Average Error: 0.0 → 0.0
Time: 12.2s
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 2019010 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))

Details

Time bar (total: 12.0s)Debug log

sample36.0ms

Algorithm
intervals

simplify10.0ms

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

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize27.0ms

Local error

Found 3 expressions with local error:

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

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
9×*-un-lft-identity
6×add-cube-cbrt
6×associate-/r/
6×fma-def
6×associate-/r*
6×add-sqr-sqrt
5×add-log-exp
4×div-inv
4×add-exp-log
4×pow1
3×log1p-expm1-u
3×add-cbrt-cube
3×expm1-log1p-u
2×distribute-lft-out
2×flip--
2×flip-+
2×flip3--
2×flip3-+
2×frac-2neg
2×clear-num
1×inv-pow
1×pow-flip
1×frac-add
1×sum-log
1×rec-exp
1×+-commutative
Counts
3 → 59
Calls
3 calls:
Slowest
10.0ms
(+ (/ 1 (- x 1)) (/ x (+ x 1)))
2.0ms
(/ 1 (- x 1))
1.0ms
(/ x (+ x 1))

series60.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
36.0ms
(+ (/ 1 (- x 1)) (/ x (+ x 1)))
14.0ms
(/ x (+ x 1))
10.0ms
(/ 1 (- x 1))

simplify1.1s

Counts
29 → 68
Calls
29 calls:
Slowest
166.0ms
(- (+ x (+ (pow x 2) 1)))
151.0ms
(- (+ (* 2 (pow x 2)) (+ (* 2 (pow x 4)) 1)))
122.0ms
(+ (* 2 (/ 1 (pow x 4))) (+ 1 (* 2 (/ 1 (pow x 2)))))
118.0ms
(- (+ x (pow x 3)) (pow x 2))
99.0ms
(+ (* 1 (+ x 1)) (* (- x 1) x))

prune480.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize32.0ms

Local error

Found 4 expressions with local error:

12.4b
(* (/ 1 (- (* x x) (* 1 1))) (+ x 1))
0.3b
(/ 1 (- (* x x) (* 1 1)))
0.0b
(/ x (+ x 1))
0.0b
(+ (* (/ 1 (- (* x x) (* 1 1))) (+ x 1)) (/ x (+ x 1)))

rewrite30.0ms

Algorithm
rewrite-expression-head
Rules
9×add-exp-log
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
7×associate-/r*
7×pow1
6×add-log-exp
6×flip-+
6×associate-/r/
6×associate-*l*
6×add-cbrt-cube
6×flip3-+
5×frac-add
4×log1p-expm1-u
4×associate-*r/
4×frac-times
4×expm1-log1p-u
3×div-inv
3×associate-*r*
2×flip--
2×prod-exp
2×associate-*l/
2×flip3--
2×distribute-lft-in
2×frac-2neg
2×clear-num
2×associate-+l+
2×rec-exp
2×distribute-rgt-in
1×difference-of-squares
1×inv-pow
1×pow-flip
1×cbrt-unprod
1×*-commutative
1×pow-prod-down
1×fma-def
1×sum-log
1×+-commutative
Counts
4 → 89
Calls
4 calls:
Slowest
13.0ms
(+ (* (/ 1 (- (* x x) (* 1 1))) (+ x 1)) (/ x (+ x 1)))
11.0ms
(* (/ 1 (- (* x x) (* 1 1))) (+ x 1))
4.0ms
(/ 1 (- (* x x) (* 1 1)))
1.0ms
(/ x (+ x 1))

series102.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
55.0ms
(+ (* (/ 1 (- (* x x) (* 1 1))) (+ x 1)) (/ x (+ x 1)))
23.0ms
(* (/ 1 (- (* x x) (* 1 1))) (+ x 1))
13.0ms
(/ 1 (- (* x x) (* 1 1)))
10.0ms
(/ x (+ x 1))

simplify8.2s

Counts
58 → 101
Calls
58 calls:
Slowest
1.3s
(* (/ 1 (- (* x x) (* 1 1))) (+ x 1))
705.0ms
(* (* (- (* x x) (* 1 1)) (- x 1)) (+ x 1))
679.0ms
(* (* (- (* x x) (* 1 1)) (+ (* x x) (- (* 1 1) (* x 1)))) (+ x 1))
636.0ms
(* (* (* (/ 1 (- (* x x) (* 1 1))) (/ 1 (- (* x x) (* 1 1)))) (/ 1 (- (* x x) (* 1 1)))) (* (* (+ x 1) (+ x 1)) (+ x 1)))
588.0ms
(* (- (* x x) (* 1 1)) (+ (* x x) (- (* 1 1) (* x 1))))

prune916.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes51.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample950.0ms

Algorithm
intervals