Average Error: 0.0 → 0.0
Time: 17.6s
Precision: 64
Internal Precision: 128
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2018362 +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: 13.6s)Debug log

start86.0ms

Algorithm
intervals

setup506.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize52.0ms

Local error

Found 4 expressions with local error:

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

rewrite12.0ms

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))))

series59.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

19.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
18.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
12.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
11.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))

simplify5.3s

Counts
12 → 64
Calls

12 calls. Slowest were:

794.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
770.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
756.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))

prune862.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize52.0ms

Local error

Found 4 expressions with local error:

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

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

5.0ms
(- (* 2 2) (* (/ (/ 2 t) (+ 1 (/ 1 t))) (/ (/ 2 t) (+ 1 (/ 1 t)))))
1.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
1.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))

series54.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

16.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
14.0ms
(- (* 2 2) (* (/ (/ 2 t) (+ 1 (/ 1 t))) (/ (/ 2 t) (+ 1 (/ 1 t)))))
12.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
11.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))

simplify5.4s

Counts
17 → 68
Calls

17 calls. Slowest were:

841.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
754.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
706.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))

prune913.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes198.0ms

Accuracy

0% (0.0b remaining)

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

bsearch4.0ms