Average Error: 0.4 → 0.4
Time: 31.6s
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\frac{\sqrt{k}}{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\]

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. Taylor expanded around inf 0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))

Details

Time bar (total: 30.9s)Debug log

sample347.0ms

Algorithm
intervals

simplify76.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

76.0ms
(* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2)))

prune25.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize30.0ms

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))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
3 → 74
Calls

3 calls. Slowest were:

6.0ms
(/ (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))

series479.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

267.0ms
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))
158.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
54.0ms
(* PI (* n 2))

simplify2.4s

Counts
67 → 83
Calls

67 calls. Slowest were:

367.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))
310.0ms
(- (log (pow (* PI (* n 2)) (- 1/2 (/ k 2)))) (log (sqrt k)))
273.0ms
(- (+ (* 1/4 (* (log (* 2 PI)) (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (* (log n) (pow k 2))))) (+ (* 1/8 (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (* (pow (log n) 2) (pow k 2)))) (+ (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (* 1/8 (* (pow (log (* 2 PI)) 2) (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (pow k 2))))))) (+ (* 1/2 (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (* (log n) k))) (* 1/2 (* (log (* 2 PI)) (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) k)))))

prune989.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize13.0ms

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)))))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
4 → 116
Calls

4 calls. Slowest were:

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

series567.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

181.0ms
(/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2))))
159.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
157.0ms
(/ 1 (/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
69.0ms
(* PI (* n 2))

simplify3.4s

Counts
95 → 128
Calls

95 calls. Slowest were:

430.0ms
(/ (sqrt k) (sqrt (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
340.0ms
(- (log (sqrt k)) (log (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
334.0ms
(- (+ (* 1/4 (* (log (* 2 PI)) (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (* (log n) (pow k 2))))) (+ (* 1/8 (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (* (pow (log n) 2) (pow k 2)))) (+ (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (* 1/8 (* (pow (log (* 2 PI)) 2) (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (pow k 2))))))) (+ (* 1/2 (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (* (log n) k))) (* 1/2 (* (log (* 2 PI)) (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) k)))))

prune1.6s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0b

localize34.0ms

Local error

Found 4 expressions with local error:

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

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 115
Calls

4 calls. Slowest were:

8.0ms
(/ (sqrt k) (pow (* 2 (* n PI)) (- 1/2 (/ k 2))))
5.0ms
(/ 1 (/ (sqrt k) (pow (* 2 (* n PI)) (- 1/2 (/ k 2)))))
3.0ms
(pow (* 2 (* n PI)) (- 1/2 (/ k 2)))

series582.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

213.0ms
(/ (sqrt k) (pow (* 2 (* n PI)) (- 1/2 (/ k 2))))
180.0ms
(/ 1 (/ (sqrt k) (pow (* 2 (* n PI)) (- 1/2 (/ k 2)))))
154.0ms
(pow (* 2 (* n PI)) (- 1/2 (/ k 2)))
34.0ms
(* n PI)

simplify3.1s

Counts
95 → 127
Calls

95 calls. Slowest were:

374.0ms
(- (log (sqrt k)) (log (pow (* 2 (* n PI)) (- 1/2 (/ k 2)))))
320.0ms
(/ (* (* (sqrt k) (sqrt k)) (sqrt k)) (* (* (pow (* 2 (* n PI)) (- 1/2 (/ k 2))) (pow (* 2 (* n PI)) (- 1/2 (/ k 2)))) (pow (* 2 (* n PI)) (- 1/2 (/ k 2)))))
317.0ms
(- (+ (* 1/4 (* (log (* 2 PI)) (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (* (log n) (pow k 2))))) (+ (* 1/8 (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (* (pow (log n) 2) (pow k 2)))) (+ (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (* 1/8 (* (pow (log (* 2 PI)) 2) (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (pow k 2))))))) (+ (* 1/2 (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (* (log n) k))) (* 1/2 (* (log (* 2 PI)) (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) k)))))

prune1.6s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

localize19.0ms

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

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 109
Calls

4 calls. Slowest were:

11.0ms
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt (sqrt k)))
4.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
2.0ms
(* PI (* n 2))

series660.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

238.0ms
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt (sqrt k)))
178.0ms
(sqrt (sqrt k))
159.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
84.0ms
(* PI (* n 2))

simplify4.3s

Counts
113 → 121
Calls

113 calls. Slowest were:

624.0ms
(- (* (log (* PI (* n 2))) (- 1/2 (/ k 2))) (log (sqrt (sqrt k))))
589.0ms
(- (* (log (* PI (* n 2))) (- 1/2 (/ k 2))) (log (sqrt (sqrt k))))
320.0ms
(- (+ (* 1/4 (* (log (* 2 PI)) (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (* (log n) (pow k 2))))) (+ (* 1/8 (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (* (pow (log n) 2) (pow k 2)))) (+ (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (* 1/8 (* (pow (log (* 2 PI)) 2) (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (pow k 2))))))) (+ (* 1/2 (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (* (log n) k))) (* 1/2 (* (log (* 2 PI)) (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) k)))))

prune1.7s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0b

regimes146.0ms

Accuracy

0% (0.3b remaining)

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

bsearch3.0ms

end0.0ms

sample8.7s

Algorithm
intervals