Average Error: 0.0 → 0.0
Time: 1.3m
Precision: 64
Internal Precision: 128
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{(\left((\left(\sqrt{2}\right) \cdot \left(\sqrt{2}\right) + \left(\frac{-2}{t + 1}\right))_*\right) \cdot \left(2 - \frac{2}{t + 1}\right) + 2)_*}\]

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

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied fma-neg0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019088 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 3"
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))