Asymptote C

Average Error: 29.5 → 0.1

Time: 15.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -17797.21049870981732965447008609771728516:\\ \;\;\;\;\left(\left(-\frac{\frac{3}{x}}{x \cdot x}\right) - \frac{1}{x \cdot x}\right) - \frac{3}{x}\\ \mathbf{elif}\;x \le 10408.71559010302371461875736713409423828:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{x + 1}, \frac{-\left(x + 1\right)}{x - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\frac{\frac{3}{x}}{x \cdot x}\right) - \frac{1}{x \cdot x}\right) - \frac{3}{x}\\ \end{array}\]

C

double f(double x) {
        double r4851035 = x;
        double r4851036 = 1.0;
        double r4851037 = r4851035 + r4851036;
        double r4851038 = r4851035 / r4851037;
        double r4851039 = r4851035 - r4851036;
        double r4851040 = r4851037 / r4851039;
        double r4851041 = r4851038 - r4851040;
        return r4851041;
}

↓

double f(double x) {
        double r4851042 = x;
        double r4851043 = -17797.210498709817;
        bool r4851044 = r4851042 <= r4851043;
        double r4851045 = 3.0;
        double r4851046 = r4851045 / r4851042;
        double r4851047 = r4851042 * r4851042;
        double r4851048 = r4851046 / r4851047;
        double r4851049 = -r4851048;
        double r4851050 = 1.0;
        double r4851051 = r4851050 / r4851047;
        double r4851052 = r4851049 - r4851051;
        double r4851053 = r4851052 - r4851046;
        double r4851054 = 10408.715590103024;
        bool r4851055 = r4851042 <= r4851054;
        double r4851056 = 1.0;
        double r4851057 = r4851042 + r4851050;
        double r4851058 = r4851056 / r4851057;
        double r4851059 = -r4851057;
        double r4851060 = r4851042 - r4851050;
        double r4851061 = r4851059 / r4851060;
        double r4851062 = fma(r4851042, r4851058, r4851061);
        double r4851063 = r4851055 ? r4851062 : r4851053;
        double r4851064 = r4851044 ? r4851053 : r4851063;
        return r4851064;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -17797.210498709817 or 10408.715590103024 < x

   1. Initial program 59.3

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

   3. Applied div-inv 59.5

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

   4. Applied fma-neg 60.3

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

   5. Taylor expanded around inf 0.3

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

   6. Simplified 0.0

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

   8. Applied associate--r+ 0.0

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

   if -17797.210498709817 < x < 10408.715590103024

   1. Initial program 0.1

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

   3. Applied div-inv 0.1

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

   4. Applied fma-neg 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -17797.21049870981732965447008609771728516:\\ \;\;\;\;\left(\left(-\frac{\frac{3}{x}}{x \cdot x}\right) - \frac{1}{x \cdot x}\right) - \frac{3}{x}\\ \mathbf{elif}\;x \le 10408.71559010302371461875736713409423828:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{x + 1}, \frac{-\left(x + 1\right)}{x - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\frac{\frac{3}{x}}{x \cdot x}\right) - \frac{1}{x \cdot x}\right) - \frac{3}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))