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Derivation

1. Initial program 0.1

   \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]

2. Initial simplification0.1

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

4. Applied add-log-exp0.1

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

5. Final simplification0.1

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

Runtime

Time bar (total: 14.1s)Debug logProfile

herbie shell --seed 2018351 
(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))))))