Asymptote C

Average Error: 29.1 → 0.1

Time: 17.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -10364.72278293059571296907961368560791016 \lor \neg \left(x \le 12015.49448950132136815227568149566650391\right):\\ \;\;\;\;-\left(\left(\frac{1}{x \cdot x} + \frac{3}{x}\right) + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{x}{{x}^{3} + {1}^{3}}\right)}^{3}, {\left(\mathsf{fma}\left(x, x, 1 \cdot \left(1 - x\right)\right)\right)}^{3}, \frac{{\left(x + 1\right)}^{3}}{x - 1} \cdot \frac{-1}{\left(x - 1\right) \cdot \left(x - 1\right)}\right) + \frac{{\left(x + 1\right)}^{3}}{x - 1} \cdot \left(\frac{-1}{\left(x - 1\right) \cdot \left(x - 1\right)} + \frac{1}{\left(x - 1\right) \cdot \left(x - 1\right)}\right)}{\mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x}, \mathsf{fma}\left(\frac{x + 1}{x - 1}, \frac{x + 1}{x - 1}, \frac{x}{x - 1}\right)\right)}\\ \end{array}\]

C

double f(double x) {
        double r94855 = x;
        double r94856 = 1.0;
        double r94857 = r94855 + r94856;
        double r94858 = r94855 / r94857;
        double r94859 = r94855 - r94856;
        double r94860 = r94857 / r94859;
        double r94861 = r94858 - r94860;
        return r94861;
}

↓

double f(double x) {
        double r94862 = x;
        double r94863 = -10364.722782930596;
        bool r94864 = r94862 <= r94863;
        double r94865 = 12015.494489501321;
        bool r94866 = r94862 <= r94865;
        double r94867 = !r94866;
        bool r94868 = r94864 || r94867;
        double r94869 = 1.0;
        double r94870 = r94862 * r94862;
        double r94871 = r94869 / r94870;
        double r94872 = 3.0;
        double r94873 = r94872 / r94862;
        double r94874 = r94871 + r94873;
        double r94875 = 3.0;
        double r94876 = pow(r94862, r94875);
        double r94877 = r94872 / r94876;
        double r94878 = r94874 + r94877;
        double r94879 = -r94878;
        double r94880 = pow(r94869, r94875);
        double r94881 = r94876 + r94880;
        double r94882 = r94862 / r94881;
        double r94883 = pow(r94882, r94875);
        double r94884 = r94869 - r94862;
        double r94885 = r94869 * r94884;
        double r94886 = fma(r94862, r94862, r94885);
        double r94887 = pow(r94886, r94875);
        double r94888 = r94862 + r94869;
        double r94889 = pow(r94888, r94875);
        double r94890 = r94862 - r94869;
        double r94891 = r94889 / r94890;
        double r94892 = -1.0;
        double r94893 = r94890 * r94890;
        double r94894 = r94892 / r94893;
        double r94895 = r94891 * r94894;
        double r94896 = fma(r94883, r94887, r94895);
        double r94897 = 1.0;
        double r94898 = r94897 / r94893;
        double r94899 = r94894 + r94898;
        double r94900 = r94891 * r94899;
        double r94901 = r94896 + r94900;
        double r94902 = r94869 + r94862;
        double r94903 = r94862 / r94902;
        double r94904 = r94888 / r94890;
        double r94905 = r94862 / r94890;
        double r94906 = fma(r94904, r94904, r94905);
        double r94907 = fma(r94903, r94903, r94906);
        double r94908 = r94901 / r94907;
        double r94909 = r94868 ? r94879 : r94908;
        return r94909;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -10364.722782930596 or 12015.494489501321 < x

   1. Initial program 59.2

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

   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 -10364.722782930596 < x < 12015.494489501321

   1. Initial program 0.1

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

   3. Applied flip3-- 0.1

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

   4. Simplified 0.1

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

   6. Applied add-cube-cbrt 0.2

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

   7. Applied *-un-lft-identity 0.2

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

   8. Applied times-frac 0.2

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

   9. Applied unpow-prod-down 0.2

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

   10. Applied flip3-+ 0.2

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

   11. Applied associate-/r/ 0.2

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

   12. Applied unpow-prod-down 0.2

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

   13. Applied prod-diff 0.2

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

   14. Simplified 0.1

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

   15. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10364.72278293059571296907961368560791016 \lor \neg \left(x \le 12015.49448950132136815227568149566650391\right):\\ \;\;\;\;-\left(\left(\frac{1}{x \cdot x} + \frac{3}{x}\right) + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{x}{{x}^{3} + {1}^{3}}\right)}^{3}, {\left(\mathsf{fma}\left(x, x, 1 \cdot \left(1 - x\right)\right)\right)}^{3}, \frac{{\left(x + 1\right)}^{3}}{x - 1} \cdot \frac{-1}{\left(x - 1\right) \cdot \left(x - 1\right)}\right) + \frac{{\left(x + 1\right)}^{3}}{x - 1} \cdot \left(\frac{-1}{\left(x - 1\right) \cdot \left(x - 1\right)} + \frac{1}{\left(x - 1\right) \cdot \left(x - 1\right)}\right)}{\mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x}, \mathsf{fma}\left(\frac{x + 1}{x - 1}, \frac{x + 1}{x - 1}, \frac{x}{x - 1}\right)\right)}\\ \end{array}\]

Reproduce

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