Average Error: 0.0 → 0.0
Time: 37.5s
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

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 add-cbrt-cube0.0

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

  4. 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: 37.5s)Debug logProfile

start31.0ms

Algorithm
intervals

setup21.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize25.0ms

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))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
3 → 59
Calls

3 calls. Slowest were:

10.0ms
(+ (/ 1 (- x 1)) (/ x (+ x 1)))
2.0ms
(/ 1 (- x 1))
1.0ms
(/ x (+ x 1))

series71.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

48.0ms
(+ (/ 1 (- x 1)) (/ x (+ x 1)))
16.0ms
(/ x (+ x 1))
7.0ms
(/ 1 (- x 1))

simplify1.0s

Counts
29 → 68
Calls

29 calls. Slowest were:

172.0ms
(- (+ (* 2 (pow x 2)) (+ (* 2 (pow x 4)) 1)))
128.0ms
(- (+ x (+ (pow x 2) 1)))
119.0ms
(+ (* 2 (/ 1 (pow x 4))) (+ 1 (* 2 (/ 1 (pow x 2)))))

prune525.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize15.0ms

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))

rewrite59.0ms

Algorithm
rewrite-expression-head
Counts
4 → 145
Calls

4 calls. Slowest were:

55.0ms
(* (* (+ (/ 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))

series86.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

36.0ms
(* (* (+ (/ 1 (- x 1)) (/ x (+ x 1))) (+ (/ 1 (- x 1)) (/ x (+ x 1)))) (+ (/ 1 (- x 1)) (/ x (+ x 1))))
18.0ms
(/ x (+ x 1))
16.0ms
(/ x (+ x 1))

simplify31.3s

Counts
159 → 157
Calls

159 calls. Slowest were:

975.0ms
(* (* (- (* (/ 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))))
955.0ms
(* (* (- (/ 1 (- x 1)) (/ x (+ x 1))) (* (- x 1) (+ x 1))) (- (/ 1 (- x 1)) (/ x (+ x 1))))
926.0ms
(* (* (+ (/ 1 (- x 1)) (/ x (+ x 1))) (- (* (/ 1 (- x 1)) (/ 1 (- x 1))) (* (/ x (+ x 1)) (/ x (+ x 1))))) (- (* (/ 1 (- x 1)) (/ 1 (- x 1))) (* (/ x (+ x 1)) (/ x (+ x 1)))))

prune2.1s

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

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