Average Error: 0.0 → 0.0
Time: 11.3s
Precision: 64
Internal Precision: 128
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{1 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)}{2 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)}\]

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

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

  2. Initial simplification0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2018362 
(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: 7.7s)Debug log

start73.0ms

Algorithm
intervals

setup131.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize40.0ms

Local error

Found 4 expressions with local error:

4.8b
(- 2 (/ 2 (+ 1 t)))
4.8b
(- 2 (/ 2 (+ 1 t)))
4.8b
(- 2 (/ 2 (+ 1 t)))
4.8b
(- 2 (/ 2 (+ 1 t)))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
4 → 44
Calls

4 calls. Slowest were:

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

series63.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

20.0ms
(- 2 (/ 2 (+ 1 t)))
16.0ms
(- 2 (/ 2 (+ 1 t)))
13.0ms
(- 2 (/ 2 (+ 1 t)))
13.0ms
(- 2 (/ 2 (+ 1 t)))

simplify3.1s

Counts
12 → 56
Calls

12 calls. Slowest were:

521.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
495.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
490.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))

prune607.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize38.0ms

Local error

Found 4 expressions with local error:

4.8b
(- 2 (/ 2 (+ 1 t)))
4.8b
(- 2 (/ 2 (+ 1 t)))
4.8b
(- 2 (/ 2 (+ 1 t)))
4.8b
(- (* 2 2) (* (/ 2 (+ 1 t)) (/ 2 (+ 1 t))))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 46
Calls

4 calls. Slowest were:

4.0ms
(- 2 (/ 2 (+ 1 t)))
2.0ms
(- (* 2 2) (* (/ 2 (+ 1 t)) (/ 2 (+ 1 t))))
0.0ms
(- 2 (/ 2 (+ 1 t)))

series61.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

20.0ms
(- (* 2 2) (* (/ 2 (+ 1 t)) (/ 2 (+ 1 t))))
16.0ms
(- 2 (/ 2 (+ 1 t)))
16.0ms
(- 2 (/ 2 (+ 1 t)))
10.0ms
(- 2 (/ 2 (+ 1 t)))

simplify2.7s

Counts
15 → 58
Calls

15 calls. Slowest were:

536.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
468.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
457.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))

prune796.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes21.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms