Average Error: 0.0 → 0.0
Time: 8.7s
Precision: 64
Internal Precision: 128
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019022 
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))

Details

Time bar (total: 8.5s)Debug log

sample27.0ms

Algorithm
intervals
Results
18.0ms256×body80valid

simplify10.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
10.0ms
(+ (/ 1 (- x 1)) (/ x (+ x 1)))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize19.0ms

Local error

Found 3 expressions with local error:

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

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
8×*-un-lft-identity
6×associate-/r*
5×add-log-exp
5×add-cube-cbrt
5×add-sqr-sqrt
4×associate-/r/
4×add-exp-log
4×pow1
3×div-inv
3×add-cbrt-cube
2×distribute-lft-out
2×flip-+
2×flip3-+
2×frac-2neg
2×clear-num
1×inv-pow
1×flip--
1×pow-flip
1×frac-add
1×flip3--
1×sum-log
1×rec-exp
1×+-commutative
Counts
3 → 47
Calls
3 calls:
Slowest
5.0ms
(+ (/ 1 (- x 1)) (/ x (+ x 1)))
1.0ms
(/ 1 (- x 1))
0.0ms
(/ x (+ x 1))

series55.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
38.0ms
(+ (/ 1 (- x 1)) (/ x (+ x 1)))
9.0ms
(/ x (+ x 1))
8.0ms
(/ 1 (- x 1))

simplify782.0ms

Counts
25 → 56
Calls
25 calls:
Slowest
132.0ms
(+ (* 1 (+ x 1)) (* (- x 1) x))
112.0ms
(- (+ (* 2 (pow x 2)) (+ (* 2 (pow x 4)) 1)))
98.0ms
(- (+ x (+ (pow x 2) 1)))
81.0ms
(+ (* 2 (/ 1 (pow x 4))) (+ 1 (* 2 (/ 1 (pow x 2)))))
56.0ms
(+ (* 2 (/ 1 (pow x 4))) (+ 1 (* 2 (/ 1 (pow x 2)))))

prune381.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize21.0ms

Local error

Found 4 expressions with local error:

0.4b
(* (cbrt (/ 1 (- x 1))) (cbrt (/ 1 (- x 1))))
0.3b
(cbrt (/ 1 (- x 1)))
0.3b
(cbrt (/ 1 (- x 1)))
0.3b
(cbrt (/ 1 (- x 1)))

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
30×cbrt-prod
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
10×associate-/r/
9×associate-*l*
9×associate-*r*
9×pow1
7×pow1/3
7×cbrt-div
6×add-exp-log
5×flip--
5×div-inv
5×flip3--
4×add-log-exp
4×add-cbrt-cube
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×associate-*r/
1×prod-exp
1×associate-*l/
1×pow-plus
1×frac-times
1×pow2
Counts
4 → 82
Calls
4 calls:
Slowest
9.0ms
(* (cbrt (/ 1 (- x 1))) (cbrt (/ 1 (- x 1))))
1.0ms
(cbrt (/ 1 (- x 1)))
1.0ms
(cbrt (/ 1 (- x 1)))
1.0ms
(cbrt (/ 1 (- x 1)))

series209.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
62.0ms
(cbrt (/ 1 (- x 1)))
59.0ms
(cbrt (/ 1 (- x 1)))
49.0ms
(* (cbrt (/ 1 (- x 1))) (cbrt (/ 1 (- x 1))))
39.0ms
(cbrt (/ 1 (- x 1)))

simplify1.4s

Counts
63 → 94
Calls
63 calls:
Slowest
214.0ms
(+ (* (pow (/ 1 (pow x 2)) 1/3) (pow (cbrt -1) 2)) (+ (* 2/3 (* (pow (/ 1 (pow x 5)) 1/3) (pow (cbrt -1) 2))) (* 5/9 (* (pow (/ 1 (pow x 8)) 1/3) (pow (cbrt -1) 2)))))
144.0ms
(+ (* 1/3 (* x (pow -1 1/3))) (+ (* 2/9 (* (pow x 2) (pow -1 1/3))) (pow -1 1/3)))
99.0ms
(+ (* 2/9 (* (pow (/ -1 (pow x 7)) 1/3) (cbrt -1))) (+ (* (pow (/ -1 x) 1/3) (cbrt -1)) (* 1/3 (* (pow (/ -1 (pow x 4)) 1/3) (cbrt -1)))))
97.0ms
(+ (* 1/3 (* x (pow -1 1/3))) (+ (* 2/9 (* (pow x 2) (pow -1 1/3))) (pow -1 1/3)))
90.0ms
(+ (* 1/3 (* x (pow -1 1/3))) (+ (* 2/9 (* (pow x 2) (pow -1 1/3))) (pow -1 1/3)))

prune897.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize14.0ms

Local error

Found 4 expressions with local error:

0.3b
(cbrt (- x 1))
0.3b
(cbrt (- x 1))
0.3b
(cbrt (- x 1))
0.2b
(* (cbrt (- x 1)) (cbrt (- x 1)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
18×cbrt-div
15×cbrt-prod
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
9×flip--
9×flip3--
9×pow1
7×pow1/3
6×add-exp-log
6×associate-*l*
6×associate-*r*
4×add-log-exp
4×add-cbrt-cube
4×frac-times
2×pow-prod-up
2×associate-*r/
2×associate-*l/
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 75
Calls
4 calls:
Slowest
5.0ms
(* (cbrt (- x 1)) (cbrt (- x 1)))
1.0ms
(cbrt (- x 1))
1.0ms
(cbrt (- x 1))
1.0ms
(cbrt (- x 1))

series193.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
60.0ms
(cbrt (- x 1))
49.0ms
(* (cbrt (- x 1)) (cbrt (- x 1)))
43.0ms
(cbrt (- x 1))
41.0ms
(cbrt (- x 1))

simplify2.7s

Counts
59 → 87
Calls
59 calls:
Slowest
307.0ms
(- (pow -1 1/3) (+ (* 1/3 (* x (pow -1 1/3))) (* 1/9 (* (pow x 2) (pow -1 1/3)))))
281.0ms
(- (pow -1 1/3) (+ (* 1/3 (* x (pow -1 1/3))) (* 1/9 (* (pow x 2) (pow -1 1/3)))))
279.0ms
(- (* (pow (pow x 2) 1/3) (pow (cbrt -1) 2)) (+ (* 2/3 (* (pow (/ 1 x) 1/3) (pow (cbrt -1) 2))) (* 1/9 (* (pow (/ 1 (pow x 4)) 1/3) (pow (cbrt -1) 2)))))
241.0ms
(- (pow -1 1/3) (+ (* 1/3 (* x (pow -1 1/3))) (* 1/9 (* (pow x 2) (pow -1 1/3)))))
225.0ms
(- (* (pow (* x -1) 1/3) (cbrt -1)) (+ (* 1/9 (* (pow (/ -1 (pow x 5)) 1/3) (cbrt -1))) (* 1/3 (* (pow (/ -1 (pow x 2)) 1/3) (cbrt -1)))))

prune823.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes66.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample857.0ms

Algorithm
intervals
Results
725.0ms8000×body80valid