Average Error: 63.0 → 0
Time: 23.9s
Precision: 64
Internal Precision: 128
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\log n + \frac{\frac{1}{2}}{n}\right) - \frac{\frac{1}{6}}{n \cdot n}\]

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0
Herbie0
\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

Derivation

  1. Initial program 63.0

    \[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]

  2. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{1}{n}\right) - \left(\frac{1}{6} \cdot \frac{1}{{n}^{2}} + \log \left(\frac{1}{n}\right)\right)\right)} - 1\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(\left(1 + \log n\right) + \left(\frac{\frac{1}{2}}{n} - \frac{\frac{1}{6}}{n \cdot n}\right)\right)} - 1\]

  4. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} \cdot \frac{1}{{n}^{2}} + \log \left(\frac{1}{n}\right)\right)}\]

  5. Simplified0

    \[\leadsto \color{blue}{\left(\log n + \frac{\frac{1}{2}}{n}\right) - \frac{\frac{1}{6}}{n \cdot n}}\]

  6. Final simplification0

    \[\leadsto \left(\log n + \frac{\frac{1}{2}}{n}\right) - \frac{\frac{1}{6}}{n \cdot n}\]

Reproduce

herbie shell --seed 1137770271 
(FPCore (n)
  :name "logs (example 3.8)"
  :pre (> n 6.8e+15)

  :herbie-target
  (- (log (+ n 1)) (- (/ 1 (* 2 n)) (- (/ 1 (* 3 (* n n))) (/ 4 (pow n 3)))))

  (- (- (* (+ n 1) (log (+ n 1))) (* n (log n))) 1))

Details

Time bar (total: 23.3s)Debug log

sample276.0ms

Algorithm
intervals
Results
106×125.0ms(body real 1280)
83×69.0ms(body real 640)
53×53.0ms(body real 320)
256×9.0ms(pre true 80)
14×5.0ms(body real 160)

simplify308.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
307.0ms
(- (- (* (+ n 1) (log (+ n 1))) (* n (log n))) 1)

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 63.0b

localize54.0ms

Local error

Found 3 expressions with local error:

5.1b
(- (* (+ n 1) (log (+ n 1))) (* n (log n)))
0.3b
(* (+ n 1) (log (+ n 1)))
0.3b
(* n (log n))

rewrite36.0ms

Algorithm
rewrite-expression-head
Rules
12×log-prod
10×add-cube-cbrt
10×*-un-lft-identity
10×add-sqr-sqrt
7×associate-*r*
6×associate--l+
6×distribute-lft-in
6×pow1
6×distribute-rgt-in
5×add-log-exp
5×add-exp-log
5×add-cbrt-cube
3×associate-*l*
2×*-commutative
2×associate-*l/
1×flip--
1×flip-+
1×cbrt-unprod
1×log-pow
1×prod-exp
1×pow-prod-down
1×diff-log
1×flip3--
1×flip3-+
1×sub-neg
Counts
3 → 55
Calls
3 calls:
Slowest
17.0ms
(- (* (+ n 1) (log (+ n 1))) (* n (log n)))
11.0ms
(* (+ n 1) (log (+ n 1)))
7.0ms
(* n (log n))

series605.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
350.0ms
(* n (log n))
195.0ms
(- (* (+ n 1) (log (+ n 1))) (* n (log n)))
60.0ms
(* (+ n 1) (log (+ n 1)))

simplify7.5s

Counts
36 → 64
Calls
36 calls:
Slowest
1.2s
(- (+ 1 (* 1/2 (/ 1 n))) (+ (* n (log (/ 1 n))) (log (/ 1 n))))
1.0s
(- (+ (* n (log -1)) (+ 1 (+ (log -1) (* 1/2 (/ 1 n))))) (+ (* n (log (/ -1 n))) (log (/ -1 n))))
907.0ms
(- (+ 1 (* 1/2 (/ 1 n))) (+ (* 1/6 (/ 1 (pow n 2))) (log (/ 1 n))))
624.0ms
(* (log (* (cbrt (+ n 1)) (cbrt (+ n 1)))) (+ n 1))
564.0ms
(- (+ n (* 1/2 (pow n 2))) (+ (* n (log n)) (* 1/6 (pow n 3))))

prune1.2s

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize68.0ms

Local error

Found 3 expressions with local error:

0.5b
(/ 1/6 (* n n))
0.0b
(- (+ (+ 1 (log n)) (- (/ 1/2 n) (/ 1/6 (* n n)))) 1)
0.0b
(+ 1 (log n))

rewrite299.0ms

Algorithm
rewrite-expression-head
Rules
4×add-sqr-sqrt
3×add-log-exp
3×add-cube-cbrt
3×add-exp-log
3×add-cbrt-cube
3×*-un-lft-identity
3×pow1
1×flip--
1×div-inv
1×flip-+
1×associate--l+
1×difference-of-sqr-1
1×flip3--
1×flip3-+
1×frac-2neg
1×sub-neg
1×associate-/r*
1×associate-+r-
1×clear-num
1×+-commutative
1×associate--l-
Counts
3 → 35
Calls
3 calls:
Slowest
293.0ms
(- (+ (+ 1 (log n)) (- (/ 1/2 n) (/ 1/6 (* n n)))) 1)
4.0ms
(+ 1 (log n))
1.0ms
(/ 1/6 (* n n))

series552.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
452.0ms
(+ 1 (log n))
61.0ms
(- (+ (+ 1 (log n)) (- (/ 1/2 n) (/ 1/6 (* n n)))) 1)
39.0ms
(/ 1/6 (* n n))

simplify1.4s

Counts
12 → 44
Calls
12 calls:
Slowest
492.0ms
(- (+ (log -1) (* 1/2 (/ 1 n))) (+ (* 1/6 (/ 1 (pow n 2))) (log (/ -1 n))))
294.0ms
(+ (sqrt (+ (+ 1 (log n)) (- (/ 1/2 n) (/ 1/6 (* n n))))) 1)
254.0ms
(- (* 1/2 (/ 1 n)) (+ (* 1/6 (/ 1 (pow n 2))) (log (/ 1 n))))
153.0ms
(- (+ (log n) (* 1/2 (/ 1 n))) (* 1/6 (/ 1 (pow n 2))))
127.0ms
(- (+ (log -1) 1) (log (/ -1 n)))

prune818.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize23.0ms

Local error

Found 1 expressions with local error:

0.5b
(/ 1/6 (* n n))

rewrite2.0ms

Algorithm
rewrite-expression-head
Rules
1×add-log-exp
1×div-inv
1×add-cube-cbrt
1×add-exp-log
1×add-cbrt-cube
1×*-un-lft-identity
1×frac-2neg
1×associate-/r*
1×clear-num
1×pow1
1×add-sqr-sqrt
Counts
1 → 11
Calls
1 calls:
Slowest
2.0ms
(/ 1/6 (* n n))

series28.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
28.0ms
(/ 1/6 (* n n))

simplify23.0ms

Counts
3 → 14
Calls
3 calls:
Slowest
8.0ms
(/ 1/6 (pow n 2))
7.0ms
(/ 1/6 (pow n 2))
7.0ms
(/ 1/6 (pow n 2))

prune237.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample9.8s

Algorithm
intervals
Results
3542×5.4s(body real 1280)
2639×2.7s(body real 640)
1399×993.0ms(body real 320)
8000×369.0ms(pre true 80)
420×182.0ms(body real 160)