Average Error: 29.6 → 0.1
Time: 23.1s
Precision: 64
Internal Precision: 128
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8120.743624075386:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{1}{N \cdot N} \cdot \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right)\\ \end{array}\]

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 < 8120.743624075386

    1. Initial program 0.1

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

    3. Applied diff-log0.1

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt0.1

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

    6. Applied log-prod0.1

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

    if 8120.743624075386 < N

    1. Initial program 59.6

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8120.743624075386:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{1}{N \cdot N} \cdot \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 323378853 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))

Details

Time bar (total: 22.3s)Debug log

sample156.0ms

Algorithm
intervals
Results
48.0ms59×body1280valid
27.0ms277×body80nan
25.0ms43×body640valid
18.0ms127×body80valid
9.0ms22×body320valid
1.0ms5×body160valid

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: 30.5b

localize32.0ms

Local error

Found 2 expressions with local error:

2.4b
(log (+ N 1))
2.0b
(- (log (+ N 1)) (log N))

rewrite30.0ms

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
27.0ms
(- (log (+ N 1)) (log N))
2.0ms
(log (+ N 1))

series153.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
107.0ms
(- (log (+ N 1)) (log N))
46.0ms
(log (+ N 1))

simplify1.3s

Counts
22 → 40
Calls
22 calls:
Slowest
366.0ms
(- (+ (/ 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))))
178.0ms
(- (+ (* 1/3 (/ 1 (pow N 3))) (/ 1 N)) (* 1/2 (/ 1 (pow N 2))))
175.0ms
(- (+ (* 1/3 (/ 1 (pow N 3))) (/ 1 N)) (* 1/2 (/ 1 (pow N 2))))
161.0ms
(- (+ N (* 1/3 (pow N 3))) (* 1/2 (pow N 2)))

prune558.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.1b

localize50.0ms

Local error

Found 4 expressions with local error:

0.7b
(/ 1 (* N N))
0.3b
(/ 1/3 N)
0.1b
(* (/ 1 (* N N)) (- 1/2 (/ 1/3 N)))
0.0b
(- (/ 1 N) (* (/ 1 (* N N)) (- 1/2 (/ 1/3 N))))

rewrite28.0ms

Algorithm
rewrite-expression-head
Rules
9×add-exp-log
7×pow1
6×add-log-exp
6×add-cube-cbrt
6×add-cbrt-cube
6×*-un-lft-identity
6×add-sqr-sqrt
5×flip--
5×frac-sub
5×flip3--
5×sub-neg
4×associate--r+
4×associate-*r/
4×associate-*l*
4×distribute-lft-in
4×frac-times
4×distribute-rgt-in
3×div-inv
3×associate-*r*
2×pow-flip
2×prod-exp
2×associate-*l/
2×frac-2neg
2×clear-num
2×rec-exp
1×inv-pow
1×cbrt-unprod
1×*-commutative
1×pow-prod-down
1×diff-log
1×associate-/r*
1×pow2
Counts
4 → 74
Calls
4 calls:
Slowest
13.0ms
(- (/ 1 N) (* (/ 1 (* N N)) (- 1/2 (/ 1/3 N))))
11.0ms
(* (/ 1 (* N N)) (- 1/2 (/ 1/3 N)))
2.0ms
(/ 1 (* N N))
0.0ms
(/ 1/3 N)

series111.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
52.0ms
(* (/ 1 (* N N)) (- 1/2 (/ 1/3 N)))
28.0ms
(- (/ 1 N) (* (/ 1 (* N N)) (- 1/2 (/ 1/3 N))))
16.0ms
(/ 1 (* N N))
14.0ms
(/ 1/3 N)

simplify9.9s

Counts
49 → 86
Calls
49 calls:
Slowest
630.0ms
(* (/ 1 (* N N)) (- (/ 1/3 N)))
629.0ms
(* (/ 1 (* N N)) (- (/ 1/3 N)))
628.0ms
(* (/ 1 (* N N)) (- (/ 1/3 N)))
627.0ms
(* (/ 1 (* N N)) (- (/ 1/3 N)))
550.0ms
(* (- (/ 1/3 N)) (/ 1 (* N N)))

prune1.0s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.1b

localize15.0ms

Local error

Found 2 expressions with local error:

