Average Error: 0.0 → 0.0
Time: 35.5s
Precision: 64
Internal Precision: 128
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\sqrt[3]{\left(\left(\frac{1}{x - 1} + \frac{x}{x + 1}\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)}\]

Error

Bits error versus x

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Your Program's Arguments

Results

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Derivation

  1. Initial program 0.0

    \[\frac{1}{x - 1} + \frac{x}{x + 1}\]

  2. Initial simplification0.0

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

  4. Applied add-cbrt-cube0.0

    \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{1}{x - 1} + \frac{x}{x + 1}\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)}}\]

  5. Final simplification0.0

    \[\leadsto \sqrt[3]{\left(\left(\frac{1}{x - 1} + \frac{x}{x + 1}\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)}\]

Runtime

Time bar (total: 35.5s)Debug logProfile

start59.0ms

Algorithm
intervals

setup35.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 3 expressions with local error:

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

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
3 → 47
Calls

3 calls. Slowest were:

7.0ms
(+ (/ 1 (- x 1)) (/ x (+ x 1)))
2.0ms
(/ 1 (- x 1))
1.0ms
(/ x (+ x 1))

series52.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

28.0ms
(+ (/ 1 (- x 1)) (/ x (+ x 1)))
16.0ms
(/ x (+ x 1))
7.0ms
(/ 1 (- x 1))

simplify760.0ms

Counts
25 → 56
Calls

25 calls. Slowest were:

130.0ms
(- (+ (* 2 (pow x 2)) (+ (* 2 (pow x 4)) 1)))
126.0ms
(+ (* 1 (+ x 1)) (* (- x 1) x))
86.0ms
(+ (* 2 (/ 1 (pow x 4))) (+ 1 (* 2 (/ 1 (pow x 2)))))

prune491.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize18.0ms

Local error

Found 4 expressions with local error:

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

rewrite46.0ms

Algorithm
rewrite-expression-head
Counts
4 → 137
Calls

4 calls. Slowest were:

42.0ms
(* (* (+ (/ 1 (- x 1)) (/ x (+ x 1))) (+ (/ 1 (- x 1)) (/ x (+ x 1)))) (+ (/ 1 (- x 1)) (/ x (+ x 1))))
0.0ms
(/ x (+ x 1))
0.0ms
(/ x (+ x 1))

series70.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

31.0ms
(* (* (+ (/ 1 (- x 1)) (/ x (+ x 1))) (+ (/ 1 (- x 1)) (/ x (+ x 1)))) (+ (/ 1 (- x 1)) (/ x (+ x 1))))
16.0ms
(/ x (+ x 1))
12.0ms
(/ x (+ x 1))

simplify29.4s

Counts
159 → 149
Calls

159 calls. Slowest were:

1.2s
(* (* (* (- x 1) (+ x 1)) (- (/ 1 (- x 1)) (/ x (+ x 1)))) (* (- x 1) (+ x 1)))
1.0s
(* (* (* (- x 1) (+ x 1)) (* (- x 1) (+ x 1))) (- (/ 1 (- x 1)) (/ x (+ x 1))))
1.0s
(* (* (- (* (/ 1 (- x 1)) (/ 1 (- x 1))) (* (/ x (+ x 1)) (/ x (+ x 1)))) (- (* (/ 1 (- x 1)) (/ 1 (- x 1))) (* (/ x (+ x 1)) (/ x (+ x 1))))) (+ (/ 1 (- x 1)) (/ x (+ x 1))))

prune2.1s

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

herbie shell --seed 2018357 
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))