Average Error: 0.4 → 0.5
Time: 31.9s
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\sqrt{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}} \cdot \sqrt{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}}\]

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 add-sqr-sqrt0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019008 +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: 31.1s)Debug log

sample303.0ms

Algorithm
intervals

simplify56.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

55.0ms
(* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2)))

prune19.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.3b

localize29.0ms

Local error

Found 3 expressions with local error:

0.5b
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
0.2b
(* PI (* n 2))
0.1b
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
3 → 74
Calls

3 calls. Slowest were:

10.0ms
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))
3.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
2.0ms
(* PI (* n 2))

series400.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

198.0ms
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))
144.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
58.0ms
(* PI (* n 2))

simplify2.4s

Counts
67 → 83
Calls

67 calls. Slowest were:

336.0ms
(/ (* (* (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (pow (* PI (* n 2)) (- 1/2 (/ k 2)))) (pow (* PI (* n 2)) (- 1/2 (/ k 2)))) (* (* (sqrt k) (sqrt k)) (sqrt k)))
298.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))
268.0ms
(- (+ (* 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)))))

prune859.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 4 expressions with local error:

0.5b
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
0.5b
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
0.2b
(* (sqrt (pow (* PI (* n 2)) (- 1/2 (/ k 2)))) (sqrt (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
0.2b
(* PI (* n 2))

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
4 → 90
Calls

4 calls. Slowest were:

20.0ms
(* (sqrt (pow (* PI (* n 2)) (- 1/2 (/ k 2)))) (sqrt (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
4.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
3.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))

series473.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

139.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
133.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
119.0ms
(* (sqrt (pow (* PI (* n 2)) (- 1/2 (/ k 2)))) (sqrt (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
83.0ms
(* PI (* n 2))

simplify3.1s

Counts
63 → 102
Calls

63 calls. Slowest were:

324.0ms
(- (+ (* 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)))))
314.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))
287.0ms
(- (+ (* 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)))))

prune1.5s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 4 expressions with local error:

0.5b
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
0.5b
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
0.3b
(* (sqrt (/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))) (sqrt (/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))))
0.2b
(* PI (* n 2))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 122
Calls

4 calls. Slowest were:

15.0ms
(* (sqrt (/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))) (sqrt (/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))))
3.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
3.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))

series616.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

235.0ms
(* (sqrt (/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))) (sqrt (/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))))
155.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
153.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
74.0ms
(* PI (* n 2))

simplify3.3s

Counts
95 → 134
Calls

95 calls. Slowest were:

339.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))
321.0ms
(- (+ (* 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)))))
320.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))

prune1.9s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 4 expressions with local error:

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

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 129
Calls

4 calls. Slowest were:

7.0ms
(* (pow PI (- 1/2 (/ k 2))) (/ (pow (* n 2) (- 1/2 (/ k 2))) (sqrt k)))
5.0ms
(/ (pow (* n 2) (- 1/2 (/ k 2))) (sqrt k))
1.0ms
(pow (* n 2) (- 1/2 (/ k 2)))

series556.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

223.0ms
(* (pow PI (- 1/2 (/ k 2))) (/ (pow (* n 2) (- 1/2 (/ k 2))) (sqrt k)))
179.0ms
(/ (pow (* n 2) (- 1/2 (/ k 2))) (sqrt k))
117.0ms
(pow (* n 2) (- 1/2 (/ k 2)))
37.0ms
(pow PI (- 1/2 (/ k 2)))

simplify6.1s

Counts
116 → 141
Calls

116 calls. Slowest were:

564.0ms
(* (* (* (pow PI (- 1/2 (/ k 2))) (pow PI (- 1/2 (/ k 2)))) (pow PI (- 1/2 (/ k 2)))) (/ (* (* (pow (* n 2) (- 1/2 (/ k 2))) (pow (* n 2) (- 1/2 (/ k 2)))) (pow (* n 2) (- 1/2 (/ k 2)))) (* (* (sqrt k) (sqrt k)) (sqrt k))))
542.0ms
(exp (* (- (log 2) (log (/ 1 n))) (- 1/2 (* 1/2 k))))
503.0ms
(+ (* (log PI) (- 1/2 (/ k 2))) (- (log (pow (* n 2) (- 1/2 (/ k 2)))) (log (sqrt k))))

prune1.7s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0b

regimes111.0ms

Accuracy

0% (0.5b remaining)

Error of 0.5b against oracle of 0.0b and baseline of 0.5b

bsearch3.0ms

end0.0ms

sample7.5s

Algorithm
intervals