Average Error: 0.0 → 0.0
Time: 6.6s
Precision: 64
Internal Precision: 128
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\frac{1}{x \cdot x - 1} \cdot \left(x + 1\right) + \frac{x}{x + 1}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\frac{1}{x - 1} + \frac{x}{x + 1}\]
  2. Using strategy rm

  3. Applied flip--0.0

    \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x + 1}}} + \frac{x}{x + 1}\]

  4. Applied associate-/r/0.0

    \[\leadsto \color{blue}{\frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(x + 1\right)} + \frac{x}{x + 1}\]

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))

Details

Time bar (total: 6.4s)Debug log

sample33.0ms

Algorithm
intervals

simplify7.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

7.0ms
(+ (/ 1 (- x 1)) (/ x (+ x 1)))

prune4.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 2 expressions with local error:

0.0b
(/ 1 (- x 1))
0.0b
(/ x (+ x 1))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
2 → 37
Calls

2 calls. Slowest were:

1.0ms
(/ 1 (- x 1))
0.0ms
(/ x (+ x 1))

series33.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

18.0ms
(/ 1 (- x 1))
15.0ms
(/ x (+ x 1))

simplify392.0ms

Counts
17 → 43
Calls

17 calls. Slowest were:

124.0ms
(- (+ x (+ (pow x 2) 1)))
88.0ms
(- (+ x (pow x 3)) (pow x 2))
31.0ms
(- (+ (/ 1 (pow x 2)) 1) (/ 1 x))

prune276.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize28.0ms

Local error

Found 3 expressions with local error:

13.7b
(* (/ 1 (- (* x x) (* 1 1))) (+ x 1))
0.1b
(/ 1 (- (* x x) (* 1 1)))
0.0b
(/ x (+ x 1))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
3 → 68
Calls

3 calls. Slowest were:

5.0ms
(* (/ 1 (- (* x x) (* 1 1))) (+ x 1))
2.0ms
(/ 1 (- (* x x) (* 1 1)))
0.0ms
(/ x (+ x 1))

series41.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

17.0ms
(/ 1 (- (* x x) (* 1 1)))
12.0ms
(/ x (+ x 1))
12.0ms
(* (/ 1 (- (* x x) (* 1 1))) (+ x 1))

simplify3.8s

Counts
41 → 77
Calls

41 calls. Slowest were:

1.2s
(* (/ 1 (- (* x x) (* 1 1))) (+ x 1))
574.0ms
(* (- (* x x) (* 1 1)) (+ (* x x) (- (* 1 1) (* x 1))))
561.0ms
(* (* (* (/ 1 (- (* x x) (* 1 1))) (/ 1 (- (* x x) (* 1 1)))) (/ 1 (- (* x x) (* 1 1)))) (* (* (+ x 1) (+ x 1)) (+ x 1)))

prune671.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample1.1s

Algorithm
intervals