Average Error: 16.8 → 16.9
Time: 53.9s
Precision: 64
Internal Precision: 128
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\left(\sqrt{1 + {\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(-2 \cdot J\right)\]

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 16.8

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

  3. Applied associate-*l*16.9

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

  4. Final simplification16.9

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

Reproduce

herbie shell --seed 2018362 
(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: 40.3s)Debug log

start351.0ms

Algorithm
intervals

setup344.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 16.5b

localize39.0ms

Local error

Found 4 expressions with local error:

9.2b
(* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
7.3b
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
0.2b
(pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)
0.1b
(* (* -2 J) (cos (/ K 2)))

rewrite38.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

20.0ms
(* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
8.0ms
(* (* -2 J) (cos (/ K 2)))
6.0ms
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))

series422.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

173.0ms
(* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
107.0ms
(pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)
102.0ms
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
39.0ms
(* (* -2 J) (cos (/ K 2)))

simplify6.9s

Counts
45 → 79
Calls

45 calls. Slowest were:

671.0ms
(* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
500.0ms
(sqrt (+ (* 1 1) (- (* (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2) (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)) (* 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))
416.0ms
(sqrt (- 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))

prune1.5s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 14.3b

localize13.0ms

Local error

Found 4 expressions with local error:

9.2b
(* (* -2 J) (* (cos (/ K 2)) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))
7.3b
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
0.2b
(pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)
0.1b
(* (* 2 J) (cos (/ K 2)))

rewrite62.0ms

Algorithm
rewrite-expression-head
Counts
4 → 62
Calls

4 calls. Slowest were:

34.0ms
(* (* -2 J) (* (cos (/ K 2)) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))
12.0ms
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
8.0ms
(* (* 2 J) (cos (/ K 2)))

series463.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

155.0ms
(* (* -2 J) (* (cos (/ K 2)) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))
133.0ms
(pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)
117.0ms
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
58.0ms
(* (* 2 J) (cos (/ K 2)))

simplify5.2s

Counts
39 → 74
Calls

39 calls. Slowest were:

568.0ms
(sqrt (+ (* 1 1) (- (* (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2) (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)) (* 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))
395.0ms
(sqrt (- 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
349.0ms
(+ (log (* -2 J)) (log (* (cos (/ K 2)) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))))

prune1.3s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 14.3b

localize20.0ms

Local error

Found 4 expressions with local error:

9.2b
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
9.2b
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
7.3b
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
7.3b
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

12.0ms
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
6.0ms
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
6.0ms
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))

series609.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

196.0ms
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
175.0ms
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
131.0ms
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
107.0ms
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))

simplify7.6s

Counts
44 → 76
Calls

44 calls. Slowest were:

325.0ms
(sqrt (* (cbrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))) (cbrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))
323.0ms
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
305.0ms
(sqrt (sqrt (+ (pow 1 3) (pow (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2) 3))))

prune2.1s

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 14.3b

localize39.0ms

Local error

Found 4 expressions with local error:

14.0b
(cbrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
14.0b
(cbrt (+ 1 (* (/ (/ U (* 2 J)) (cos (/ K 2))) (/ (/ U (* 2 J)) (cos (/ K 2))))))
9.2b
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
9.2b
(sqrt (* (fabs (cbrt (+ 1 (* (/ (/ U (* 2 J)) (cos (/ K 2))) (/ (/ U (* 2 J)) (cos (/ K 2))))))) (sqrt (cbrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

13.0ms
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
7.0ms
(cbrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
2.0ms
(sqrt (* (fabs (cbrt (+ 1 (* (/ (/ U (* 2 J)) (cos (/ K 2))) (/ (/ U (* 2 J)) (cos (/ K 2))))))) (sqrt (cbrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))))

series890.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

259.0ms
(sqrt (* (fabs (cbrt (+ 1 (* (/ (/ U (* 2 J)) (cos (/ K 2))) (/ (/ U (* 2 J)) (cos (/ K 2))))))) (sqrt (cbrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))))
221.0ms
(cbrt (+ 1 (* (/ (/ U (* 2 J)) (cos (/ K 2))) (/ (/ U (* 2 J)) (cos (/ K 2))))))
209.0ms
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
201.0ms
(cbrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))

simplify9.2s

Counts
35 → 67
Calls

35 calls. Slowest were:

1.3s
(sqrt (* (fabs (cbrt (+ 1 (* (/ (/ U (* 2 J)) (cos (/ K 2))) (/ (/ U (* 2 J)) (cos (/ K 2))))))) (sqrt (cbrt (+ (pow 1 3) (pow (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2) 3))))))
1.1s
(cbrt (- (* 1 1) (* (* (/ (/ U (* 2 J)) (cos (/ K 2))) (/ (/ U (* 2 J)) (cos (/ K 2)))) (* (/ (/ U (* 2 J)) (cos (/ K 2))) (/ (/ U (* 2 J)) (cos (/ K 2)))))))
834.0ms
(cbrt (+ (pow 1 3) (pow (* (/ (/ U (* 2 J)) (cos (/ K 2))) (/ (/ U (* 2 J)) (cos (/ K 2)))) 3)))

prune2.2s

Pruning

11 alts after pruning (8 fresh and 3 done)

Merged error: 14.3b

regimes985.0ms

Accuracy

0% (4.5b remaining)

Error of 16.9b against oracle of 12.4b and baseline of 16.9b

bsearch4.0ms