Average Error: 0.0 → 0.0
Time: 12.5s
Precision: 64
Internal Precision: 128
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{(\left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right) + 2)_*}\]

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. Initial simplification0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2018365 +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)))))))))

Details

Time bar (total: 8.3s)Debug log

start54.0ms

Algorithm
intervals

setup1.2s

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize36.0ms

Local error

Found 4 expressions with local error:

3.3b
(- 2 (/ 2 (+ 1 t)))
3.3b
(- 2 (/ 2 (+ 1 t)))
0.0b
(/ 2 (+ 1 t))
0.0b
(/ 2 (+ 1 t))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

1.0ms
(- 2 (/ 2 (+ 1 t)))
1.0ms
(/ 2 (+ 1 t))
1.0ms
(/ 2 (+ 1 t))

series59.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

21.0ms
(- 2 (/ 2 (+ 1 t)))
16.0ms
(- 2 (/ 2 (+ 1 t)))
11.0ms
(/ 2 (+ 1 t))
10.0ms
(/ 2 (+ 1 t))

simplify6.1s

Counts
22 → 72
Calls

22 calls. Slowest were:

809.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
755.0ms
(- (+ (* 2 (/ 1 t)) (* 2 (/ 1 (pow t 3)))) (* 2 (/ 1 (pow t 2))))
709.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))

prune840.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0.0b