Average Error: 17.0 → 7.8
Time: 28.2s
Precision: 64
Internal Precision: 128
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*\right) \cdot \left(J \cdot -2\right)\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 17.0

    \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]

  2. Simplified7.8

    \[\leadsto \color{blue}{\sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^* \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)}\]
  3. Using strategy rm

  4. Applied associate-*r*7.8

    \[\leadsto \color{blue}{\left(\sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^* \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(-2 \cdot J\right)}\]

  5. Final simplification7.8

    \[\leadsto \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*\right) \cdot \left(J \cdot -2\right)\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  (* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))

Details

Time bar (total: 27.2s)Debug log

sample219.0ms

Algorithm
intervals

simplify302.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

302.0ms
(* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))

prune30.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 7.1b

localize54.0ms

Local error

Found 3 expressions with local error:

7.1b
(* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (* (cos (/ K 2)) (* -2 J)))
0.1b
(/ (/ (/ U 2) J) (cos (/ K 2)))
0.1b
(* (cos (/ K 2)) (* -2 J))

rewrite33.0ms

Algorithm
rewrite-expression-head
Counts
3 → 71
Calls

3 calls. Slowest were:

15.0ms
(* (cos (/ K 2)) (* -2 J))
11.0ms
(* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (* (cos (/ K 2)) (* -2 J)))
6.0ms
(/ (/ (/ U 2) J) (cos (/ K 2)))

series397.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

253.0ms
(* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (* (cos (/ K 2)) (* -2 J)))
107.0ms
(/ (/ (/ U 2) J) (cos (/ K 2)))
37.0ms
(* (cos (/ K 2)) (* -2 J))

simplify2.5s

Counts
56 → 80
Calls

56 calls. Slowest were:

568.0ms
(/ (* (* (/ (/ U 2) J) (/ (/ U 2) J)) (/ (/ U 2) J)) (* (* (cos (/ K 2)) (cos (/ K 2))) (cos (/ K 2))))
232.0ms
(+ (* 1/16 (/ (* (pow K 2) U) J)) (* 1/2 (/ U J)))
217.0ms
(- (log (/ (/ U 2) J)) (log (cos (/ K 2))))

prune1.0s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 3.5b

localize13.0ms

Local error

Found 3 expressions with local error:

7.1b
(* (* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (cos (/ K 2))) (* -2 J))
0.1b
(/ (/ (/ U 2) J) (cos (/ K 2)))
0.0b
(* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (cos (/ K 2)))

rewrite45.0ms

Algorithm
rewrite-expression-head
Counts
3 → 71
Calls

3 calls. Slowest were:

23.0ms
(* (* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (cos (/ K 2))) (* -2 J))
10.0ms
(/ (/ (/ U 2) J) (cos (/ K 2)))
10.0ms
(* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (cos (/ K 2)))

series477.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

249.0ms
(* (* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (cos (/ K 2))) (* -2 J))
151.0ms
(* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (cos (/ K 2)))
77.0ms
(/ (/ (/ U 2) J) (cos (/ K 2)))

simplify3.1s

Counts
57 → 80
Calls

57 calls. Slowest were:

839.0ms
(* (* (* (* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (cos (/ K 2))) (* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (cos (/ K 2)))) (* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (cos (/ K 2)))) (* (* (* -2 J) (* -2 J)) (* -2 J)))
466.0ms
(/ (* (* (/ (/ U 2) J) (/ (/ U 2) J)) (/ (/ U 2) J)) (* (* (cos (/ K 2)) (cos (/ K 2))) (cos (/ K 2))))
228.0ms
(+ (* 1/16 (/ (* (pow K 2) U) J)) (* 1/2 (/ U J)))

prune1.5s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 3.5b

localize22.0ms

Local error

Found 4 expressions with local error:

14.2b
(cbrt (/ (/ U 2) J))
14.2b
(cbrt (/ (/ U 2) J))
14.2b
(cbrt (/ (/ U 2) J))
7.1b
(* (hypot 1 (/ (* (cbrt (/ (/ U 2) J)) (cbrt (/ (/ U 2) J))) (/ (cos (/ K 2)) (cbrt (/ (/ U 2) J))))) (* (cos (/ K 2)) (* -2 J)))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

14.0ms
(* (hypot 1 (/ (* (cbrt (/ (/ U 2) J)) (cbrt (/ (/ U 2) J))) (/ (cos (/ K 2)) (cbrt (/ (/ U 2) J))))) (* (cos (/ K 2)) (* -2 J)))
1.0ms
(cbrt (/ (/ U 2) J))
1.0ms
(cbrt (/ (/ U 2) J))

series799.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

443.0ms
(* (hypot 1 (/ (* (cbrt (/ (/ U 2) J)) (cbrt (/ (/ U 2) J))) (/ (cos (/ K 2)) (cbrt (/ (/ U 2) J))))) (* (cos (/ K 2)) (* -2 J)))
146.0ms
(cbrt (/ (/ U 2) J))
106.0ms
(cbrt (/ (/ U 2) J))
105.0ms
(cbrt (/ (/ U 2) J))

simplify2.1s

Counts
36 → 77
Calls

36 calls. Slowest were:

205.0ms
(* (cbrt 1/2) (exp (* 1/3 (- (log (/ -1 J)) (log (/ -1 U))))))
204.0ms
(* (cbrt 1/2) (exp (* 1/3 (- (log (/ 1 J)) (log (/ 1 U))))))
203.0ms
(* (cbrt 1/2) (exp (* 1/3 (- (log (/ 1 J)) (log (/ 1 U))))))

prune1.2s

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 3.5b

localize17.0ms

Local error

Found 4 expressions with local error:

7.1b
(* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (* (* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2)))) (* (cbrt (cos (/ K 2))) (* -2 J))))
0.3b
(cbrt (cos (/ K 2)))
0.3b
(cbrt (cos (/ K 2)))
0.3b
(cbrt (cos (/ K 2)))

rewrite83.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

78.0ms
(* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (* (* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2)))) (* (cbrt (cos (/ K 2))) (* -2 J))))
1.0ms
(cbrt (cos (/ K 2)))
1.0ms
(cbrt (cos (/ K 2)))

series759.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

259.0ms
(* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (* (* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2)))) (* (cbrt (cos (/ K 2))) (* -2 J))))
180.0ms
(cbrt (cos (/ K 2)))
177.0ms
(cbrt (cos (/ K 2)))
142.0ms
(cbrt (cos (/ K 2)))

simplify4.4s

Counts
39 → 80
Calls

39 calls. Slowest were:

380.0ms
(+ (log (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2))))) (+ (+ (log (cbrt (cos (/ K 2)))) (log (cbrt (cos (/ K 2))))) (+ (log (cbrt (cos (/ K 2)))) (log (* -2 J)))))
377.0ms
(+ (log (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2))))) (+ (+ (log (cbrt (cos (/ K 2)))) (log (cbrt (cos (/ K 2))))) (log (* (cbrt (cos (/ K 2))) (* -2 J)))))
348.0ms
(* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (* (* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2)))) (* (cbrt (cos (/ K 2))) (* -2 J))))

prune1.1s

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 3.5b

regimes373.0ms

Accuracy

0% (3.8b remaining)

Error of 7.8b against oracle of 3.9b and baseline of 7.8b

bsearch3.0ms

end0.0ms

sample6.5s

Algorithm
intervals