Asymptote C

Average Error: 29.0 → 0.1

Time: 22.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -20084.46570443186283227987587451934814453 \lor \neg \left(x \le 12201.68751023736513161566108465194702148\right):\\ \;\;\;\;-\left(\left(\frac{3}{x} + \frac{3}{{x}^{3}}\right) + \frac{1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{x + 1}, \left(-\mathsf{fma}\left(x, x, 1 \cdot \left(1 + x\right)\right)\right) \cdot \frac{1 + x}{{x}^{3} - {1}^{3}}\right) + \mathsf{fma}\left(-\mathsf{fma}\left(x, x, 1 \cdot \left(1 + x\right)\right), \frac{1 + x}{{x}^{3} - {1}^{3}}, \mathsf{fma}\left(x, x, 1 \cdot \left(1 + x\right)\right) \cdot \frac{1 + x}{{x}^{3} - {1}^{3}}\right)\\ \end{array}\]

C

double f(double x) {
        double r96006 = x;
        double r96007 = 1.0;
        double r96008 = r96006 + r96007;
        double r96009 = r96006 / r96008;
        double r96010 = r96006 - r96007;
        double r96011 = r96008 / r96010;
        double r96012 = r96009 - r96011;
        return r96012;
}

↓

double f(double x) {
        double r96013 = x;
        double r96014 = -20084.465704431863;
        bool r96015 = r96013 <= r96014;
        double r96016 = 12201.687510237365;
        bool r96017 = r96013 <= r96016;
        double r96018 = !r96017;
        bool r96019 = r96015 || r96018;
        double r96020 = 3.0;
        double r96021 = r96020 / r96013;
        double r96022 = 3.0;
        double r96023 = pow(r96013, r96022);
        double r96024 = r96020 / r96023;
        double r96025 = r96021 + r96024;
        double r96026 = 1.0;
        double r96027 = r96013 * r96013;
        double r96028 = r96026 / r96027;
        double r96029 = r96025 + r96028;
        double r96030 = -r96029;
        double r96031 = 1.0;
        double r96032 = r96013 + r96026;
        double r96033 = r96031 / r96032;
        double r96034 = r96026 + r96013;
        double r96035 = r96026 * r96034;
        double r96036 = fma(r96013, r96013, r96035);
        double r96037 = -r96036;
        double r96038 = pow(r96026, r96022);
        double r96039 = r96023 - r96038;
        double r96040 = r96034 / r96039;
        double r96041 = r96037 * r96040;
        double r96042 = fma(r96013, r96033, r96041);
        double r96043 = r96036 * r96040;
        double r96044 = fma(r96037, r96040, r96043);
        double r96045 = r96042 + r96044;
        double r96046 = r96019 ? r96030 : r96045;
        return r96046;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -20084.465704431863 or 12201.687510237365 < x

   1. Initial program 59.2

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

   2. Simplified 59.2

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

   4. Applied add-cbrt-cube 60.4

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

   5. Applied add-cbrt-cube 62.0

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

   6. Applied cbrt-undiv 62.0

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

   7. Simplified 59.2

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

   8. Taylor expanded around inf 0.3

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

   9. Simplified 0.0

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

   if -20084.465704431863 < x < 12201.687510237365

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   4. Applied add-cbrt-cube 0.1

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

   5. Applied add-cbrt-cube 0.1

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

   6. Applied cbrt-undiv 0.1

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

   7. Simplified 0.1

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

   9. Applied flip3-- 0.1

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

   10. Applied associate-/r/ 0.1

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

   11. Applied div-inv 0.1

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

   12. Applied unpow-prod-down 0.1

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

   13. Applied cbrt-prod 0.1

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

   14. Applied prod-diff 0.1

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

   15. Simplified 0.1

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

   16. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -20084.46570443186283227987587451934814453 \lor \neg \left(x \le 12201.68751023736513161566108465194702148\right):\\ \;\;\;\;-\left(\left(\frac{3}{x} + \frac{3}{{x}^{3}}\right) + \frac{1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{x + 1}, \left(-\mathsf{fma}\left(x, x, 1 \cdot \left(1 + x\right)\right)\right) \cdot \frac{1 + x}{{x}^{3} - {1}^{3}}\right) + \mathsf{fma}\left(-\mathsf{fma}\left(x, x, 1 \cdot \left(1 + x\right)\right), \frac{1 + x}{{x}^{3} - {1}^{3}}, \mathsf{fma}\left(x, x, 1 \cdot \left(1 + x\right)\right) \cdot \frac{1 + x}{{x}^{3} - {1}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))