Average Error: 0.4 → 0.4
Time: 26.2s
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {k}^{\frac{-1}{2}}\]

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 pow1/20.4

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

  4. Applied pow-flip0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019010 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))

Details

Time bar (total: 25.4s)Debug log

sample358.0ms

Algorithm
intervals

simplify90.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
90.0ms
(* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2)))

prune24.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize33.0ms

Local error

Found 3 expressions with local error:

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
18×times-frac
15×add-cube-cbrt
15×*-un-lft-identity
15×add-sqr-sqrt
10×unpow-prod-up
9×add-exp-log
6×associate-/l*
5×add-cbrt-cube
5×unpow-prod-down
5×sub-neg
4×pow1
3×add-log-exp
3×div-exp
3×associate-/r*
3×pow-unpow
2×pow-sub
2×pow-exp
2×pow-to-exp
1×associate-/l/
1×div-inv
1×*-commutative
1×associate-*r*
1×frac-2neg
1×pow-pow
1×clear-num
1×cbrt-undiv
Counts
3 → 68
Calls
3 calls:
Slowest
5.0ms
(/ (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))

series416.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
205.0ms
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))
137.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
73.0ms
(* PI (* n 2))

simplify3.3s

Counts
67 → 77
Calls
67 calls:
Slowest
642.0ms
(- (* (log (* PI (* n 2))) (- 1/2 (/ k 2))) (log (sqrt k)))
622.0ms
(- (* (log (* PI (* n 2))) (- 1/2 (/ k 2))) (log (sqrt k)))
256.0ms
(- (+ (* 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)))))
244.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))
238.0ms
(/ (* (* (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)))

prune850.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0b

localize45.0ms

Local error

Found 4 expressions with local error:

0.4b
(pow (* (* 2 PI) n) (/ (- 1 k) 2))
0.3b
(/ 1 (sqrt k))
0.2b
(* (* 2 PI) n)
0.2b
(* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2)))

rewrite28.0ms

Algorithm
rewrite-expression-head
Rules
16×add-exp-log
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
6×prod-exp
6×add-cbrt-cube
6×pow1
5×associate-*l*
4×add-log-exp
4×associate-*r*
4×pow-unpow
4×rec-exp
3×pow-sub
3×div-inv
3×pow-exp
3×div-sub
3×associate-/r*
3×pow-to-exp
2×pow-flip
2×*-commutative
2×unpow-prod-down
1×inv-pow
1×cbrt-unprod
1×associate-*r/
1×associate-*l/
1×pow1/2
1×frac-2neg
1×pow-pow
1×clear-num
1×frac-times
Counts
4 → 68
Calls
4 calls:
Slowest
17.0ms
(* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2)))
5.0ms
(pow (* (* 2 PI) n) (/ (- 1 k) 2))
4.0ms
(* (* 2 PI) n)
1.0ms
(/ 1 (sqrt k))

series357.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
165.0ms
(* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2)))
125.0ms
(pow (* (* 2 PI) n) (/ (- 1 k) 2))
55.0ms
(* (* 2 PI) n)
11.0ms
(/ 1 (sqrt k))

simplify3.2s

Counts
42 → 80
Calls
42 calls:
Slowest
507.0ms
(* (log (* (* 2 PI) n)) (/ (- 1 k) 2))
490.0ms
(* (* (* (/ 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))))
364.0ms
(cbrt (pow (* (* 2 PI) n) (/ (- 1 k) 2)))
319.0ms
(+ (log (/ 1 (sqrt k))) (log (pow (* (* 2 PI) n) (/ (- 1 k) 2))))
250.0ms
(+ (- (log (sqrt k))) (log (pow (* (* 2 PI) n) (/ (- 1 k) 2))))

prune811.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0b

localize13.0ms

Local error

Found 3 expressions with local error:

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

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
11×add-exp-log
6×add-cube-cbrt
6×prod-exp
6×*-un-lft-identity
6×pow-to-exp
6×add-sqr-sqrt
5×add-cbrt-cube
4×associate-*l*
4×associate-*r*
4×pow-unpow
4×pow1
3×add-log-exp
3×pow-exp
2×pow-sub
2×*-commutative
2×div-sub
2×unpow-prod-down
1×div-inv
1×cbrt-unprod
1×associate-*r/
1×pow-pow
Counts
3 → 48
Calls
3 calls:
Slowest
7.0ms
(* (pow k -1/2) (pow (* (* 2 PI) n) (/ (- 1 k) 2)))
3.0ms
(pow (* (* 2 PI) n) (/ (- 1 k) 2))
2.0ms
(* (* 2 PI) n)

series389.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
190.0ms
(* (pow k -1/2) (pow (* (* 2 PI) n) (/ (- 1 k) 2)))
115.0ms
(pow (* (* 2 PI) n) (/ (- 1 k) 2))
84.0ms
(* (* 2 PI) n)

simplify2.2s

Counts
31 → 57
Calls
31 calls:
Slowest
550.0ms
(* (log (* (* 2 PI) n)) (/ (- 1 k) 2))
221.0ms
(exp (* 1/2 (* (- 1 k) (- (log (* -2 PI)) (log (/ -1 n))))))
220.0ms
(- (+ (* 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)))))
210.0ms
(+ (* (log k) -1/2) (log (pow (* (* 2 PI) n) (/ (- 1 k) 2))))
186.0ms
(+ (log (pow k -1/2)) (log (pow (* (* 2 PI) n) (/ (- 1 k) 2))))

prune757.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0b

localize15.0ms

Local error

Found 3 expressions with local error:

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

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
18×times-frac
15×add-cube-cbrt
15×*-un-lft-identity
15×add-sqr-sqrt
10×unpow-prod-up
9×add-exp-log
6×associate-/l*
5×add-cbrt-cube
5×unpow-prod-down
5×sub-neg
4×pow1
3×add-log-exp
3×div-exp
3×associate-/r*
3×pow-unpow
2×pow-sub
2×pow-exp
2×pow-to-exp
1×associate-/l/
1×div-inv
1×*-commutative
1×frac-2neg
1×pow-pow
1×clear-num
1×cbrt-undiv
Counts
3 → 67
Calls
3 calls:
Slowest
9.0ms
(/ (pow (* (* n PI) 2) (- 1/2 (/ k 2))) (sqrt k))
2.0ms
(pow (* (* n PI) 2) (- 1/2 (/ k 2)))
0.0ms
(* n PI)

series394.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
219.0ms
(/ (pow (* (* n PI) 2) (- 1/2 (/ k 2))) (sqrt k))
144.0ms
(pow (* (* n PI) 2) (- 1/2 (/ k 2)))
30.0ms
(* n PI)

simplify3.0s

Counts
67 → 76
Calls
67 calls:
Slowest
527.0ms
(- (* (log (* (* n PI) 2)) (- 1/2 (/ k 2))) (log (sqrt k)))
518.0ms
(- (* (log (* (* n PI) 2)) (- 1/2 (/ k 2))) (log (sqrt k)))
257.0ms
(- (+ (* 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)))))
253.0ms
(/ (* (* (pow (* (* n PI) 2) (- 1/2 (/ k 2))) (pow (* (* n PI) 2) (- 1/2 (/ k 2)))) (pow (* (* n PI) 2) (- 1/2 (/ k 2)))) (* (* (sqrt k) (sqrt k)) (sqrt k)))
249.0ms
(* (log (* (* n PI) 2)) (- 1/2 (/ k 2)))

prune764.0ms

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0b

regimes182.0ms

Accuracy

0% (0.3b remaining)

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

bsearch1.0ms

end0.0ms

sample8.1s

Algorithm
intervals