Average Error: 16.9 → 17.0
Time: 45.0s
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 16.9

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

  3. Applied associate-*l*17.0

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

  4. Final simplification17.0

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

Reproduce

herbie shell --seed 2019005 
(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: 43.2s)Debug log

sample168.0ms

Algorithm
intervals

simplify374.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

374.0ms
(* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))

prune17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 18.4b

localize67.0ms

Local error

Found 4 expressions with local error:

9.6b
(* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
8.7b
(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)))

rewrite64.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

33.0ms
(* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
13.0ms
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
9.0ms
(* (* -2 J) (cos (/ K 2)))

series365.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

125.0ms
(* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
120.0ms
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
81.0ms
(pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)
39.0ms
(* (* -2 J) (cos (/ K 2)))

simplify6.6s

Counts
45 → 79
Calls

45 calls. Slowest were:

579.0ms
(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)))))
402.0ms
(sqrt (- 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
338.0ms
(+ (+ (log (* -2 J)) (log (cos (/ K 2)))) (log (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))

prune1.3s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 13.8b

localize6.0ms

Local error

Found 4 expressions with local error:

9.7b
(* (* -2 J) (* (cos (/ K 2)) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))
8.7b
(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)))

rewrite46.0ms

Algorithm
rewrite-expression-head
Counts
4 → 62
Calls

4 calls. Slowest were:

18.0ms
(* (* -2 J) (* (cos (/ K 2)) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))
11.0ms
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
9.0ms
(* (* 2 J) (cos (/ K 2)))

series408.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

145.0ms
(* (* -2 J) (* (cos (/ K 2)) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))
109.0ms
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
98.0ms
(pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)
55.0ms
(* (* 2 J) (cos (/ K 2)))

simplify5.2s

Counts
39 → 74
Calls

39 calls. Slowest were:

583.0ms
(+ (log (* -2 J)) (+ (log (cos (/ K 2))) (log (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))))
523.0ms
(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)))))
408.0ms
(sqrt (- 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))

prune1.1s

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 13.8b

localize10.0ms

Local error

Found 4 expressions with local error:

9.7b
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
9.7b
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
8.7b
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
8.7b
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

8.0ms
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
7.0ms
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
7.0ms
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))

series664.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

233.0ms
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
205.0ms
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
120.0ms
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
105.0ms
(sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))

simplify7.2s

Counts
44 → 76
Calls

44 calls. Slowest were:

320.0ms
(sqrt (sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))
312.0ms
(sqrt (sqrt (* (cbrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))) (cbrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))))
309.0ms
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))

prune1.5s

Pruning

13 alts after pruning (11 fresh and 2 done)

Merged error: 13.8b

localize32.0ms

Local error

Found 4 expressions with local error:

16.3b
(cbrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
16.3b
(cbrt (+ 1 (* (/ (/ U (* 2 J)) (cos (/ K 2))) (/ (/ U (* 2 J)) (cos (/ K 2))))))
9.7b
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
9.7b
(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))))))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

7.0ms
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
4.0ms
(cbrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
3.0ms
(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))))))

series993.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

362.0ms
(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))))))
230.0ms
(cbrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))
206.0ms
(sqrt (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))
195.0ms
(cbrt (+ 1 (* (/ (/ U (* 2 J)) (cos (/ K 2))) (/ (/ U (* 2 J)) (cos (/ K 2))))))

simplify9.8s

Counts
35 → 67
Calls

35 calls. Slowest were:

1.5s
(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))))))
1.2s
(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)))))))
875.0ms
(cbrt (+ (pow 1 3) (pow (* (/ (/ U (* 2 J)) (cos (/ K 2))) (/ (/ U (* 2 J)) (cos (/ K 2)))) 3)))

prune1.4s

Pruning

13 alts after pruning (10 fresh and 3 done)

Merged error: 13.8b

regimes560.0ms

Accuracy

0% (4.9b remaining)

Error of 17.0b against oracle of 12.0b and baseline of 17.0b

bsearch2.0ms

end0.0ms

sample5.2s

Algorithm
intervals