Average Error: 0.0 → 0.0
Time: 38.4s
Precision: 64
Internal Precision: 128
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\frac{\left(4 - \frac{2}{1 + t} \cdot \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)}{2 + \frac{2}{1 + t}} + 2}\]

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. Simplified0.0

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

  4. Applied flip--0.0

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

  5. Applied associate-*r/0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019089 
(FPCore (t)
  :name "Kahan p13 Example 3"
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))