Asymptote C

Average Error: 29.7 → 0.1

Time: 17.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -13754.17570154071472643408924341201782227 \lor \neg \left(x \le 11917.97074271185010729823261499404907227\right):\\ \;\;\;\;-\left(\left(\frac{1}{x \cdot x} + \frac{3}{x}\right) + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{1 + x}, \frac{-\left(1 + x\right)}{x - 1}\right) + \left(\frac{1 + x}{{\left(\sqrt[3]{x - 1}\right)}^{3}} + \left(-\frac{1 + x}{{\left(\sqrt[3]{x - 1}\right)}^{3}}\right)\right)\\ \end{array}\]

C

double f(double x) {
        double r96698 = x;
        double r96699 = 1.0;
        double r96700 = r96698 + r96699;
        double r96701 = r96698 / r96700;
        double r96702 = r96698 - r96699;
        double r96703 = r96700 / r96702;
        double r96704 = r96701 - r96703;
        return r96704;
}

↓

double f(double x) {
        double r96705 = x;
        double r96706 = -13754.175701540715;
        bool r96707 = r96705 <= r96706;
        double r96708 = 11917.97074271185;
        bool r96709 = r96705 <= r96708;
        double r96710 = !r96709;
        bool r96711 = r96707 || r96710;
        double r96712 = 1.0;
        double r96713 = r96705 * r96705;
        double r96714 = r96712 / r96713;
        double r96715 = 3.0;
        double r96716 = r96715 / r96705;
        double r96717 = r96714 + r96716;
        double r96718 = 3.0;
        double r96719 = pow(r96705, r96718);
        double r96720 = r96715 / r96719;
        double r96721 = r96717 + r96720;
        double r96722 = -r96721;
        double r96723 = 1.0;
        double r96724 = r96712 + r96705;
        double r96725 = r96723 / r96724;
        double r96726 = -r96724;
        double r96727 = r96705 - r96712;
        double r96728 = r96726 / r96727;
        double r96729 = fma(r96705, r96725, r96728);
        double r96730 = cbrt(r96727);
        double r96731 = pow(r96730, r96718);
        double r96732 = r96724 / r96731;
        double r96733 = -r96732;
        double r96734 = r96732 + r96733;
        double r96735 = r96729 + r96734;
        double r96736 = r96711 ? r96722 : r96735;
        return r96736;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -13754.175701540715 or 11917.97074271185 < 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

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

   if -13754.175701540715 < x < 11917.97074271185

   1. Initial program 0.1

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

   3. Applied add-cube-cbrt 0.2

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

   4. Applied add-cube-cbrt 0.2

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

   5. Applied times-frac 0.2

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

   6. Applied add-sqr-sqrt 31.9

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

   7. Applied prod-diff 31.9

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

   8. Simplified 0.2

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

   9. Simplified 0.2

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

   11. Applied div-inv 0.2

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

   12. Applied fma-neg 0.2

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

   13. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13754.17570154071472643408924341201782227 \lor \neg \left(x \le 11917.97074271185010729823261499404907227\right):\\ \;\;\;\;-\left(\left(\frac{1}{x \cdot x} + \frac{3}{x}\right) + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{1 + x}, \frac{-\left(1 + x\right)}{x - 1}\right) + \left(\frac{1 + x}{{\left(\sqrt[3]{x - 1}\right)}^{3}} + \left(-\frac{1 + x}{{\left(\sqrt[3]{x - 1}\right)}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))