Asymptote C

Average Error: 29.4 → 0.1

Time: 13.2s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -10924.7946830091769 \lor \neg \left(x \le 10746.1267471036172\right):\\ \;\;\;\;\mathsf{fma}\left(-\frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt[3]{x}}, \frac{\frac{\sqrt{1}}{x}}{\sqrt[3]{x}}, \frac{3}{x} \cdot \left(-1 + \frac{\frac{-1}{x}}{x}\right)\right) + \mathsf{fma}\left(-1 + \frac{\frac{-1}{x}}{x}, \frac{3}{x}, \frac{3}{x} \cdot \left(1 + \frac{1}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{x \cdot x - 1 \cdot 1}, x - 1, -\left(x + 1\right) \cdot \frac{x + 1}{x \cdot x - 1 \cdot 1}\right) + \frac{x + 1}{x \cdot x - 1 \cdot 1} \cdot \left(\left(-\left(x + 1\right)\right) + \left(x + 1\right)\right)\\ \end{array}\]

C

double f(double x) {
        double r147538 = x;
        double r147539 = 1.0;
        double r147540 = r147538 + r147539;
        double r147541 = r147538 / r147540;
        double r147542 = r147538 - r147539;
        double r147543 = r147540 / r147542;
        double r147544 = r147541 - r147543;
        return r147544;
}

↓

double f(double x) {
        double r147545 = x;
        double r147546 = -10924.794683009177;
        bool r147547 = r147545 <= r147546;
        double r147548 = 10746.126747103617;
        bool r147549 = r147545 <= r147548;
        double r147550 = !r147549;
        bool r147551 = r147547 || r147550;
        double r147552 = 1.0;
        double r147553 = sqrt(r147552);
        double r147554 = cbrt(r147545);
        double r147555 = r147553 / r147554;
        double r147556 = r147555 / r147554;
        double r147557 = -r147556;
        double r147558 = r147553 / r147545;
        double r147559 = r147558 / r147554;
        double r147560 = 3.0;
        double r147561 = r147560 / r147545;
        double r147562 = -1.0;
        double r147563 = r147562 / r147545;
        double r147564 = r147563 / r147545;
        double r147565 = r147562 + r147564;
        double r147566 = r147561 * r147565;
        double r147567 = fma(r147557, r147559, r147566);
        double r147568 = 1.0;
        double r147569 = r147545 * r147545;
        double r147570 = r147568 / r147569;
        double r147571 = r147568 + r147570;
        double r147572 = r147561 * r147571;
        double r147573 = fma(r147565, r147561, r147572);
        double r147574 = r147567 + r147573;
        double r147575 = r147552 * r147552;
        double r147576 = r147569 - r147575;
        double r147577 = r147545 / r147576;
        double r147578 = r147545 - r147552;
        double r147579 = r147545 + r147552;
        double r147580 = r147579 / r147576;
        double r147581 = r147579 * r147580;
        double r147582 = -r147581;
        double r147583 = fma(r147577, r147578, r147582);
        double r147584 = -r147579;
        double r147585 = r147584 + r147579;
        double r147586 = r147580 * r147585;
        double r147587 = r147583 + r147586;
        double r147588 = r147551 ? r147574 : r147587;
        return r147588;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -10924.794683009177 or 10746.126747103617 < 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

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

   5. Applied div-inv 0.0

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

   6. Applied div-inv 0.3

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

   7. Applied distribute-lft-out 0.3

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

   8. Applied *-un-lft-identity 0.3

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

   9. Applied add-cube-cbrt 0.3

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

   10. Applied add-sqr-sqrt 0.3

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

   11. Applied times-frac 0.3

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

   12. Applied times-frac 0.3

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

   13. Applied distribute-lft-neg-in 0.3

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

   14. Applied prod-diff 0.3

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

   15. Simplified 0.2

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

   16. Simplified 0.0

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

   if -10924.794683009177 < x < 10746.126747103617

   1. Initial program 0.1

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

   3. Applied flip-- 0.1

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

   4. Applied associate-/r/ 0.1

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

   5. Applied flip-+ 0.1

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

   6. Applied associate-/r/ 0.1

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

   7. Applied prod-diff 0.1

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

   8. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10924.7946830091769 \lor \neg \left(x \le 10746.1267471036172\right):\\ \;\;\;\;\mathsf{fma}\left(-\frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt[3]{x}}, \frac{\frac{\sqrt{1}}{x}}{\sqrt[3]{x}}, \frac{3}{x} \cdot \left(-1 + \frac{\frac{-1}{x}}{x}\right)\right) + \mathsf{fma}\left(-1 + \frac{\frac{-1}{x}}{x}, \frac{3}{x}, \frac{3}{x} \cdot \left(1 + \frac{1}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{x \cdot x - 1 \cdot 1}, x - 1, -\left(x + 1\right) \cdot \frac{x + 1}{x \cdot x - 1 \cdot 1}\right) + \frac{x + 1}{x \cdot x - 1 \cdot 1} \cdot \left(\left(-\left(x + 1\right)\right) + \left(x + 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))