Average Error: 0.0 → 0.0
Time: 11.5s
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

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

Runtime

Time bar (total: 11.5s)Debug log

start105.0ms

Algorithm
intervals

setup1.2s

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize33.0ms

Local error

Found 4 expressions with local error:

3.8b
(- 2 (/ 2 (+ 1 t)))
3.8b
(- 2 (/ 2 (+ 1 t)))
0.0b
(fma (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))) 2)
0.0b
(/ 2 (+ 1 t))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
4 → 53
Calls

4 calls. Slowest were:

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

series83.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

41.0ms
(fma (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))) 2)
16.0ms
(- 2 (/ 2 (+ 1 t)))
15.0ms
(- 2 (/ 2 (+ 1 t)))

simplify5.1s

Counts
17 → 65
Calls

17 calls. Slowest were:

822.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
789.0ms
(- (+ (* 2 (/ 1 t)) (* 2 (/ 1 (pow t 3)))) (* 2 (/ 1 (pow t 2))))
769.0ms
(- (+ (* 2 (/ 1 t)) (* 2 (/ 1 (pow t 3)))) (* 2 (/ 1 (pow t 2))))

prune634.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

herbie shell --seed 2018359 +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)))))))))