Average Error: 0.0 → 0.0
Time: 14.4s
Precision: 64
Internal Precision: 128
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{(\left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right) + 1)_*}{(\left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right) + 2)_*}\]

Error

Bits error versus t

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 2"
  (/ (+ 1 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))) (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))))))

Details

Time bar (total: 13.7s)Debug log

sample47.0ms

Algorithm
intervals

simplify109.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

108.0ms
(/ (+ 1 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))) (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))

prune9.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize22.0ms

Local error

Found 4 expressions with local error:

3.2b
(- 2 (/ 2 (+ 1 t)))
3.2b
(- 2 (/ 2 (+ 1 t)))
3.2b
(- 2 (/ 2 (+ 1 t)))
3.2b
(- 2 (/ 2 (+ 1 t)))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

1.0ms
(- 2 (/ 2 (+ 1 t)))
0.0ms
(- 2 (/ 2 (+ 1 t)))
0.0ms
(- 2 (/ 2 (+ 1 t)))

series49.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

16.0ms
(- 2 (/ 2 (+ 1 t)))
14.0ms
(- 2 (/ 2 (+ 1 t)))
10.0ms
(- 2 (/ 2 (+ 1 t)))
9.0ms
(- 2 (/ 2 (+ 1 t)))

simplify5.3s

Counts
12 → 64
Calls

12 calls. Slowest were:

787.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
736.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
711.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))

prune651.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize46.0ms

Local error

Found 4 expressions with local error:

3.2b
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
3.2b
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
3.2b
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
3.2b
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

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

series70.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

19.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
17.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
17.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
16.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))

simplify5.2s

Counts
12 → 64
Calls

12 calls. Slowest were:

815.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
785.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
749.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))

prune562.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes100.0ms

Accuracy

0% (0.0b remaining)

Error of 0.0b against oracle of 0.0b and baseline of 0.0b

bsearch1.0ms

end0.0ms

sample1.5s

Algorithm
intervals