Average Error: 30.0 → 0.1
Time: 19.7s
Precision: 64
Internal Precision: 128
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 4534.829363575865:\\ \;\;\;\;\sqrt{\log \left(1 + N\right)} \cdot \sqrt{\log \left(1 + N\right)} - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{1}{N \cdot N} \cdot \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right)\\ \end{array}\]

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if N < 4534.829363575865

    1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt0.1

      \[\leadsto \color{blue}{\sqrt{\log \left(N + 1\right)} \cdot \sqrt{\log \left(N + 1\right)}} - \log N\]

    if 4534.829363575865 < N

    1. Initial program 59.4

      \[\log \left(N + 1\right) - \log N\]

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{N} - \frac{1}{N \cdot N} \cdot \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4534.829363575865:\\ \;\;\;\;\sqrt{\log \left(1 + N\right)} \cdot \sqrt{\log \left(1 + N\right)} - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{1}{N \cdot N} \cdot \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 1137770271 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))

Details

Time bar (total: 19.0s)Debug log

sample178.0ms

Algorithm
intervals
Results
53×46.0ms(body real 1280)
44×26.0ms(body real 640)
249×26.0ms(body nan 80)
505×20.0ms(pre true 80)
125×19.0ms(body real 80)
25×11.0ms(body real 320)
9×2.0ms(body real 160)

simplify9.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
9.0ms
(- (log (+ N 1)) (log N))

prune12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 30.6b

localize33.0ms

Local error

Found 2 expressions with local error:

3.3b
(log (+ N 1))
2.1b
(- (log (+ N 1)) (log N))

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
7×*-un-lft-identity
6×log-prod
6×add-sqr-sqrt
4×add-cube-cbrt
4×log-div
4×pow1
3×add-exp-log
3×associate--l+
2×add-log-exp
2×distribute-lft-out--
2×flip-+
2×log-pow
2×add-cbrt-cube
2×flip3-+
2×associate--l-
1×difference-of-squares
1×flip--
1×rem-log-exp
1×diff-log
1×flip3--
1×sub-neg
Counts
2 → 34
Calls
2 calls:
Slowest
11.0ms
(- (log (+ N 1)) (log N))
2.0ms
(log (+ N 1))

series166.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
119.0ms
(- (log (+ N 1)) (log N))
47.0ms
(log (+ N 1))

simplify1.5s

Counts
22 → 40
Calls
22 calls:
Slowest
405.0ms
(- (+ (/ 1 N) (log -1)) (+ (* 1/2 (/ 1 (pow N 2))) (log (/ -1 N))))
242.0ms
(- (/ 1 N) (+ (* 1/2 (/ 1 (pow N 2))) (log (/ 1 N))))
197.0ms
(- (+ (* 1/3 (/ 1 (pow N 3))) (/ 1 N)) (* 1/2 (/ 1 (pow N 2))))
197.0ms
(- (+ (* 1/3 (/ 1 (pow N 3))) (/ 1 N)) (* 1/2 (/ 1 (pow N 2))))
182.0ms
(- (+ N (* 1/3 (pow N 3))) (* 1/2 (pow N 2)))

prune644.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize55.0ms

Local error

Found 3 expressions with local error:

0.4b
(/ 1/3 N)
0.3b
(/ 1 (* N N))
0.1b
(* (/ 1 (* N N)) (- 1/2 (/ 1/3 N)))

rewrite15.0ms

Algorithm
rewrite-expression-head
Rules
8×add-exp-log
6×pow1
5×add-cube-cbrt
5×add-cbrt-cube
5×*-un-lft-identity
5×add-sqr-sqrt
4×associate-*l*
3×add-log-exp
3×div-inv
3×associate-*r*
2×flip--
2×pow-flip
2×associate-*r/
2×prod-exp
2×flip3--
2×distribute-lft-in
2×frac-2neg
2×sub-neg
2×clear-num
2×rec-exp
2×frac-times
2×distribute-rgt-in
1×inv-pow
1×cbrt-unprod
1×*-commutative
1×associate-*l/
1×pow-prod-down
1×associate-/r*
1×pow2
Counts
3 → 53
Calls
3 calls:
Slowest
11.0ms
(* (/ 1 (* N N)) (- 1/2 (/ 1/3 N)))
2.0ms
(/ 1 (* N N))
0.0ms
(/ 1/3 N)

