Average Error: 0.0 → 0.0
Time: 6.5s
Precision: 64
Internal Precision: 128
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{1 + \frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}}{2 + \frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}}\]

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

    \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]

  2. Final simplification0.0

    \[\leadsto \frac{1 + \frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}}{2 + \frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}}\]

Reproduce

herbie shell --seed 2019008 
(FPCore (t)
  :name "Kahan p13 Example 1"
  (/ (+ 1 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))) (+ 2 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))))))

Details

Time bar (total: 6.2s)Debug log

sample36.0ms

Algorithm
intervals

simplify460.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

460.0ms
(/ (+ 1 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))) (+ 2 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize34.0ms

Local error

Found 4 expressions with local error:

0.0b
(/ (* 2 t) (+ 1 t))
0.0b
(/ (* 2 t) (+ 1 t))
0.0b
(/ (* 2 t) (+ 1 t))
0.0b
(/ (* 2 t) (+ 1 t))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
4 → 84
Calls

4 calls. Slowest were:

8.0ms
(/ (* 2 t) (+ 1 t))
3.0ms
(/ (* 2 t) (+ 1 t))
3.0ms
(/ (* 2 t) (+ 1 t))

series61.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

17.0ms
(/ (* 2 t) (+ 1 t))
17.0ms
(/ (* 2 t) (+ 1 t))
16.0ms
(/ (* 2 t) (+ 1 t))
11.0ms
(/ (* 2 t) (+ 1 t))

simplify3.7s

Counts
64 → 96
Calls

64 calls. Slowest were:

587.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
479.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
430.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))

prune743.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample1.1s

Algorithm
intervals