Average Error: 0.0 → 0.0
Time: 19.0s
Precision: 64
Internal Precision: 128
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\sqrt[3]{\frac{1}{x - 1} + \frac{x}{x + 1}} \cdot \left(\sqrt[3]{\frac{1}{x - 1} + \frac{x}{x + 1}} \cdot \sqrt[3]{\frac{1}{x - 1} + \frac{x}{x + 1}}\right)\]

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-cube-cbrt0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))

Details

Time bar (total: 16.7s)Debug log

start42.0ms

Algorithm
intervals

setup12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize13.0ms

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))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
3 → 59
Calls

3 calls. Slowest were:

6.0ms
(+ (/ 1 (- x 1)) (/ x (+ x 1)))
1.0ms
(/ 1 (- x 1))
0.0ms
(/ x (+ x 1))

series72.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

40.0ms
(+ (/ 1 (- x 1)) (/ x (+ x 1)))
22.0ms
(/ x (+ x 1))
11.0ms
(/ 1 (- x 1))

simplify1.0s

Counts
29 → 68
Calls

29 calls. Slowest were:

158.0ms
(- (+ (* 2 (pow x 2)) (+ (* 2 (pow x 4)) 1)))
129.0ms
(+ (* 2 (/ 1 (pow x 4))) (+ 1 (* 2 (/ 1 (pow x 2)))))
122.0ms
(- (+ x (+ (pow x 2) 1)))

prune550.0ms

Pruning

2 alts after pruning (2 fresh and 0 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)))) (cbrt (+ (/ 1 (- x 1)) (/ x (+ x 1)))))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 109
Calls

4 calls. Slowest were:

17.0ms
(* (cbrt (+ (/ 1 (- x 1)) (/ x (+ x 1)))) (cbrt (+ (/ 1 (- x 1)) (/ x (+ x 1)))))
1.0ms
(/ 1 (- x 1))
1.0ms
(/ 1 (- x 1))

series195.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

170.0ms
(* (cbrt (+ (/ 1 (- x 1)) (/ x (+ x 1)))) (cbrt (+ (/ 1 (- x 1)) (/ x (+ x 1)))))
11.0ms
(/ 1 (- x 1))
7.0ms
(/ 1 (- x 1))
7.0ms
(/ 1 (- x 1))

simplify10.1s

Counts
78 → 121
Calls

78 calls. Slowest were:

751.0ms
(* (cbrt (- (/ 1 (- x 1)) (/ x (+ x 1)))) (cbrt (* (- x 1) (+ x 1))))
746.0ms
(* (cbrt (* (- x 1) (+ x 1))) (cbrt (- (/ 1 (- x 1)) (/ x (+ x 1)))))
583.0ms
(* (cbrt (- (* (/ 1 (- x 1)) (/ 1 (- x 1))) (* (/ x (+ x 1)) (/ x (+ x 1))))) (cbrt (- (* (/ 1 (- x 1)) (/ 1 (- x 1))) (* (/ x (+ x 1)) (/ x (+ x 1))))))

prune1.4s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize13.0ms

Local error

Found 4 expressions with local error:

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

rewrite71.0ms

Algorithm
rewrite-expression-head
Counts
4 → 79
Calls

4 calls. Slowest were:

31.0ms
(log (exp (+ (/ 1 (- x 1)) (/ x (+ x 1)))))
26.0ms
(exp (+ (/ 1 (- x 1)) (/ x (+ x 1))))
10.0ms
(+ (/ 1 (- x 1)) (/ x (+ x 1)))

series166.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

84.0ms
(exp (+ (/ 1 (- x 1)) (/ x (+ x 1))))
37.0ms
(+ (/ 1 (- x 1)) (/ x (+ x 1)))
33.0ms
(log (exp (+ (/ 1 (- x 1)) (/ x (+ x 1)))))
11.0ms
(/ 1 (- x 1))

simplify2.0s

Counts
45 → 91
Calls

45 calls. Slowest were:

259.0ms
(- (+ (* 2/3 (* (pow x 6) (exp -1))) (exp -1)) (* 2 (* (pow x 2) (exp -1))))
208.0ms
(- (+ (* 2 (pow x 2)) (+ (* 2 (pow x 4)) 1)))
195.0ms
(- (+ x (+ (pow x 2) 1)))

prune924.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes63.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms