Average Error: 0.0 → 0.0
Time: 31.2s
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 2019007 +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: 30.8s)Debug log

sample68.0ms

Algorithm
intervals

simplify161.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

160.0ms
(/ (+ 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)))))))

prune15.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize48.0ms

Local error

Found 4 expressions with local error:

4.6b
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
4.6b
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
4.6b
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
4.6b
(- 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))))

series80.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

24.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
19.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
18.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
18.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))

simplify4.7s

Counts
12 → 64
Calls

12 calls. Slowest were:

804.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
689.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
610.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))

prune629.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize42.0ms

Local error

Found 4 expressions with local error:

4.6b
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
4.6b
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
4.6b
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
0.1b
(/ (+ 1 (* (fma (* t t) (fma 2 t -2) (* 2 t)) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))) (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

9.0ms
(/ (+ 1 (* (fma (* t t) (fma 2 t -2) (* 2 t)) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))) (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))
1.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
1.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))

series149.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

109.0ms
(/ (+ 1 (* (fma (* t t) (fma 2 t -2) (* 2 t)) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))) (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))
15.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
13.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
11.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))

simplify15.1s

Counts
41 → 84
Calls

41 calls. Slowest were:

1.2s
(/ 1 (sqrt (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))))))
1.0s
(cbrt (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))
1.0s
(+ (pow 1 3) (pow (* (fma (* t t) (fma 2 t -2) (* 2 t)) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))) 3))

prune1.1s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize35.0ms

Local error

Found 4 expressions with local error:

4.6b
(- 2 (/ 2 (+ t 1)))
4.6b
(- -2 (/ -2 (+ t 1)))
4.6b
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
4.6b
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))

rewrite2.0ms

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

series60.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

17.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
17.0ms
(- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))
14.0ms
(- -2 (/ -2 (+ t 1)))
11.0ms
(- 2 (/ 2 (+ t 1)))

simplify5.2s

Counts
12 → 64
Calls

12 calls. Slowest were:

1.2s
(- (* 2 (pow t 2)) (+ (* 2 t) (* 2 (pow t 3))))
718.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
688.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))

prune1.2s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0.0b

regimes192.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms

end0.0ms

sample2.0s

Algorithm
intervals