Asymptote C

Average Error: 29.4 → 0.1

Time: 8.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -11931.4119690201605 \lor \neg \left(x \le 12679.85899306799\right):\\ \;\;\;\;\left(-\frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right) - \left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right)\right)}\\ \end{array}\]

C

double f(double x) {
        double r173746 = x;
        double r173747 = 1.0;
        double r173748 = r173746 + r173747;
        double r173749 = r173746 / r173748;
        double r173750 = r173746 - r173747;
        double r173751 = r173748 / r173750;
        double r173752 = r173749 - r173751;
        return r173752;
}

↓

double f(double x) {
        double r173753 = x;
        double r173754 = -11931.41196902016;
        bool r173755 = r173753 <= r173754;
        double r173756 = 12679.85899306799;
        bool r173757 = r173753 <= r173756;
        double r173758 = !r173757;
        bool r173759 = r173755 || r173758;
        double r173760 = 1.0;
        double r173761 = r173753 * r173753;
        double r173762 = r173760 / r173761;
        double r173763 = -r173762;
        double r173764 = 3.0;
        double r173765 = 3.0;
        double r173766 = pow(r173753, r173765);
        double r173767 = r173764 / r173766;
        double r173768 = r173764 / r173753;
        double r173769 = r173767 + r173768;
        double r173770 = r173763 - r173769;
        double r173771 = r173753 - r173760;
        double r173772 = r173771 * r173771;
        double r173773 = r173761 * r173772;
        double r173774 = r173753 + r173760;
        double r173775 = r173774 * r173774;
        double r173776 = r173775 * r173775;
        double r173777 = r173773 - r173776;
        double r173778 = r173753 / r173774;
        double r173779 = r173774 / r173771;
        double r173780 = r173778 + r173779;
        double r173781 = r173775 * r173772;
        double r173782 = r173780 * r173781;
        double r173783 = r173777 / r173782;
        double r173784 = r173759 ? r173770 : r173783;
        return r173784;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -11931.41196902016 or 12679.85899306799 < 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 0.0

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

   if -11931.41196902016 < x < 12679.85899306799

   1. Initial program 0.1

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

   3. Applied flip-- 0.1

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

   5. Applied frac-times 0.1

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

   6. Applied frac-times 0.1

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

   7. Applied frac-sub 0.1

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

   8. Applied associate-/l/ 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11931.4119690201605 \lor \neg \left(x \le 12679.85899306799\right):\\ \;\;\;\;\left(-\frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right) - \left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))