Asymptote C

Average Error: 29.3 → 0.2

Time: 46.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -13193.961968205371 \lor \neg \left(x \le 9758.9774509112976\right):\\ \;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\frac{\mathsf{fma}\left(1, {\left({\left(\frac{x}{x + 1}\right)}^{3}\right)}^{3}, -{\left(\frac{x + 1}{x - 1}\right)}^{3} \cdot \left({\left(\frac{x + 1}{x - 1}\right)}^{3} \cdot {\left(\frac{x + 1}{x - 1}\right)}^{3}\right)\right) + \mathsf{fma}\left(-{\left(\frac{x + 1}{x - 1}\right)}^{3}, {\left(\frac{x + 1}{x - 1}\right)}^{3} \cdot {\left(\frac{x + 1}{x - 1}\right)}^{3}, {\left(\frac{x + 1}{x - 1}\right)}^{3} \cdot \left({\left(\frac{x + 1}{x - 1}\right)}^{3} \cdot {\left(\frac{x + 1}{x - 1}\right)}^{3}\right)\right)}{\mathsf{fma}\left({\left(\frac{x + 1}{x - 1}\right)}^{3}, {\left(\frac{x}{x + 1}\right)}^{3} + {\left(\frac{x + 1}{x - 1}\right)}^{3}, {\left(\frac{x}{x + 1}\right)}^{6}\right)}}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \frac{x}{x + 1} + \frac{x + 1}{x - 1}, \frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}\right)}^{3}}\\ \end{array}\]

C

double f(double x) {
        double r239313 = x;
        double r239314 = 1.0;
        double r239315 = r239313 + r239314;
        double r239316 = r239313 / r239315;
        double r239317 = r239313 - r239314;
        double r239318 = r239315 / r239317;
        double r239319 = r239316 - r239318;
        return r239319;
}

↓

double f(double x) {
        double r239320 = x;
        double r239321 = -13193.96196820537;
        bool r239322 = r239320 <= r239321;
        double r239323 = 9758.977450911298;
        bool r239324 = r239320 <= r239323;
        double r239325 = !r239324;
        bool r239326 = r239322 || r239325;
        double r239327 = 1.0;
        double r239328 = -r239327;
        double r239329 = 2.0;
        double r239330 = pow(r239320, r239329);
        double r239331 = r239328 / r239330;
        double r239332 = 3.0;
        double r239333 = 1.0;
        double r239334 = r239333 / r239320;
        double r239335 = 3.0;
        double r239336 = pow(r239320, r239335);
        double r239337 = r239333 / r239336;
        double r239338 = r239332 * r239337;
        double r239339 = fma(r239332, r239334, r239338);
        double r239340 = r239331 - r239339;
        double r239341 = r239320 + r239327;
        double r239342 = r239320 / r239341;
        double r239343 = pow(r239342, r239335);
        double r239344 = pow(r239343, r239335);
        double r239345 = r239320 - r239327;
        double r239346 = r239341 / r239345;
        double r239347 = pow(r239346, r239335);
        double r239348 = r239347 * r239347;
        double r239349 = r239347 * r239348;
        double r239350 = -r239349;
        double r239351 = fma(r239333, r239344, r239350);
        double r239352 = -r239347;
        double r239353 = fma(r239352, r239348, r239349);
        double r239354 = r239351 + r239353;
        double r239355 = r239343 + r239347;
        double r239356 = 6.0;
        double r239357 = pow(r239342, r239356);
        double r239358 = fma(r239347, r239355, r239357);
        double r239359 = r239354 / r239358;
        double r239360 = r239342 + r239346;
        double r239361 = r239342 * r239342;
        double r239362 = fma(r239346, r239360, r239361);
        double r239363 = r239359 / r239362;
        double r239364 = pow(r239363, r239335);
        double r239365 = cbrt(r239364);
        double r239366 = r239326 ? r239340 : r239365;
        return r239366;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -13193.96196820537 or 9758.977450911298 < x

   1. Initial program 59.3

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

   2. Taylor expanded around inf 0.3

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

   3. Simplified 0.3

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

   if -13193.96196820537 < x < 9758.977450911298

   1. Initial program 0.1

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

   3. Applied flip3-- 0.1

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

   4. Simplified 0.1

      \[\leadsto \frac{{\left(\frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \frac{x}{x + 1} + \frac{x + 1}{x - 1}, \frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}}\]
   5. Using strategy rm

