Average Error: 0.0 → 0.0
Time: 5.9s
Precision: 64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{\mathsf{fma}\left(2 - \frac{2}{t + 1}, 2 - \frac{2}{t + 1}, 1\right)}{\mathsf{fma}\left(2 - \frac{2}{t + 1}, 2 - \frac{2}{t + 1}, 2\right)}\]
\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
\frac{\mathsf{fma}\left(2 - \frac{2}{t + 1}, 2 - \frac{2}{t + 1}, 1\right)}{\mathsf{fma}\left(2 - \frac{2}{t + 1}, 2 - \frac{2}{t + 1}, 2\right)}
double f(double t) {
        double r607337 = 1.0;
        double r607338 = 2.0;
        double r607339 = t;
        double r607340 = r607338 / r607339;
        double r607341 = r607337 / r607339;
        double r607342 = r607337 + r607341;
        double r607343 = r607340 / r607342;
        double r607344 = r607338 - r607343;
        double r607345 = r607344 * r607344;
        double r607346 = r607337 + r607345;
        double r607347 = r607338 + r607345;
        double r607348 = r607346 / r607347;
        return r607348;
}
↓
double f(double t) {
        double r607349 = 2.0;
        double r607350 = t;
        double r607351 = 1.0;
        double r607352 = r607350 + r607351;
        double r607353 = r607349 / r607352;
        double r607354 = r607349 - r607353;
        double r607355 = fma(r607354, r607354, r607351);
        double r607356 = fma(r607354, r607354, r607349);
        double r607357 = r607355 / r607356;
        return r607357;
}

Error

Bits error versus t

Derivation

  1. Initial program 0.0

    \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2 - \frac{2}{t + 1}, 2 - \frac{2}{t + 1}, 1\right)}{\mathsf{fma}\left(2 - \frac{2}{t + 1}, 2 - \frac{2}{t + 1}, 2\right)}}\]

  3. Final simplification0.0

    \[\leadsto \frac{\mathsf{fma}\left(2 - \frac{2}{t + 1}, 2 - \frac{2}{t + 1}, 1\right)}{\mathsf{fma}\left(2 - \frac{2}{t + 1}, 2 - \frac{2}{t + 1}, 2\right)}\]

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 2"
  (/ (+ 1 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))) (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))))))