Average Error: 0.0 → 0.0
Time: 16.7s
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

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

Runtime

Time bar (total: 16.7s)Debug logProfile

start86.0ms

Algorithm
intervals

setup620.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize19.0ms

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))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 124
Calls

4 calls. Slowest were:

11.0ms
(* (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t)))
7.0ms
(* (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t)))
3.0ms
(/ (* t 2) (+ 1 t))

series57.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

17.0ms
(/ (* t 2) (+ 1 t))
15.0ms
(/ (* t 2) (+ 1 t))
13.0ms
(* (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t)))

simplify11.2s

Counts
104 → 136
Calls

104 calls. Slowest were:

834.0ms
(* (* (* (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t))) (/ (* t 2) (+ 1 t))) (* (* (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t))) (/ (* t 2) (+ 1 t))))
778.0ms
(* (* (* (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t))) (/ (* t 2) (+ 1 t))) (* (* (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t))) (/ (* t 2) (+ 1 t))))
586.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))

prune1.5s

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0.0b

herbie shell --seed 2018357 
(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))))))