Average Error: 0.4 → 0.5
Time: 37.6s
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left({k}^{\frac{-1}{2}} \cdot \left({\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {2}^{\left(\frac{1 - k}{2}\right)}\right)\right) \cdot {n}^{\left(\frac{1 - k}{2}\right)}\]

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. Using strategy rm

  3. Applied unpow-prod-down0.5

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

  4. Applied associate-*r*0.5

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

  6. Applied unpow-prod-down0.5

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

  8. Applied pow1/20.5

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

  9. Applied pow-flip0.5

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

  10. Simplified0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2018365 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))

Details

Time bar (total: 23.5s)Debug log

start878.0ms

Algorithm
intervals

setup104.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.4b

localize44.0ms

Local error

Found 3 expressions with local error:

0.9b
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
0.2b
(* PI (* n 2))
0.2b
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))

rewrite16.0ms

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))

series433.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

230.0ms
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))
153.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
50.0ms
(* PI (* n 2))

simplify3.5s

Counts
67 → 77
Calls

67 calls. Slowest were:

681.0ms
(- (* (log (* PI (* n 2))) (- 1/2 (/ k 2))) (log (sqrt k)))
666.0ms
(- (* (log (* PI (* n 2))) (- 1/2 (/ k 2))) (log (sqrt k)))
302.0ms
(/ (* (* (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)))

prune1.1s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize38.0ms

Local error

Found 4 expressions with local error:

0.9b
(pow (* (* 2 PI) n) (/ (- 1 k) 2))
0.2b
(* (* 2 PI) n)
0.2b
(/ 1 (sqrt k))
0.2b
(* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2)))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

16.0ms
(* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2)))
5.0ms
(pow (* (* 2 PI) n) (/ (- 1 k) 2))
4.0ms
(* (* 2 PI) n)

series428.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

183.0ms
(* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2)))
162.0ms
(pow (* (* 2 PI) n) (/ (- 1 k) 2))
70.0ms
(* (* 2 PI) n)
12.0ms
(/ 1 (sqrt k))

simplify3.1s

Counts
42 → 80
Calls

42 calls. Slowest were:

569.0ms
(* (log (* (* 2 PI) n)) (/ (- 1 k) 2))
487.0ms
(* (* (* (/ 1 (sqrt k)) (/ 1 (sqrt k))) (/ 1 (sqrt k))) (* (* (pow (* (* 2 PI) n) (/ (- 1 k) 2)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))
394.0ms
(+ (log (/ 1 (sqrt k))) (log (pow (* (* 2 PI) n) (/ (- 1 k) 2))))

prune1.1s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize31.0ms

Local error

Found 4 expressions with local error:

0.6b
(pow n (/ (- 1 k) 2))
0.5b
(pow (* 2 PI) (/ (- 1 k) 2))
0.3b
(* (/ 1 (sqrt k)) (pow (* 2 PI) (/ (- 1 k) 2)))
0.2b
(/ 1 (sqrt k))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

9.0ms
(* (/ 1 (sqrt k)) (pow (* 2 PI) (/ (- 1 k) 2)))
2.0ms
(pow (* 2 PI) (/ (- 1 k) 2))
2.0ms
(pow n (/ (- 1 k) 2))

series288.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

133.0ms
(* (/ 1 (sqrt k)) (pow (* 2 PI) (/ (- 1 k) 2)))
76.0ms
(pow n (/ (- 1 k) 2))
59.0ms
(pow (* 2 PI) (/ (- 1 k) 2))
20.0ms
(/ 1 (sqrt k))

simplify3.6s

Counts
47 → 84
Calls

47 calls. Slowest were:

513.0ms
(* (* (* (/ 1 (sqrt k)) (/ 1 (sqrt k))) (/ 1 (sqrt k))) (* (* (pow (* 2 PI) (/ (- 1 k) 2)) (pow (* 2 PI) (/ (- 1 k) 2))) (pow (* 2 PI) (/ (- 1 k) 2))))
442.0ms
(+ (log (/ 1 (sqrt k))) (* (log (* 2 PI)) (/ (- 1 k) 2)))
404.0ms
(+ (- (log (sqrt k))) (* (log (* 2 PI)) (/ (- 1 k) 2)))

prune1.3s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0b

localize22.0ms

Local error

Found 4 expressions with local error:

0.6b
(pow n (/ (- 1 k) 2))
0.5b
(pow PI (/ (- 1 k) 2))
0.3b
(* (/ 1 (sqrt k)) (* (pow 2 (/ (- 1 k) 2)) (pow PI (/ (- 1 k) 2))))
0.2b
(/ 1 (sqrt k))

rewrite42.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

37.0ms
(* (/ 1 (sqrt k)) (* (pow 2 (/ (- 1 k) 2)) (pow PI (/ (- 1 k) 2))))
2.0ms
(pow n (/ (- 1 k) 2))
2.0ms
(pow PI (/ (- 1 k) 2))

series313.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

204.0ms
(* (/ 1 (sqrt k)) (* (pow 2 (/ (- 1 k) 2)) (pow PI (/ (- 1 k) 2))))
63.0ms
(pow n (/ (- 1 k) 2))
28.0ms
(pow PI (/ (- 1 k) 2))
17.0ms
(/ 1 (sqrt k))

simplify5.4s

Counts
53 → 88
Calls

53 calls. Slowest were:

591.0ms
(* (* (* (/ 1 (sqrt k)) (/ 1 (sqrt k))) (/ 1 (sqrt k))) (* (* (* (pow 2 (/ (- 1 k) 2)) (pow PI (/ (- 1 k) 2))) (* (pow 2 (/ (- 1 k) 2)) (pow PI (/ (- 1 k) 2)))) (* (pow 2 (/ (- 1 k) 2)) (pow PI (/ (- 1 k) 2)))))
498.0ms
(* (* (* (/ 1 (sqrt k)) (/ 1 (sqrt k))) (/ 1 (sqrt k))) (* (* (* (pow 2 (/ (- 1 k) 2)) (pow 2 (/ (- 1 k) 2))) (pow 2 (/ (- 1 k) 2))) (* (* (pow PI (/ (- 1 k) 2)) (pow PI (/ (- 1 k) 2))) (pow PI (/ (- 1 k) 2)))))
474.0ms
(+ (log (/ 1 (sqrt k))) (+ (log (pow 2 (/ (- 1 k) 2))) (* (log PI) (/ (- 1 k) 2))))

prune1.5s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

regimes237.0ms

Accuracy

0% (0.4b remaining)

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

bsearch3.0ms