Average Error: 0.4 → 0.4
Time: 30.7s
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 clear-num0.4

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

  5. Taylor expanded around -inf 0.4

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

  6. 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 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))

Details

Time bar (total: 29.9s)Debug log

sample284.0ms

Algorithm
intervals

simplify56.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

56.0ms
(* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2)))

prune27.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize28.0ms

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))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
3 → 68
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))

series820.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

446.0ms
(* PI (* n 2))
211.0ms
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))
163.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))

simplify3.3s

Counts
67 → 77
Calls

67 calls. Slowest were:

634.0ms
(- (* (log (* PI (* n 2))) (- 1/2 (/ k 2))) (log (sqrt k)))
604.0ms
(- (* (log (* PI (* n 2))) (- 1/2 (/ k 2))) (log (sqrt k)))
282.0ms
(- (+ (* 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)))))

prune819.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize10.0ms

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)))))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 108
Calls

4 calls. Slowest were:

9.0ms
(/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2))))
5.0ms
(/ 1 (/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
4.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))

series506.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

159.0ms
(/ 1 (/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
157.0ms
(/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2))))
109.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
80.0ms
(* PI (* n 2))

simplify3.2s

Counts
95 → 120
Calls

95 calls. Slowest were:

345.0ms
(- (log (sqrt k)) (* (log (* PI (* n 2))) (- 1/2 (/ k 2))))
338.0ms
(- (log (sqrt k)) (* (log (* PI (* n 2))) (- 1/2 (/ k 2))))
288.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))

prune1.5s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0b

localize31.0ms

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 → 107
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)))))
4.0ms
(pow (* 2 (* n PI)) (- 1/2 (/ k 2)))

series589.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

205.0ms
(/ 1 (/ (sqrt k) (pow (* 2 (* n PI)) (- 1/2 (/ k 2)))))
193.0ms
(/ (sqrt k) (pow (* 2 (* n PI)) (- 1/2 (/ k 2))))
141.0ms
(pow (* 2 (* n PI)) (- 1/2 (/ k 2)))
50.0ms
(* n PI)

simplify3.1s

Counts
95 → 119
Calls

95 calls. Slowest were:

391.0ms
(- (log (sqrt k)) (* (log (* 2 (* n PI))) (- 1/2 (/ k 2))))
309.0ms
(- (log (sqrt k)) (* (log (* 2 (* n PI))) (- 1/2 (/ k 2))))
295.0ms
(* (log (* 2 (* n PI))) (- 1/2 (/ k 2)))

prune1.4s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

localize8.0ms

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))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 101
Calls

4 calls. Slowest were:

6.0ms
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt (sqrt k)))
2.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
1.0ms
(* PI (* n 2))

series631.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

233.0ms
(sqrt (sqrt k))
212.0ms
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt (sqrt k)))
119.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
67.0ms
(* PI (* n 2))

simplify4.1s

Counts
113 → 113
Calls

113 calls. Slowest were:

609.0ms
(- (* (log (* PI (* n 2))) (- 1/2 (/ k 2))) (log (sqrt (sqrt k))))
592.0ms
(- (* (log (* PI (* n 2))) (- 1/2 (/ k 2))) (log (sqrt (sqrt k))))
302.0ms
(- (+ (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (pow (/ 1 k) 1/4)) (+ (* 1/8 (* (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (pow (log n) 2)) (pow (pow k 7) 1/4))) (+ (* 1/4 (* (* (log (* 2 PI)) (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (log n))) (pow (pow k 7) 1/4))) (* 1/8 (* (* (pow (log (* 2 PI)) 2) (exp (* 1/2 (+ (log n) (log (* 2 PI)))))) (pow (pow k 7) 1/4)))))) (+ (* 1/2 (* (* (log (* 2 PI)) (exp (* 1/2 (+ (log n) (log (* 2 PI)))))) (pow (pow k 3) 1/4))) (* 1/2 (* (* (exp (* 1/2 (+ (log n) (log (* 2 PI))))) (log n)) (pow (pow k 3) 1/4)))))

prune1.5s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0b

regimes119.0ms

Accuracy

0% (0.3b remaining)

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

bsearch2.0ms

end0.0ms

sample7.7s

Algorithm
intervals