Asymptote C

Average Error: 29.7 → 0.1

Time: 26.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -15309.12152708683424862101674079895019531 \lor \neg \left(x \le 12455.49900284703107899986207485198974609\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{{x}^{3}} + \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}{\mathsf{fma}\left(\frac{x}{x + 1}, \frac{x}{x + 1}, \frac{x + 1}{x - 1} \cdot \left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right)\right)} \cdot \mathsf{fma}\left(\frac{x + 1}{x - 1}, \frac{x}{x + 1} + \frac{x + 1}{x - 1}, {\left(\sqrt[3]{\frac{x}{x + 1}}\right)}^{6}\right)\\ \end{array}\]

C

double f(double x) {
        double r129121 = x;
        double r129122 = 1.0;
        double r129123 = r129121 + r129122;
        double r129124 = r129121 / r129123;
        double r129125 = r129121 - r129122;
        double r129126 = r129123 / r129125;
        double r129127 = r129124 - r129126;
        return r129127;
}

↓

double f(double x) {
        double r129128 = x;
        double r129129 = -15309.121527086834;
        bool r129130 = r129128 <= r129129;
        double r129131 = 12455.499002847031;
        bool r129132 = r129128 <= r129131;
        double r129133 = !r129132;
        bool r129134 = r129130 || r129133;
        double r129135 = 1.0;
        double r129136 = -r129135;
        double r129137 = r129128 * r129128;
        double r129138 = r129136 / r129137;
        double r129139 = 3.0;
        double r129140 = 3.0;
        double r129141 = pow(r129128, r129140);
        double r129142 = r129139 / r129141;
        double r129143 = r129139 / r129128;
        double r129144 = r129142 + r129143;
        double r129145 = r129138 - r129144;
        double r129146 = r129128 + r129135;
        double r129147 = r129128 / r129146;
        double r129148 = r129128 - r129135;
        double r129149 = r129146 / r129148;
        double r129150 = r129147 - r129149;
        double r129151 = r129147 + r129149;
        double r129152 = r129149 * r129151;
        double r129153 = fma(r129147, r129147, r129152);
        double r129154 = r129150 / r129153;
        double r129155 = cbrt(r129147);
        double r129156 = 6.0;
        double r129157 = pow(r129155, r129156);
        double r129158 = fma(r129149, r129151, r129157);
        double r129159 = r129154 * r129158;
        double r129160 = r129134 ? r129145 : r129159;
        return r129160;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -15309.121527086834 or 12455.499002847031 < x

   1. Initial program 59.4

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

   3. Applied add-log-exp 59.4

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

   4. Applied add-log-exp 59.4

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

   5. Applied diff-log 59.4

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

   6. Simplified 59.4

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

   7. Taylor expanded around inf 0.3

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

   8. Simplified 0.0

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

   if -15309.121527086834 < x < 12455.499002847031

   1. Initial program 0.1

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

   3. Applied add-log-exp 0.1

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

   4. Applied add-log-exp 0.1

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

   5. Applied diff-log 0.1

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

   6. Simplified 0.1

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

   8. Applied flip3-- 0.1

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

   9. Simplified 0.1

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

   11. Applied *-un-lft-identity 0.1

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

   12. Applied difference-cubes 0.1

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

   13. Applied times-frac 0.1

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

   14. Applied exp-prod 0.1

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

   15. Applied log-pow 0.1

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

   16. Simplified 0.1

       \[\leadsto \frac{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}{\mathsf{fma}\left(\frac{x}{x + 1}, \frac{x}{x + 1}, \frac{x + 1}{x - 1} \cdot \left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right)\right)} \cdot \color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \frac{x}{x + 1} + \frac{x + 1}{x - 1}, {\left(\sqrt[3]{\frac{x}{x + 1}}\right)}^{6}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -15309.12152708683424862101674079895019531 \lor \neg \left(x \le 12455.49900284703107899986207485198974609\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{{x}^{3}} + \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}{\mathsf{fma}\left(\frac{x}{x + 1}, \frac{x}{x + 1}, \frac{x + 1}{x - 1} \cdot \left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right)\right)} \cdot \mathsf{fma}\left(\frac{x + 1}{x - 1}, \frac{x}{x + 1} + \frac{x + 1}{x - 1}, {\left(\sqrt[3]{\frac{x}{x + 1}}\right)}^{6}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))