Average Error: 0.1 → 0.1
Time: 22.5s
Precision: 64
Internal Precision: 128
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{(\left(\frac{t \cdot 2}{1 + t}\right) \cdot \left(\frac{t \cdot 2}{1 + t}\right) + 1)_*}{(\left(\frac{t \cdot 2}{1 + t}\right) \cdot \left(\frac{t \cdot 2}{1 + t}\right) + 2)_*}\]

Error

Bits error versus t

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

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

  3. Final simplification0.1

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

Runtime

Time bar (total: 22.5s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.10.10.10.00%
herbie shell --seed 2018351 +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))))))