Average Error: 0.4 → 0.4
Time: 1.0m
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\frac{\sqrt{k}}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\]

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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019002 +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: 59.3s)Debug log

sample439.0ms

Algorithm
intervals

simplify68.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

67.0ms
(* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2)))

prune23.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize57.0ms

Local error

Found 3 expressions with local error:

0.4b
(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))

series433.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

211.0ms
(/ (pow (* PI (* n 2)) (- 1/2 (/ k 2))) (sqrt k))
139.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
82.0ms
(* PI (* n 2))

simplify2.4s

Counts
67 → 83
Calls

67 calls. Slowest were:

319.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))
308.0ms
(- (+ (* 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)))))
239.0ms
(/ (* (* (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)))

prune990.0ms

Pruning

8 alts after pruning (8 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)))
0.3b
(* 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)))))

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)))

series589.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

185.0ms
(/ 1 (/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
163.0ms
(/ (sqrt k) (pow (* PI (* n 2)) (- 1/2 (/ k 2))))
163.0ms
(pow (* PI (* n 2)) (- 1/2 (/ k 2)))
77.0ms
(* PI (* n 2))

simplify3.1s

Counts
95 → 128
Calls

95 calls. Slowest were:

379.0ms
(- (log (sqrt k)) (log (pow (* PI (* n 2)) (- 1/2 (/ k 2)))))
368.0ms
(* (log (* PI (* n 2))) (- 1/2 (/ k 2)))
290.0ms
(- (+ (* 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)))))

prune1.5s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0b

localize39.0ms

Local error

Found 4 expressions with local error:

0.5b
(pow PI (- 1/2 (/ k 2)))
0.3b
(pow (* n 2) (- 1/2 (/ k 2)))
0.1b
(/ (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)))))

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
4 → 254
Calls

4 calls. Slowest were:

12.0ms
(/ (/ 1 (/ 1 (pow PI (- 1/2 (/ k 2))))) (/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2)))))
8.0ms
(/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2))))
3.0ms
(pow (* n 2) (- 1/2 (/ k 2)))

series566.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

305.0ms
(/ (/ 1 (/ 1 (pow PI (- 1/2 (/ k 2))))) (/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2)))))
120.0ms
(/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2))))
101.0ms
(pow (* n 2) (- 1/2 (/ k 2)))
39.0ms
(pow PI (- 1/2 (/ k 2)))

simplify32.5s

Counts
351 → 266
Calls

351 calls. Slowest were:

783.0ms
(/ (/ 1 (/ 1 (pow PI (- 1/2 (/ k 2))))) (/ (cbrt (sqrt k)) (pow (* n 2) (- (/ k 2)))))
661.0ms
(- (- (log (/ 1 (pow PI (- 1/2 (/ k 2)))))) (- (log (sqrt k)) (log (pow (* n 2) (- 1/2 (/ k 2))))))
575.0ms
(/ (/ 1 (/ 1 (pow PI (- 1/2 (/ k 2))))) (/ (sqrt k) (sqrt (pow (* n 2) (- 1/2 (/ k 2))))))

prune3.8s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0b

localize13.0ms

Local error

Found 4 expressions with local error:

0.5b
(pow PI (- 1/2 (/ k 2)))
0.3b
(pow (* n 2) (- 1/2 (/ k 2)))
0.1b
(/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2))))
0.1b
(/ 1 (/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2)))))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
4 → 121
Calls

4 calls. Slowest were:

8.0ms
(/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2))))
4.0ms
(/ 1 (/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2)))))
3.0ms
(pow (* n 2) (- 1/2 (/ k 2)))

series374.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

131.0ms
(/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2))))
124.0ms
(/ 1 (/ (sqrt k) (pow (* n 2) (- 1/2 (/ k 2)))))
95.0ms
(pow (* n 2) (- 1/2 (/ k 2)))
24.0ms
(pow PI (- 1/2 (/ k 2)))

simplify2.9s

Counts
101 → 133
Calls

101 calls. Slowest were:

577.0ms
(exp (* (- (log 2) (log (/ 1 n))) (- 1/2 (* 1/2 k))))
312.0ms
(* (log (* n 2)) (- 1/2 (/ k 2)))
280.0ms
(- (+ (pow PI 1/2) (* 1/8 (* (* (pow (log PI) 2) (pow k 2)) (sqrt PI)))) (* 1/2 (* (* (log PI) k) (sqrt PI))))

prune1.9s

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0b

regimes154.0ms

Accuracy

0% (0.3b remaining)

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

bsearch3.0ms

end0.0ms

sample7.4s

Algorithm
intervals