   6. Applied add-cbrt-cube 0.1

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

   7. Applied add-cbrt-cube 0.1

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

   8. Applied cbrt-undiv 0.1

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

   9. Simplified 0.1

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{{\left(\frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \frac{x}{x + 1} + \frac{x + 1}{x - 1}, \frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}\right)}^{3}}}\]
   10. Using strategy rm

   11. Applied flip3-- 0.1

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

   12. Simplified 0.1

       \[\leadsto \sqrt[3]{{\left(\frac{\frac{{\left({\left(\frac{x}{x + 1}\right)}^{3}\right)}^{3} - {\left({\left(\frac{x + 1}{x - 1}\right)}^{3}\right)}^{3}}{\color{blue}{\mathsf{fma}\left({\left(\frac{x + 1}{x - 1}\right)}^{3}, {\left(\frac{x}{x + 1}\right)}^{3} + {\left(\frac{x + 1}{x - 1}\right)}^{3}, {\left(\frac{x}{x + 1}\right)}^{6}\right)}}}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \frac{x}{x + 1} + \frac{x + 1}{x - 1}, \frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}\right)}^{3}}\]
   13. Using strategy rm

   14. Applied unpow3 0.1

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

   15. Applied *-un-lft-identity 0.1

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

   16. Applied *-un-lft-identity 0.1

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

   17. Applied times-frac 0.1

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

   18. Applied unpow-prod-down 0.1

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

   19. Applied unpow-prod-down 0.1

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

   20. Applied prod-diff 0.1

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

   21. Simplified 0.1

       \[\leadsto \sqrt[3]{{\left(\frac{\frac{\color{blue}{\mathsf{fma}\left(1, {\left({\left(\frac{x}{x + 1}\right)}^{3}\right)}^{3}, -{\left(\frac{x + 1}{x - 1}\right)}^{3} \cdot \left({\left(\frac{x + 1}{x - 1}\right)}^{3} \cdot {\left(\frac{x + 1}{x - 1}\right)}^{3}\right)\right)} + \mathsf{fma}\left(-{\left(\frac{x + 1}{x - 1}\right)}^{3}, {\left(\frac{x + 1}{x - 1}\right)}^{3} \cdot {\left(\frac{x + 1}{x - 1}\right)}^{3}, {\left(\frac{x + 1}{x - 1}\right)}^{3} \cdot \left({\left(\frac{x + 1}{x - 1}\right)}^{3} \cdot {\left(\frac{x + 1}{x - 1}\right)}^{3}\right)\right)}{\mathsf{fma}\left({\left(\frac{x + 1}{x - 1}\right)}^{3}, {\left(\frac{x}{x + 1}\right)}^{3} + {\left(\frac{x + 1}{x - 1}\right)}^{3}, {\left(\frac{x}{x + 1}\right)}^{6}\right)}}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \frac{x}{x + 1} + \frac{x + 1}{x - 1}, \frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}\right)}^{3}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13193.961968205371 \lor \neg \left(x \le 9758.9774509112976\right):\\ \;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\frac{\mathsf{fma}\left(1, {\left({\left(\frac{x}{x + 1}\right)}^{3}\right)}^{3}, -{\left(\frac{x + 1}{x - 1}\right)}^{3} \cdot \left({\left(\frac{x + 1}{x - 1}\right)}^{3} \cdot {\left(\frac{x + 1}{x - 1}\right)}^{3}\right)\right) + \mathsf{fma}\left(-{\left(\frac{x + 1}{x - 1}\right)}^{3}, {\left(\frac{x + 1}{x - 1}\right)}^{3} \cdot {\left(\frac{x + 1}{x - 1}\right)}^{3}, {\left(\frac{x + 1}{x - 1}\right)}^{3} \cdot \left({\left(\frac{x + 1}{x - 1}\right)}^{3} \cdot {\left(\frac{x + 1}{x - 1}\right)}^{3}\right)\right)}{\mathsf{fma}\left({\left(\frac{x + 1}{x - 1}\right)}^{3}, {\left(\frac{x}{x + 1}\right)}^{3} + {\left(\frac{x + 1}{x - 1}\right)}^{3}, {\left(\frac{x}{x + 1}\right)}^{6}\right)}}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \frac{x}{x + 1} + \frac{x + 1}{x - 1}, \frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))