Asymptote C

Average Error: 29.3 → 0.1

Time: 19.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -11166.74680765492303180508315563201904297 \lor \neg \left(x \le 17625.54287631354236509650945663452148438\right):\\ \;\;\;\;\mathsf{fma}\left(-\frac{\sqrt{3}}{x}, \frac{\sqrt{3}}{x \cdot x}, -\frac{\frac{1}{x} + 3}{x}\right) + \left(\frac{1}{x} + \frac{-1}{x}\right) \cdot \left(\frac{1}{x} + 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{1 \cdot 1 - x \cdot x}, 1 - x, \frac{-\left(x + 1\right)}{x - 1}\right)\\ \end{array}\]

C

double f(double x) {
        double r175890 = x;
        double r175891 = 1.0;
        double r175892 = r175890 + r175891;
        double r175893 = r175890 / r175892;
        double r175894 = r175890 - r175891;
        double r175895 = r175892 / r175894;
        double r175896 = r175893 - r175895;
        return r175896;
}

↓

double f(double x) {
        double r175897 = x;
        double r175898 = -11166.746807654923;
        bool r175899 = r175897 <= r175898;
        double r175900 = 17625.542876313542;
        bool r175901 = r175897 <= r175900;
        double r175902 = !r175901;
        bool r175903 = r175899 || r175902;
        double r175904 = 3.0;
        double r175905 = sqrt(r175904);
        double r175906 = r175905 / r175897;
        double r175907 = -r175906;
        double r175908 = r175897 * r175897;
        double r175909 = r175905 / r175908;
        double r175910 = 1.0;
        double r175911 = r175910 / r175897;
        double r175912 = r175911 + r175904;
        double r175913 = r175912 / r175897;
        double r175914 = -r175913;
        double r175915 = fma(r175907, r175909, r175914);
        double r175916 = 1.0;
        double r175917 = r175916 / r175897;
        double r175918 = -1.0;
        double r175919 = r175918 / r175897;
        double r175920 = r175917 + r175919;
        double r175921 = r175920 * r175912;
        double r175922 = r175915 + r175921;
        double r175923 = r175910 * r175910;
        double r175924 = r175923 - r175908;
        double r175925 = r175897 / r175924;
        double r175926 = r175910 - r175897;
        double r175927 = r175897 + r175910;
        double r175928 = -r175927;
        double r175929 = r175897 - r175910;
        double r175930 = r175928 / r175929;
        double r175931 = fma(r175925, r175926, r175930);
        double r175932 = r175903 ? r175922 : r175931;
        return r175932;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -11166.746807654923 or 17625.542876313542 < x

   1. Initial program 59.4

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

   2. Simplified 59.4

      \[\leadsto \color{blue}{\frac{x}{1 + x} - \frac{1 + x}{x - 1}}\]

   3. Taylor expanded around inf 0.3

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

   4. Simplified 0.0

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

   6. Applied div-inv 0.0

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

   7. Applied div-inv 0.3

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

   8. Applied distribute-rgt-out 0.3

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

   9. Applied cube-mult 0.3

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

   10. Applied add-sqr-sqrt 0.3

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

   11. Applied times-frac 0.3

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

   12. Applied distribute-lft-neg-in 0.3

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

   13. Applied prod-diff 0.3

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

   14. Simplified 0.2

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

   15. Simplified 0.0

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

   if -11166.746807654923 < x < 17625.542876313542

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{x}{1 + x} - \frac{1 + x}{x - 1}}\]
   3. Using strategy rm

   4. Applied flip-+ 0.1

      \[\leadsto \frac{x}{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 - x}}} - \frac{1 + x}{x - 1}\]

   5. Applied associate-/r/ 0.1

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

   6. Applied fma-neg 0.1

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

   7. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11166.74680765492303180508315563201904297 \lor \neg \left(x \le 17625.54287631354236509650945663452148438\right):\\ \;\;\;\;\mathsf{fma}\left(-\frac{\sqrt{3}}{x}, \frac{\sqrt{3}}{x \cdot x}, -\frac{\frac{1}{x} + 3}{x}\right) + \left(\frac{1}{x} + \frac{-1}{x}\right) \cdot \left(\frac{1}{x} + 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{1 \cdot 1 - x \cdot x}, 1 - x, \frac{-\left(x + 1\right)}{x - 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))