Average Error: 0.0 → 0.0
Time: 3.8m
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{1 + \frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}}{4 - \left(\frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}\right) \cdot \left(\frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}\right)} \cdot \left(2 - \frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}\right)\]
\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}
↓
\frac{1 + \frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}}{4 - \left(\frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}\right) \cdot \left(\frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}\right)} \cdot \left(2 - \frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}\right)
double f(double t) {
        double r27348342 = 1.0;
        double r27348343 = 2.0;
        double r27348344 = t;
        double r27348345 = r27348343 * r27348344;
        double r27348346 = r27348342 + r27348344;
        double r27348347 = r27348345 / r27348346;
        double r27348348 = r27348347 * r27348347;
        double r27348349 = r27348342 + r27348348;
        double r27348350 = r27348343 + r27348348;
        double r27348351 = r27348349 / r27348350;
        return r27348351;
}
↓
double f(double t) {
        double r27348352 = 1.0;
        double r27348353 = t;
        double r27348354 = 2.0;
        double r27348355 = r27348353 * r27348354;
        double r27348356 = r27348352 + r27348353;
        double r27348357 = r27348355 / r27348356;
        double r27348358 = r27348357 * r27348357;
        double r27348359 = r27348352 + r27348358;
        double r27348360 = 4.0;
        double r27348361 = r27348358 * r27348358;
        double r27348362 = r27348360 - r27348361;
        double r27348363 = r27348359 / r27348362;
        double r27348364 = r27348354 - r27348358;
        double r27348365 = r27348363 * r27348364;
        return r27348365;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
  2. Using strategy rm

  3. Applied flip-+0.0

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

  4. Applied associate-/r/0.0

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

  5. Final simplification0.0

    \[\leadsto \frac{1 + \frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}}{4 - \left(\frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}\right) \cdot \left(\frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}\right)} \cdot \left(2 - \frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}\right)\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 1"
  (/ (+ 1 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))) (+ 2 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))))))