Average Error: 63.0 → 0.0
Time: 11.4s
Precision: 64
Internal Precision: 128
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(1 + \log n\right) + \left(\frac{\frac{1}{2}}{n} - \frac{\frac{1}{6}}{n \cdot n}\right)\right) - 1\]

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019021 
(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: 10.7s)Debug log

sample218.0ms

Algorithm
intervals
Results
128.0ms123×body1280valid
50.0ms80×body640valid
18.0ms39×body320valid
7.0ms256×pre80true
4.0ms14×body160valid

simplify224.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
224.0ms
(- (- (* (+ n 1) (log (+ n 1))) (* n (log n))) 1)

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 63.0b

localize40.0ms

Local error

Found 3 expressions with local error:

5.5b
(- (* (+ n 1) (log (+ n 1))) (* n (log n)))
0.3b
(* (+ n 1) (log (+ n 1)))
0.3b
(* n (log n))

rewrite24.0ms

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
13.0ms
(- (* (+ n 1) (log (+ n 1))) (* n (log n)))
8.0ms
(* (+ n 1) (log (+ n 1)))
2.0ms
(* n (log n))

series366.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
201.0ms
(* n (log n))
122.0ms
(- (* (+ n 1) (log (+ n 1))) (* n (log n)))
44.0ms
(* (+ n 1) (log (+ n 1)))

simplify3.4s

Counts
36 → 64
Calls
36 calls:
Slowest
576.0ms
(- (+ 1 (* 1/2 (/ 1 n))) (+ (* n (log (/ 1 n))) (log (/ 1 n))))
450.0ms
(- (+ (* n (log -1)) (+ 1 (+ (log -1) (* 1/2 (/ 1 n))))) (+ (* n (log (/ -1 n))) (log (/ -1 n))))
339.0ms
(- (+ 1 (* 1/2 (/ 1 n))) (+ (* 1/6 (/ 1 (pow n 2))) (log (/ 1 n))))
234.0ms
(* (* (* (+ n 1) (+ n 1)) (+ n 1)) (* (* (log (+ n 1)) (log (+ n 1))) (log (+ n 1))))
220.0ms
(- (+ n (* 1/2 (pow n 2))) (+ (* n (log n)) (* 1/6 (pow n 3))))

prune561.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize17.0ms

Local error

Found 1 expressions with local error:

0.9b
(/ 1/6 (* n n))

rewrite1.0ms

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
0.0ms
(/ 1/6 (* n n))

series18.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
18.0ms
(/ 1/6 (* n n))

simplify20.0ms

Counts
3 → 14
Calls
3 calls:
Slowest
7.0ms
(/ 1/6 (pow n 2))
6.0ms
(/ 1/6 (pow n 2))
6.0ms
(/ 1/6 (pow n 2))

prune133.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample5.7s

Algorithm
intervals
Results
3.2s3768×body1280valid
1.7s2514×body640valid
433.0ms1272×body320valid
161.0ms8000×pre80true
133.0ms446×body160valid