Asymptote A

Average Error: 14.4 → 0.1

Time: 15.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -197.6959578773428347631124779582023620605 \lor \neg \left(x \le 221.5934540176410223466518800705671310425\right):\\ \;\;\;\;\left(-\frac{2}{{x}^{6}}\right) - \left(\frac{\frac{2}{x}}{x} + \frac{2}{{x}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{1}{{x}^{3} - {1}^{3}}, \mathsf{fma}\left(x, x, \left(x + 1\right) \cdot 1\right), \frac{1}{x + 1}\right) + \frac{1}{{x}^{3} - {1}^{3}} \cdot \left(\left(-\mathsf{fma}\left(x, x, \left(x + 1\right) \cdot 1\right)\right) + \mathsf{fma}\left(x, x, \left(x + 1\right) \cdot 1\right)\right)\\ \end{array}\]

C

double f(double x) {
        double r121661 = 1.0;
        double r121662 = x;
        double r121663 = r121662 + r121661;
        double r121664 = r121661 / r121663;
        double r121665 = r121662 - r121661;
        double r121666 = r121661 / r121665;
        double r121667 = r121664 - r121666;
        return r121667;
}

↓

double f(double x) {
        double r121668 = x;
        double r121669 = -197.69595787734283;
        bool r121670 = r121668 <= r121669;
        double r121671 = 221.59345401764102;
        bool r121672 = r121668 <= r121671;
        double r121673 = !r121672;
        bool r121674 = r121670 || r121673;
        double r121675 = 2.0;
        double r121676 = 6.0;
        double r121677 = pow(r121668, r121676);
        double r121678 = r121675 / r121677;
        double r121679 = -r121678;
        double r121680 = r121675 / r121668;
        double r121681 = r121680 / r121668;
        double r121682 = 4.0;
        double r121683 = pow(r121668, r121682);
        double r121684 = r121675 / r121683;
        double r121685 = r121681 + r121684;
        double r121686 = r121679 - r121685;
        double r121687 = 1.0;
        double r121688 = 3.0;
        double r121689 = pow(r121668, r121688);
        double r121690 = pow(r121687, r121688);
        double r121691 = r121689 - r121690;
        double r121692 = r121687 / r121691;
        double r121693 = -r121692;
        double r121694 = r121668 + r121687;
        double r121695 = r121694 * r121687;
        double r121696 = fma(r121668, r121668, r121695);
        double r121697 = r121687 / r121694;
        double r121698 = fma(r121693, r121696, r121697);
        double r121699 = -r121696;
        double r121700 = r121699 + r121696;
        double r121701 = r121692 * r121700;
        double r121702 = r121698 + r121701;
        double r121703 = r121674 ? r121686 : r121702;
        return r121703;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -197.69595787734283 or 221.59345401764102 < x

   1. Initial program 29.1

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

   2. Taylor expanded around inf 0.8

      \[\leadsto \color{blue}{-\left(2 \cdot \frac{1}{{x}^{6}} + \left(2 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{{x}^{4}}\right)\right)}\]

   3. Simplified 0.8

      \[\leadsto \color{blue}{\left(-\frac{2}{{x}^{6}}\right) - \left(\frac{2}{x \cdot x} + \frac{2}{{x}^{4}}\right)}\]
   4. Using strategy rm

   5. Applied associate-/r* 0.1

      \[\leadsto \left(-\frac{2}{{x}^{6}}\right) - \left(\color{blue}{\frac{\frac{2}{x}}{x}} + \frac{2}{{x}^{4}}\right)\]

   if -197.69595787734283 < x < 221.59345401764102

   1. Initial program 0.0

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

   3. Applied flip3-- 0.0

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

   4. Applied associate-/r/ 0.0

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

   5. Applied add-sqr-sqrt 0.2

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

   6. Applied prod-diff 0.2

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

   7. Simplified 0.0

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

   8. Simplified 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -197.6959578773428347631124779582023620605 \lor \neg \left(x \le 221.5934540176410223466518800705671310425\right):\\ \;\;\;\;\left(-\frac{2}{{x}^{6}}\right) - \left(\frac{\frac{2}{x}}{x} + \frac{2}{{x}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{1}{{x}^{3} - {1}^{3}}, \mathsf{fma}\left(x, x, \left(x + 1\right) \cdot 1\right), \frac{1}{x + 1}\right) + \frac{1}{{x}^{3} - {1}^{3}} \cdot \left(\left(-\mathsf{fma}\left(x, x, \left(x + 1\right) \cdot 1\right)\right) + \mathsf{fma}\left(x, x, \left(x + 1\right) \cdot 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))