Average Error: 0.0 → 0.0
Time: 39.3s
Precision: 64
Internal Precision: 128
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \log_* (1 + (e^{\frac{1}{2 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)}} - 1)^*)\]

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

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

  2. Initial simplification0.0

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

  4. Applied fma-udef0.0

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

  6. Applied log1p-expm1-u0.0

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

  7. Final simplification0.0

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

Runtime

Time bar (total: 39.3s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.00.00.00.0100%
herbie shell --seed 2018352 +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)))))))))