Kahan p13 Example 3

Average Error: 0.0 → 0.0

Time: 9.9s

Precision: 64

Math

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

↓

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

C

double f(double t) {
        double r1721334 = 1.0;
        double r1721335 = 2.0;
        double r1721336 = t;
        double r1721337 = r1721335 / r1721336;
        double r1721338 = r1721334 / r1721336;
        double r1721339 = r1721334 + r1721338;
        double r1721340 = r1721337 / r1721339;
        double r1721341 = r1721335 - r1721340;
        double r1721342 = r1721341 * r1721341;
        double r1721343 = r1721335 + r1721342;
        double r1721344 = r1721334 / r1721343;
        double r1721345 = r1721334 - r1721344;
        return r1721345;
}

↓

double f(double t) {
        double r1721346 = 1.0;
        double r1721347 = 2.0;
        double r1721348 = t;
        double r1721349 = r1721346 * r1721348;
        double r1721350 = r1721349 + r1721346;
        double r1721351 = r1721347 / r1721350;
        double r1721352 = r1721347 - r1721351;
        double r1721353 = r1721352 * r1721352;
        double r1721354 = r1721353 * r1721353;
        double r1721355 = r1721347 * r1721353;
        double r1721356 = r1721354 - r1721355;
        double r1721357 = r1721347 * r1721347;
        double r1721358 = r1721356 + r1721357;
        double r1721359 = r1721357 * r1721347;
        double r1721360 = r1721353 * r1721354;
        double r1721361 = r1721359 + r1721360;
        double r1721362 = r1721346 / r1721361;
        double r1721363 = r1721358 * r1721362;
        double r1721364 = r1721346 - r1721363;
        return r1721364;
}

Error

Bits error versus t

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{1 - \frac{1}{2 + \left(2 - \frac{2}{1 + 1 \cdot t}\right) \cdot \left(2 - \frac{2}{1 + 1 \cdot t}\right)}}\]
3. Using strategy rm

4. Applied flip3-+ 0.0

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

5. Applied associate-/r/ 0.0

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

6. Simplified 0.0

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

7. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019174 
(FPCore (t)
  :name "Kahan p13 Example 3"
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))