2.7b
(log (/ (+ N 1) N))
0.0b
(/ (+ N 1) N)

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
4×add-cube-cbrt
4×*-un-lft-identity
4×log-prod
4×add-sqr-sqrt
3×associate-/l*
3×add-exp-log
3×pow1
2×associate-/l/
2×add-log-exp
2×div-inv
2×add-cbrt-cube
1×flip-+
1×log-pow
1×rem-log-exp
1×flip3-+
1×frac-2neg
1×clear-num
1×log-div
Counts
2 → 29
Calls
2 calls:
Slowest
5.0ms
(/ (+ N 1) N)
2.0ms
(log (/ (+ N 1) N))

series109.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
66.0ms
(log (/ (+ N 1) N))
43.0ms
(/ (+ N 1) N)

simplify512.0ms

Counts
16 → 35
Calls
16 calls:
Slowest
179.0ms
(- (+ (* 1/3 (/ 1 (pow N 3))) (/ 1 N)) (* 1/2 (/ 1 (pow N 2))))
175.0ms
(- (+ (* 1/3 (/ 1 (pow N 3))) (/ 1 N)) (* 1/2 (/ 1 (pow N 2))))
91.0ms
(- N (+ (log N) (* 1/2 (pow N 2))))
14.0ms
(- (* N N) (* 1 1))
7.0ms
(+ (pow N 3) (pow 1 3))

prune408.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize23.0ms

Local error

Found 4 expressions with local error:

2.4b
(log (sqrt (/ (+ N 1) N)))
2.4b
(log (sqrt (/ (+ N 1) N)))
0.1b
(sqrt (/ (+ N 1) N))
0.1b
(sqrt (/ (+ N 1) N))

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
16×sqrt-prod
14×log-prod
12×add-sqr-sqrt
10×add-cube-cbrt
10×*-un-lft-identity
6×add-exp-log
6×pow1
4×add-log-exp
4×div-inv
4×log-pow
4×sqrt-div
4×pow1/2
4×add-cbrt-cube
2×rem-log-exp
2×log-div
2×rem-sqrt-square
Counts
4 → 64
Calls
4 calls:
Slowest
3.0ms
(log (sqrt (/ (+ N 1) N)))
2.0ms
(log (sqrt (/ (+ N 1) N)))
2.0ms
(sqrt (/ (+ N 1) N))
2.0ms
(sqrt (/ (+ N 1) N))

series226.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
83.0ms
(log (sqrt (/ (+ N 1) N)))
80.0ms
(log (sqrt (/ (+ N 1) N)))
31.0ms
(sqrt (/ (+ N 1) N))
31.0ms
(sqrt (/ (+ N 1) N))

simplify1.4s

Counts
42 → 76
Calls
42 calls:
Slowest
185.0ms
(- (+ (* 1/6 (/ 1 (pow N 3))) (* 1/2 (/ 1 N))) (* 1/4 (/ 1 (pow N 2))))
184.0ms
(- (+ (* 1/6 (/ 1 (pow N 3))) (* 1/2 (/ 1 N))) (* 1/4 (/ 1 (pow N 2))))
182.0ms
(- (+ (* 1/6 (/ 1 (pow N 3))) (* 1/2 (/ 1 N))) (* 1/4 (/ 1 (pow N 2))))
181.0ms
(- (+ (* 1/6 (/ 1 (pow N 3))) (* 1/2 (/ 1 N))) (* 1/4 (/ 1 (pow N 2))))
126.0ms
(- (+ (* 1/2 (/ 1 N)) 1) (* 1/8 (/ 1 (pow N 2))))

prune1.0s

Pruning

5 alts after pruning (2 fresh and 3 done)

Merged error: 0.0b

regimes68.0ms

Accuracy

100% (0.0b remaining)

Error of 0.1b against oracle of 0.0b and baseline of 30.8b

bsearch93.0ms

end0.0ms

sample5.0s

Algorithm
intervals
Results
1.6s1798×body1280valid
1.4s1274×body640valid
884.0ms8083×body80nan
666.0ms4064×body80valid
249.0ms593×body320valid
72.0ms271×body160valid