series91.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
54.0ms
(* (/ 1 (* N N)) (- 1/2 (/ 1/3 N)))
20.0ms
(/ 1/3 N)
16.0ms
(/ 1 (* N N))

simplify5.0s

Counts
31 → 62
Calls
31 calls:
Slowest
694.0ms
(* (/ 1 (* N N)) (- (/ 1/3 N)))
680.0ms
(* (- (/ 1/3 N)) (/ 1 (* N N)))
678.0ms
(* (/ 1 (* N N)) (- (/ 1/3 N)))
564.0ms
(* (- (/ 1/3 N)) (/ 1 (* N N)))
371.0ms
(* (/ 1 (* N N)) (- 1/2 (/ 1/3 N)))

prune776.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize26.0ms

Local error

Found 4 expressions with local error:

15.7b
(- (* (sqrt (log (+ N 1))) (sqrt (log (+ N 1)))) (log N))
3.3b
(log (+ N 1))
3.3b
(log (+ N 1))
0.3b
(* (sqrt (log (+ N 1))) (sqrt (log (+ N 1))))

rewrite35.0ms

Algorithm
rewrite-expression-head
Rules
13×pow1
11×add-sqr-sqrt
10×add-cube-cbrt
10×*-un-lft-identity
8×sqrt-prod
8×add-exp-log
7×associate-*l*
7×associate-*r*
6×add-cbrt-cube
6×log-prod
5×add-log-exp
4×log-pow
4×pow1/2
4×log-div
2×flip-+
2×pow-prod-up
2×pow-prod-down
2×rem-log-exp
2×flip3-+
1×difference-of-squares
1×flip--
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×diff-log
1×sqrt-unprod
1×flip3--
1×pow-plus
1×rem-square-sqrt
1×sub-neg
1×pow2
Counts
4 → 73
Calls
4 calls:
Slowest
17.0ms
(- (* (sqrt (log (+ N 1))) (sqrt (log (+ N 1)))) (log N))
13.0ms
(* (sqrt (log (+ N 1))) (sqrt (log (+ N 1))))
2.0ms
(log (+ N 1))
2.0ms
(log (+ N 1))

series256.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
113.0ms
(- (* (sqrt (log (+ N 1))) (sqrt (log (+ N 1)))) (log N))
50.0ms
(log (+ N 1))
47.0ms
(* (sqrt (log (+ N 1))) (sqrt (log (+ N 1))))
46.0ms
(log (+ N 1))

simplify3.0s

Counts
51 → 85
Calls
51 calls:
Slowest
374.0ms
(- (+ (/ 1 N) (log -1)) (+ (* 1/2 (/ 1 (pow N 2))) (log (/ -1 N))))
370.0ms
(- (+ (/ 1 N) (log -1)) (+ (* 1/2 (/ 1 (pow N 2))) (log (/ -1 N))))
365.0ms
(- (+ (/ 1 N) (log -1)) (+ (* 1/2 (/ 1 (pow N 2))) (log (/ -1 N))))
222.0ms
(- (/ 1 N) (+ (* 1/2 (/ 1 (pow N 2))) (log (/ 1 N))))
220.0ms
(- (/ 1 N) (+ (* 1/2 (/ 1 (pow N 2))) (log (/ 1 N))))

prune1.9s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes66.0ms

Accuracy

100% (0.0b remaining)

Error of 0.1b against oracle of 0.1b and baseline of 30.3b

bsearch125.0ms

end0.0ms

sample5.2s

Algorithm
intervals
Results
1723×1.6s(body real 1280)
7935×973.0ms(body nan 80)
1253×822.0ms(body real 640)
4003×658.0ms(body real 80)
15935×426.0ms(pre true 80)
695×341.0ms(body real 320)
326×91.0ms(body real 160)