Average Error: 0.0 → 0.0
Time: 14.4s
Precision: 64
Internal Precision: 128
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{1 + \frac{2 \cdot t}{t + 1} \cdot \frac{2 \cdot t}{t + 1}}{2 + \frac{2 \cdot t}{t + 1} \cdot \frac{2 \cdot t}{t + 1}}\]

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. Initial simplification0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2018360 
(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: 11.8s)Debug log

start74.0ms

Algorithm
intervals

setup591.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize40.0ms

Local error

Found 4 expressions with local error:

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

rewrite42.0ms

Algorithm
rewrite-expression-head
Counts
4 → 124
Calls

4 calls. Slowest were:

14.0ms
(* (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t)))
14.0ms
(* (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t)))
6.0ms
(/ (* t 2) (+ 1 t))

series75.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

20.0ms
(* (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t)))
20.0ms
(* (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t)))
18.0ms
(/ (* t 2) (+ 1 t))
17.0ms
(/ (* t 2) (+ 1 t))

simplify9.6s

Counts
104 → 136
Calls

104 calls. Slowest were:

699.0ms
(* (* (* (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t))) (/ (* t 2) (+ 1 t))) (* (* (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t))) (/ (* t 2) (+ 1 t))))
680.0ms
(* (* (* (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t))) (/ (* t 2) (+ 1 t))) (* (* (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t))) (/ (* t 2) (+ 1 t))))
572.0ms
(* (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t)))

prune1.4s

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0.0b