Average Error: 0.4 → 0.5
Time: 27.9s
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\left(\sqrt{k} \cdot \sqrt{\frac{1}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\right) \cdot \sqrt{\frac{1}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}}\]

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 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. Using strategy rm

  6. Applied div-inv0.4

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

  8. Applied add-sqr-sqrt0.5

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

  9. Applied associate-*r*0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019007 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))

Details

Time bar (total: 26.9s)Debug log

sample362.0ms

Algorithm
intervals

simplify65.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

64.0ms
(* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2)))

prune18.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize21.0ms

Local error

Found 3 expressions with local error:

0.4b
(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))

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))

series453.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

229.0ms
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))
156.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
68.0ms
(* PI (* n 2))

simplify3.4s

Counts
67 → 77
Calls

67 calls. Slowest were:

620.0ms
(- (* (log (* PI (* n 2))) (- 1/2 (/ k 2))) (log (sqrt k)))
614.0ms
(- (* (log (* PI (* n 2))) (- 1/2 (/ k 2))) (log (sqrt k)))
329.0ms
(- (+ (* 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)))))

prune927.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 4 expressions with local error:

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

rewrite10.0ms

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)))

series584.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

183.0ms
(/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2))))
167.0ms
(/ 1 (/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
154.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
79.0ms
(* PI (* n 2))

simplify3.2s

Counts
95 → 120
Calls

95 calls. Slowest were:

392.0ms
(- (log (sqrt k)) (* (log (* PI (* n 2))) (- 1/2 (/ k 2))))
371.0ms
(- (log (sqrt k)) (* (log (* PI (* n 2))) (- 1/2 (/ k 2))))
264.0ms
(- (+ (* 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)))))

prune1.8s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 4 expressions with local error:

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

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

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

series574.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

207.0ms
(/ 1 (pow (* PI (* n 2)) (- 1/2 (/ k 2))))
165.0ms
(* (sqrt k) (/ 1 (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
129.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
72.0ms
(* PI (* n 2))

simplify2.9s

Counts
44 → 84
Calls

44 calls. Slowest were:

416.0ms
(* (* (* (sqrt k) (sqrt k)) (sqrt k)) (* (* (/ 1 (pow (* PI (* n 2)) (- 1/2 (/ k 2)))) (/ 1 (pow (* PI (* n 2)) (- 1/2 (/ k 2))))) (/ 1 (pow (* PI (* n 2)) (- 1/2 (/ k 2))))))
259.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))
256.0ms
(+ (log (sqrt k)) (- (log (pow (* PI (* n 2)) (- 1/2 (/ k 2))))))

prune1.1s

Pruning

9 alts after pruning (9 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)))
0.4b
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
0.2b
(* PI (* n 2))
0.2b
(* PI (* n 2))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

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

series427.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

144.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
137.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
82.0ms
(* PI (* n 2))
63.0ms
(* PI (* n 2))

simplify1.8s

Counts
30 → 64
Calls

30 calls. Slowest were:

291.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))
273.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))
241.0ms
(- (+ (* 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)))))

prune997.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0b

regimes137.0ms

Accuracy

0% (0.5b remaining)

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

bsearch2.0ms

end0.0ms

sample8.1s

Algorithm
intervals