Average Error: 0.4 → 0.4
Time: 58.8s
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[{\left(2 \cdot n\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)} \cdot \frac{\frac{1}{\frac{1}{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}}{\sqrt{k}}\]

Error

Bits error versus k

Bits error versus n

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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 *-un-lft-identity0.4

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

  5. Applied associate-/l*0.4

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

  7. Applied unpow-prod-down0.5

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

  8. Applied *-un-lft-identity0.5

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

  9. Applied times-frac0.4

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

  10. Applied associate-/r*0.4

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

  12. Applied associate-/r/0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019005 +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: 58.0s)Debug log

sample319.0ms

Algorithm
intervals

simplify78.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

77.0ms
(* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2)))

prune12.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.3b

localize40.0ms

Local error

Found 3 expressions with local error:

1.4b
(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 → 74
Calls

3 calls. Slowest were:

10.0ms
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))
4.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
2.0ms
(* PI (* n 2))

series429.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

196.0ms
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))
167.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
66.0ms
(* PI (* n 2))

simplify2.4s

Counts
67 → 83
Calls

67 calls. Slowest were:

322.0ms
(- (+ (* 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)))))
302.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))
278.0ms
(/ (* (* (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)))

prune877.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 4 expressions with local error:

1.4b
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
0.3b
(* PI (* n 2))
0.1b
(/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2))))
0.1b
(/ 1 (/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 116
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)))

series564.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

197.0ms
(/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2))))
166.0ms
(/ 1 (/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
129.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
71.0ms
(* PI (* n 2))

simplify3.4s

Counts
95 → 128
Calls

95 calls. Slowest were:

569.0ms
(exp (* (- 1/2 (* 1/2 k)) (- (log (* 2 PI)) (log (/ 1 n)))))
340.0ms
(- (log (sqrt k)) (log (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
323.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))

prune1.6s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 4 expressions with local error:

1.4b
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
0.3b
(/ 1 (sqrt k))
0.3b
(* PI (* n 2))
0.2b
(/ (/ 1 (sqrt k)) (/ 1 (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 101
Calls

4 calls. Slowest were:

9.0ms
(/ (/ 1 (sqrt k)) (/ 1 (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
2.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
1.0ms
(* PI (* n 2))

series399.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

191.0ms
(/ (/ 1 (sqrt k)) (/ 1 (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
140.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
57.0ms
(* PI (* n 2))
12.0ms
(/ 1 (sqrt k))

simplify5.3s

Counts
83 → 113
Calls

83 calls. Slowest were:

1.2s
(/ (* (* (/ 1 (sqrt k)) (/ 1 (sqrt k))) (/ 1 (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))))))
431.0ms
(- (- (log (sqrt k))) (- (log (pow (* PI (* n 2)) (- 1/2 (/ k 2))))))
360.0ms
(- (log (/ 1 (sqrt k))) (- (log (pow (* PI (* n 2)) (- 1/2 (/ k 2))))))

prune1.4s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.0b

localize21.0ms

Local error

Found 4 expressions with local error:

1.3b
(pow (* n 2) (- 1/2 (/ k 2)))
0.4b
(pow PI (- 1/2 (/ k 2)))
0.2b
(/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2))))
0.1b
(/ (/ 1 (/ 1 (pow PI (- 1/2 (/ k 2))))) (/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2)))))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 254
Calls

4 calls. Slowest were:

6.0ms
(/ (/ 1 (/ 1 (pow PI (- 1/2 (/ k 2))))) (/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2)))))
4.0ms
(/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2))))
1.0ms
(pow (* n 2) (- 1/2 (/ k 2)))

series512.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

274.0ms
(/ (/ 1 (/ 1 (pow PI (- 1/2 (/ k 2))))) (/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2)))))
114.0ms
(/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2))))
91.0ms
(pow (* n 2) (- 1/2 (/ k 2)))
32.0ms
(pow PI (- 1/2 (/ k 2)))

simplify29.1s

Counts
351 → 266
Calls

351 calls. Slowest were:

749.0ms
(/ (/ 1 (/ 1 (pow PI (- 1/2 (/ k 2))))) (/ (sqrt (sqrt k)) (pow (* n 2) (- 1/2 (/ k 2)))))
610.0ms
(- (- (log (/ 1 (pow PI (- 1/2 (/ k 2)))))) (- (log (sqrt k)) (log (pow (* n 2) (- 1/2 (/ k 2))))))
542.0ms
(exp (* (- (log 2) (log (/ 1 n))) (- 1/2 (* 1/2 k))))

prune3.5s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

regimes109.0ms

Accuracy

0% (0.4b remaining)

Error of 0.4b against oracle of 0.0b and baseline of 0.4b

bsearch3.0ms

end0.0ms

sample7.8s

Algorithm
intervals