Error

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. Simplified0.0

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

4. Applied *-un-lft-identity0.0

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

5. Applied times-frac0.0

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

6. Simplified0.0

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

7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019002 +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))))))

Details

Time bar (total: 28.5s)Debug log