Average Error: 0.0 → 0.0
Time: 21.9s
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. 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

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

Reproduce

herbie shell --seed 2018360 
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))

Details

Time bar (total: 19.8s)Debug log

start59.0ms

Algorithm
intervals

setup24.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize41.0ms

Local error

Found 3 expressions with local error:

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

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
3 → 47
Calls

3 calls. Slowest were:

8.0ms
(+ (/ 1 (- x 1)) (/ x (+ x 1)))
2.0ms
(/ 1 (- x 1))
1.0ms
(/ x (+ x 1))

series68.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

40.0ms
(+ (/ 1 (- x 1)) (/ x (+ x 1)))
16.0ms
(/ x (+ x 1))
12.0ms
(/ 1 (- x 1))

simplify740.0ms

Counts
25 → 56
Calls

25 calls. Slowest were:

138.0ms
(+ (* 1 (+ x 1)) (* (- x 1) x))
131.0ms
(- (+ (* 2 (pow x 2)) (+ (* 2 (pow x 4)) 1)))
75.0ms
(- (+ x (+ (pow x 2) 1)))

prune422.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize7.0ms

Local error

Found 4 expressions with local error:

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

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 62
Calls

4 calls. Slowest were:

12.0ms
(exp (+ (/ 1 (- x 1)) (/ x (+ x 1))))
6.0ms
(+ (/ 1 (- x 1)) (/ x (+ x 1)))
1.0ms
(/ 1 (- x 1))

series113.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

72.0ms
(exp (+ (/ 1 (- x 1)) (/ x (+ x 1))))
23.0ms
(+ (/ 1 (- x 1)) (/ x (+ x 1)))
11.0ms
(/ x (+ x 1))
7.0ms
(/ 1 (- x 1))

simplify1.1s

Counts
36 → 74
Calls

36 calls. Slowest were:

214.0ms
(- (+ (* 2/3 (* (pow x 6) (exp -1))) (exp -1)) (* 2 (* (pow x 2) (exp -1))))
147.0ms
(+ (* 1 (+ x 1)) (* (- x 1) x))
120.0ms
(- (+ (* 2 (pow x 2)) (+ (* 2 (pow x 4)) 1)))

prune738.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize9.0ms

Local error

Found 4 expressions with local error:

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

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 127
Calls

4 calls. Slowest were:

25.0ms
(cbrt (* (* (+ (/ 1 (- x 1)) (/ x (+ x 1))) (+ (/ 1 (- x 1)) (/ x (+ x 1)))) (+ (/ 1 (- x 1)) (/ x (+ x 1)))))
1.0ms
(/ 1 (- x 1))
1.0ms
(/ 1 (- x 1))

series84.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

60.0ms
(cbrt (* (* (+ (/ 1 (- x 1)) (/ x (+ x 1))) (+ (/ 1 (- x 1)) (/ x (+ x 1)))) (+ (/ 1 (- x 1)) (/ x (+ x 1)))))
10.0ms
(/ 1 (- x 1))
7.0ms
(/ 1 (- x 1))
6.0ms
(/ 1 (- x 1))

simplify14.5s

Counts
95 → 139
Calls

95 calls. Slowest were:

1.3s
(cbrt (* (* (+ (/ 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))))))
1.2s
(cbrt (* (* (+ (* 1 (+ x 1)) (* (- x 1) x)) (+ (* 1 (+ x 1)) (* (- x 1) x))) (+ (* 1 (+ x 1)) (* (- x 1) x))))
740.0ms
(cbrt (* (* (- (* (/ 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)))))

prune1.8s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes56.0ms

Accuracy

0% (0.0b remaining)

Error of 0.0b against oracle of 0.0b and baseline of 0.0b

bsearch2.0ms