Average Error: 17.3 → 7.6
Time: 21.9s
Precision: 64
Internal Precision: 128
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*\right) \cdot \left(J \cdot -2\right)\]

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 17.3

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

  2. Simplified7.6

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

  4. Applied associate-*r*7.6

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

  5. Final simplification7.6

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

Reproduce

herbie shell --seed 2019008 +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: 20.4s)Debug log

sample168.0ms

Algorithm
intervals

simplify217.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

217.0ms
(* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))

prune16.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 6.5b

localize24.0ms

Local error

Found 4 expressions with local error:

6.4b
(* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (* (cos (/ K 2)) (* -2 J)))
0.1b
(* (cos (/ K 2)) (* -2 J))
0.1b
(/ (/ (/ U 2) J) (cos (/ K 2)))
0.0b
(hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2))))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 81
Calls

4 calls. Slowest were:

10.0ms
(* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (* (cos (/ K 2)) (* -2 J)))
5.0ms
(/ (/ (/ U 2) J) (cos (/ K 2)))
4.0ms
(* (cos (/ K 2)) (* -2 J))

series432.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

184.0ms
(* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (* (cos (/ K 2)) (* -2 J)))
147.0ms
(hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2))))
67.0ms
(/ (/ (/ U 2) J) (cos (/ K 2)))
34.0ms
(* (cos (/ K 2)) (* -2 J))

simplify1.8s

Counts
59 → 93
Calls

59 calls. Slowest were:

325.0ms
(/ (* (* (/ (/ U 2) J) (/ (/ U 2) J)) (/ (/ U 2) J)) (* (* (cos (/ K 2)) (cos (/ K 2))) (cos (/ K 2))))
167.0ms
(+ (* 1/16 (/ (* (pow K 2) U) J)) (* 1/2 (/ U J)))
127.0ms
(* (* (* (cos (/ K 2)) (cos (/ K 2))) (cos (/ K 2))) (* (* (* -2 J) (* -2 J)) (* -2 J)))

prune897.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 2.4b

localize8.0ms

Local error

Found 4 expressions with local error:

6.4b
(* (* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (cos (/ K 2))) (* -2 J))
0.1b
(/ (/ (/ U 2) J) (cos (/ K 2)))
0.1b
(* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (cos (/ K 2)))
0.0b
(hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2))))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 81
Calls

4 calls. Slowest were:

11.0ms
(* (* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (cos (/ K 2))) (* -2 J))
5.0ms
(* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (cos (/ K 2)))
5.0ms
(/ (/ (/ U 2) J) (cos (/ K 2)))

series520.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

182.0ms
(* (* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (cos (/ K 2))) (* -2 J))
138.0ms
(hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2))))
135.0ms
(* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (cos (/ K 2)))
66.0ms
(/ (/ (/ U 2) J) (cos (/ K 2)))

simplify2.1s

Counts
60 → 93
Calls

60 calls. Slowest were:

639.0ms
(* (* (* (* (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)))
370.0ms
(/ (* (* (/ (/ U 2) J) (/ (/ U 2) J)) (/ (/ U 2) J)) (* (* (cos (/ K 2)) (cos (/ K 2))) (cos (/ K 2))))
122.0ms
(+ (* 1/16 (/ (* (pow K 2) U) J)) (* 1/2 (/ U J)))

prune756.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 2.4b

localize22.0ms

Local error

Found 4 expressions with local error:

6.4b
(* (hypot 1 (* 1/2 (/ U (* (cos (* 1/2 K)) J)))) (* (cos (/ K 2)) (* -2 J)))
0.2b
(* 1/2 (/ U (* (cos (* 1/2 K)) J)))
0.1b
(* (cos (/ K 2)) (* -2 J))
0.1b
(* (cos (* 1/2 K)) J)

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

11.0ms
(* (hypot 1 (* 1/2 (/ U (* (cos (* 1/2 K)) J)))) (* (cos (/ K 2)) (* -2 J)))
5.0ms
(* (cos (/ K 2)) (* -2 J))
4.0ms
(* (cos (* 1/2 K)) J)

series303.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

182.0ms
(* (hypot 1 (* 1/2 (/ U (* (cos (* 1/2 K)) J)))) (* (cos (/ K 2)) (* -2 J)))
54.0ms
(* 1/2 (/ U (* (cos (* 1/2 K)) J)))
36.0ms
(* (cos (/ K 2)) (* -2 J))
31.0ms
(* (cos (* 1/2 K)) J)

simplify1.5s

Counts
33 → 77
Calls

33 calls. Slowest were:

270.0ms
(* 1/2 (/ U (* (cos (* 1/2 K)) J)))
149.0ms
(- J (* 1/8 (* (pow K 2) J)))
127.0ms
(* (* (* (cos (/ K 2)) (cos (/ K 2))) (cos (/ K 2))) (* (* (* -2 J) (* -2 J)) (* -2 J)))

prune742.0ms

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 2.4b

localize13.0ms

Local error

Found 4 expressions with local error:

6.4b
(* (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)))

rewrite44.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

41.0ms
(* (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)))

series525.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

181.0ms
(* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (* (* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2)))) (* (cbrt (cos (/ K 2))) (* -2 J))))
115.0ms
(cbrt (cos (/ K 2)))
114.0ms
(cbrt (cos (/ K 2)))
114.0ms
(cbrt (cos (/ K 2)))

simplify3.2s

Counts
39 → 80
Calls

39 calls. Slowest were:

269.0ms
(* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (* (* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2)))) (* (cbrt (cos (/ K 2))) (* -2 J))))
252.0ms
(+ (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)))))
224.0ms
(+ (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)))))

prune937.0ms

Pruning

10 alts after pruning (7 fresh and 3 done)

Merged error: 2.3b

regimes364.0ms

Accuracy

0% (3.8b remaining)

Error of 7.6b against oracle of 3.7b and baseline of 7.6b

bsearch3.0ms

end0.0ms

sample5.8s

Algorithm
intervals