Average Error: 0.4 → 0.4
Time: 58.9s
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\frac{\sqrt{k}}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\]

Error

Bits error versus k

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.4

    \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\color{blue}{1 \cdot {\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}}\]

  5. Applied associate-/l*0.4

    \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{k}}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}}\]

  6. Final simplification0.4

    \[\leadsto \frac{1}{\frac{\sqrt{k}}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\]

Reproduce

herbie shell --seed 2019002 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))

Details

Time bar (total: 58.1s)Debug log

sample326.0ms

Algorithm
intervals

simplify67.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

66.0ms
(* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2)))

prune24.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize47.0ms

Local error

Found 3 expressions with local error:

0.4b
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
0.3b
(* PI (* n 2))
0.2b
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
3 → 68
Calls

3 calls. Slowest were:

5.0ms
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))
2.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
1.0ms
(* PI (* n 2))

series395.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

221.0ms
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))
124.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
50.0ms
(* PI (* n 2))

simplify3.8s

Counts
67 → 77
Calls

67 calls. Slowest were:

957.0ms
(- (* (log (* PI (* n 2))) (- 1/2 (/ k 2))) (log (sqrt k)))
645.0ms
(- (* (log (* PI (* n 2))) (- 1/2 (/ k 2))) (log (sqrt k)))
313.0ms
(- (log (pow (* PI (* n 2)) (- 1/2 (/ k 2)))) (log (sqrt k)))

prune821.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0b

localize6.0ms

Local error

Found 4 expressions with local error:

0.4b
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
0.3b
(* PI (* n 2))
0.2b
(/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2))))
0.1b
(/ 1 (/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 108
Calls

4 calls. Slowest were:

4.0ms
(/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2))))
2.0ms
(/ 1 (/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
2.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))

series559.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

183.0ms
(/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2))))
161.0ms
(/ 1 (/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
158.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
57.0ms
(* PI (* n 2))

simplify3.4s

Counts
95 → 120
Calls

95 calls. Slowest were:

450.0ms
(- (log (sqrt k)) (* (log (* PI (* n 2))) (- 1/2 (/ k 2))))
375.0ms
(- (log (sqrt k)) (* (log (* PI (* n 2))) (- 1/2 (/ k 2))))
344.0ms
(- (log (sqrt k)) (log (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))

prune1.3s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0b

localize25.0ms

Local error

Found 4 expressions with local error:

0.5b
(pow PI (- 1/2 (/ k 2)))
0.3b
(pow (* n 2) (- 1/2 (/ k 2)))
0.1b
(/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2))))
0.1b
(/ (/ 1 (/ 1 (pow PI (- 1/2 (/ k 2))))) (/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2)))))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 246
Calls

4 calls. Slowest were:

11.0ms
(/ (/ 1 (/ 1 (pow PI (- 1/2 (/ k 2))))) (/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2)))))
8.0ms
(/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2))))
2.0ms
(pow (* n 2) (- 1/2 (/ k 2)))

series483.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

248.0ms
(/ (/ 1 (/ 1 (pow PI (- 1/2 (/ k 2))))) (/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2)))))
128.0ms
(/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2))))
83.0ms
(pow (* n 2) (- 1/2 (/ k 2)))
24.0ms
(pow PI (- 1/2 (/ k 2)))

simplify31.0s

Counts
351 → 258
Calls

351 calls. Slowest were:

905.0ms
(/ (/ 1 1) (/ (* (cbrt (sqrt k)) (cbrt (sqrt k))) (* (cbrt (pow (* n 2) (- 1/2 (/ k 2)))) (cbrt (pow (* n 2) (- 1/2 (/ k 2)))))))
506.0ms
(- (log (/ 1 (/ 1 (pow PI (- 1/2 (/ k 2)))))) (- (log (sqrt k)) (log (pow (* n 2) (- 1/2 (/ k 2))))))
490.0ms
(/ (/ 1 (/ 1 (pow PI (- 1/2 (/ k 2))))) (/ (sqrt (sqrt k)) (pow (* n 2) (- 1/2 (/ k 2)))))

prune3.5s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.5b
(pow PI (- 1/2 (/ k 2)))
0.3b
(pow (* n 2) (- 1/2 (/ k 2)))
0.1b
(/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2))))
0.1b
(/ 1 (/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2)))))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
4 → 113
Calls

4 calls. Slowest were:

8.0ms
(/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2))))
4.0ms
(/ 1 (/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2)))))
2.0ms
(pow (* n 2) (- 1/2 (/ k 2)))

series408.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

144.0ms
(/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2))))
122.0ms
(/ 1 (/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2)))))
101.0ms
(pow (* n 2) (- 1/2 (/ k 2)))
41.0ms
(pow PI (- 1/2 (/ k 2)))

simplify2.3s

Counts
101 → 125
Calls

101 calls. Slowest were:

341.0ms
(exp (* (- (log 2) (log (/ 1 n))) (- 1/2 (* 1/2 k))))
226.0ms
(- (+ (pow PI 1/2) (* 1/8 (* (* (pow (log PI) 2) (pow k 2)) (sqrt PI)))) (* 1/2 (* (* (log PI) k) (sqrt PI))))
202.0ms
(/ (* (* (sqrt k) (sqrt k)) (sqrt k)) (* (* (pow (* n 2) (- 1/2 (/ k 2))) (pow (* n 2) (- 1/2 (/ k 2)))) (pow (* n 2) (- 1/2 (/ k 2)))))

prune1.7s

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0b

regimes170.0ms

Accuracy

0% (0.3b remaining)

Error of 0.4b against oracle of 0.1b and baseline of 0.4b

bsearch3.0ms

end0.0ms

sample7.7s

Algorithm
intervals