Average Error: 0.4 → 0.5
Time: 40.2s
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. Initial simplification0.3

    \[\leadsto \frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}\]
  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 2018360 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))

Details

Time bar (total: 26.0s)Debug log

start472.0ms

Algorithm
intervals

setup88.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.3b

localize22.0ms

Local error

Found 3 expressions with local error:

0.7b
(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))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
3 → 68
Calls

3 calls. Slowest were:

9.0ms
(/ (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))

series426.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

231.0ms
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))
146.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
49.0ms
(* PI (* n 2))

simplify3.2s

Counts
67 → 77
Calls

67 calls. Slowest were:

575.0ms
(- (* (log (* PI (* n 2))) (- 1/2 (/ k 2))) (log (sqrt k)))
565.0ms
(- (* (log (* PI (* n 2))) (- 1/2 (/ k 2))) (log (sqrt k)))
348.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))

prune1.1s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 4 expressions with local error:

0.7b
(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)))))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 108
Calls

4 calls. Slowest were:

8.0ms
(/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2))))
4.0ms
(/ 1 (/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
3.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))

series600.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

201.0ms
(/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2))))
164.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
150.0ms
(/ 1 (/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
84.0ms
(* PI (* n 2))

simplify2.8s

Counts
95 → 120
Calls

95 calls. Slowest were:

347.0ms
(- (log (sqrt k)) (* (log (* PI (* n 2))) (- 1/2 (/ k 2))))
297.0ms
(- (log (sqrt k)) (log (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
290.0ms
(- (log (sqrt k)) (* (log (* PI (* n 2))) (- 1/2 (/ k 2))))

prune1.8s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 4 expressions with local error:

0.6b
(pow (* n 2) (- 1/2 (/ k 2)))
0.5b
(pow PI (- 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)))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 121
Calls

4 calls. Slowest were:

13.0ms
(* (pow PI (- 1/2 (/ k 2))) (/ (pow (* n 2) (- 1/2 (/ k 2))) (sqrt k)))
9.0ms
(/ (pow (* n 2) (- 1/2 (/ k 2))) (sqrt k))
3.0ms
(pow (* n 2) (- 1/2 (/ k 2)))

series595.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

275.0ms
(* (pow PI (- 1/2 (/ k 2))) (/ (pow (* n 2) (- 1/2 (/ k 2))) (sqrt k)))
181.0ms
(/ (pow (* n 2) (- 1/2 (/ k 2))) (sqrt k))
98.0ms
(pow (* n 2) (- 1/2 (/ k 2)))
40.0ms
(pow PI (- 1/2 (/ k 2)))

simplify5.6s

Counts
116 → 133
Calls

116 calls. Slowest were:

1.2s
(+ (log (pow PI (- 1/2 (/ k 2)))) (log (/ (pow (* n 2) (- 1/2 (/ k 2))) (sqrt k))))
492.0ms
(+ (* (log PI) (- 1/2 (/ k 2))) (- (log (pow (* n 2) (- 1/2 (/ k 2)))) (log (sqrt k))))
369.0ms
(+ (log (pow PI (- 1/2 (/ k 2)))) (- (log (pow (* n 2) (- 1/2 (/ k 2)))) (log (sqrt k))))

prune2.0s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.0b

localize34.0ms

Local error

Found 4 expressions with local error:

0.6b
(pow n (- 1/2 (/ k 2)))
0.5b
(pow PI (- 1/2 (/ k 2)))
0.2b
(/ (* (pow n (- 1/2 (/ k 2))) (pow 2 (- 1/2 (/ k 2)))) (sqrt k))
0.2b
(* (pow n (- 1/2 (/ k 2))) (pow 2 (- 1/2 (/ k 2))))

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

15.0ms
(/ (* (pow n (- 1/2 (/ k 2))) (pow 2 (- 1/2 (/ k 2)))) (sqrt k))
10.0ms
(* (pow n (- 1/2 (/ k 2))) (pow 2 (- 1/2 (/ k 2))))
1.0ms
(pow n (- 1/2 (/ k 2)))

series533.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

260.0ms
(/ (* (pow n (- 1/2 (/ k 2))) (pow 2 (- 1/2 (/ k 2)))) (sqrt k))
163.0ms
(* (pow n (- 1/2 (/ k 2))) (pow 2 (- 1/2 (/ k 2))))
85.0ms
(pow n (- 1/2 (/ k 2)))
24.0ms
(pow PI (- 1/2 (/ k 2)))

simplify4.7s

Counts
64 → 94
Calls

64 calls. Slowest were:

480.0ms
(exp (* (- 1/2 (* 1/2 k)) (- (log -1) (log (/ -1 n)))))
420.0ms
(- (+ (* (log n) (- 1/2 (/ k 2))) (log (pow 2 (- 1/2 (/ k 2))))) (log (sqrt k)))
395.0ms
(- (+ (log (pow n (- 1/2 (/ k 2)))) (log (pow 2 (- 1/2 (/ k 2))))) (log (sqrt k)))

prune1.7s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0b

regimes160.0ms

Accuracy

0% (0.4b remaining)

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

bsearch3.0ms