814297 = 13687.362331495488;
        bool r7814298 = r7814285 <= r7814297;
        double r7814299 = 1.0;
        double r7814300 = r7814299 + r7814285;
        double r7814301 = sqrt(r7814300);
        double r7814302 = r7814291 * r7814301;
        double r7814303 = r7814302 * r7814302;
        double r7814304 = r7814301 * r7814301;
        double r7814305 = r7814302 * r7814301;
        double r7814306 = r7814304 - r7814305;
        double r7814307 = r7814303 + r7814306;
        double r7814308 = r7814299 - r7814285;
        double r7814309 = r7814308 + r7814290;
        double r7814310 = r7814285 / r7814301;
        double r7814311 = r7814290 - r7814299;
        double r7814312 = r7814310 * r7814311;
        double r7814313 = r7814309 * r7814312;
        double r7814314 = r7814307 * r7814313;
        double r7814315 = 3.0;
        double r7814316 = pow(r7814301, r7814315);
        double r7814317 = pow(r7814302, r7814315);
        double r7814318 = r7814316 + r7814317;
        double r7814319 = r7814300 * r7814318;
        double r7814320 = r7814314 - r7814319;
        double r7814321 = r7814285 - r7814299;
        double r7814322 = r7814321 * r7814301;
        double r7814323 = r7814322 * r7814309;
        double r7814324 = r7814307 * r7814300;
        double r7814325 = r7814323 * r7814324;
        double r7814326 = r7814320 / r7814325;
        double r7814327 = r7814298 ? r7814326 : r7814296;
        double r7814328 = r7814287 ? r7814296 : r7814327;
        return r7814328;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -1.008231828666988 or 13687.362331495488 < x

   1. Initial program 58.9

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]

   2. Taylor expanded around inf 0.5

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

   3. Simplified 0.2

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

   if -1.008231828666988 < x < 13687.362331495488

   1. Initial program 0.1

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
   2. Using strategy rm

   3. Applied flip3-+ 0.1

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

   4. Applied associate-/l/ 0.1

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

   5. Applied add-sqr-sqrt 0.1

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

   6. Applied associate-/r* 0.1

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

   7. Applied frac-sub 0.1

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

   8. Simplified 0.1

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

   9. Simplified 0.1

      \[\leadsto \frac{\left(\frac{x}{\sqrt{1 + x}} \cdot \left(x - 1\right)\right) \cdot \left(x \cdot x + \left(1 - x\right)\right) - \left(\sqrt{1 + x} \cdot \left(\left(x \cdot x\right) \cdot x\right) + \sqrt{1 + x}\right)}{\color{blue}{\left(\sqrt{1 + x} \cdot \left(x - 1\right)\right) \cdot \left(x \cdot x + \left(1 - x\right)\right)}}\]
   10. Using strategy rm

   11. Applied flip3-+ 0.1

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

   12. Applied flip-- 0.1

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

   13. Applied associate-*r/ 0.1

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

   14. Applied associate-*l/ 0.1

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

   15. Applied frac-sub 0.1

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

   16. Applied associate-/l/ 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.008231828666988:\\ \;\;\;\;\frac{-3}{x} + \left(\frac{-3}{x \cdot \left(x \cdot x\right)} + \frac{-1}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 13687.362331495488:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \sqrt{1 + x}\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \sqrt{1 + x}\right) + \left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \sqrt{1 + x}\right) \cdot \sqrt{1 + x}\right)\right) \cdot \left(\left(\left(1 - x\right) + x \cdot x\right) \cdot \left(\frac{x}{\sqrt{1 + x}} \cdot \left(x \cdot x - 1\right)\right)\right) - \left(1 + x\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{3} + {\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \sqrt{1 + x}\right)}^{3}\right)}{\left(\left(\left(x - 1\right) \cdot \sqrt{1 + x}\right) \cdot \left(\left(1 - x\right) + x \cdot x\right)\right) \cdot \left(\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \sqrt{1 + x}\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \sqrt{1 + x}\right) + \left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \sqrt{1 + x}\right) \cdot \sqrt{1 + x}\right)\right) \cdot \left(1 + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \left(\frac{-3}{x \cdot \left(x \cdot x\right)} + \frac{-1}{x \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))