Average Error: 0.4 → 0.5
Time: 27.9s
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}} \cdot \sqrt{\frac{{\left(\sqrt[3]{\left(2 \cdot n\right) \cdot \pi} \cdot \left(\sqrt[3]{\left(2 \cdot n\right) \cdot \pi} \cdot \sqrt[3]{\left(2 \cdot n\right) \cdot \pi}\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}}\]

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. 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 add-sqr-sqrt0.5

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

  6. Applied add-cube-cbrt0.5

    \[\leadsto \sqrt{\frac{{\color{blue}{\left(\left(\sqrt[3]{\pi \cdot \left(n \cdot 2\right)} \cdot \sqrt[3]{\pi \cdot \left(n \cdot 2\right)}\right) \cdot \sqrt[3]{\pi \cdot \left(n \cdot 2\right)}\right)}}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}} \cdot \sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}}\]

  7. Final simplification0.5

    \[\leadsto \sqrt{\frac{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}} \cdot \sqrt{\frac{{\left(\sqrt[3]{\left(2 \cdot n\right) \cdot \pi} \cdot \left(\sqrt[3]{\left(2 \cdot n\right) \cdot \pi} \cdot \sqrt[3]{\left(2 \cdot n\right) \cdot \pi}\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}}\]

Reproduce

herbie shell --seed 2019007 +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: 26.8s)Debug log

sample324.0ms

Algorithm
intervals

simplify85.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune12.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

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))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
3 → 74
Calls

3 calls. Slowest were:

8.0ms
(/ (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))

series448.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

242.0ms
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))
144.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
62.0ms
(* PI (* n 2))

simplify2.3s

Counts
67 → 83
Calls

67 calls. Slowest were:

372.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))
287.0ms
(/ (* (* (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)))
279.0ms
(- (log (pow (* PI (* n 2)) (- 1/2 (/ k 2)))) (log (sqrt k)))

prune1.0s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize8.0ms

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)))))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 116
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)))))
2.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))

series606.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

191.0ms
(/ 1 (/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
179.0ms
(/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2))))
150.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
86.0ms
(* PI (* n 2))

simplify3.3s

Counts
95 → 128
Calls

95 calls. Slowest were:

376.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))
371.0ms
(- (+ (* 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)))))
323.0ms
(- (log (sqrt k)) (log (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))

prune2.1s

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))))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 80
Calls

4 calls. Slowest were:

10.0ms
(* (sqrt k) (/ 1 (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
4.0ms
(/ 1 (pow (* PI (* n 2)) (- 1/2 (/ k 2))))
3.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))

series636.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

184.0ms
(/ 1 (pow (* PI (* n 2)) (- 1/2 (/ k 2))))
184.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
181.0ms
(* (sqrt k) (/ 1 (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
87.0ms
(* PI (* n 2))

simplify3.1s

Counts
44 → 92
Calls

44 calls. Slowest were:

348.0ms
(* (* (* (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))))))
337.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))
321.0ms
(+ (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))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

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

series401.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

146.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
130.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
69.0ms
(* PI (* n 2))
56.0ms
(* PI (* n 2))

simplify2.2s

Counts
30 → 72
Calls

30 calls. Slowest were:

358.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))
353.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))
313.0ms
(- (+ (* 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)))))

prune1.1s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0b

regimes186.0ms

Accuracy

0% (0.5b remaining)

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

bsearch2.0ms

end0.0ms

sample7.7s

Algorithm
intervals