Average Error: 16.9 → 0.5
Time: 25.3s
Precision: 64
Internal Precision: 128
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[(\left(\left(\ell \cdot J\right) \cdot 2\right) \cdot \left(\cos \left(\frac{K}{2}\right)\right) + U)_*\]

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 16.9

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

  2. Taylor expanded around 0 0.5

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

  4. Applied fma-def0.5

    \[\leadsto \color{blue}{(\left(2 \cdot \left(J \cdot \ell\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right)\right) + U)_*}\]

  5. Final simplification0.5

    \[\leadsto (\left(\left(\ell \cdot J\right) \cdot 2\right) \cdot \left(\cos \left(\frac{K}{2}\right)\right) + U)_*\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))

Details

Time bar (total: 24.2s)Debug log

sample280.0ms

Algorithm
intervals

simplify278.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

278.0ms
(+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U)

prune16.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 16.9b

localize27.0ms

Local error

Found 3 expressions with local error:

6.1b
(- (exp l) (exp (- l)))
0.0b
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
0.0b
(* J (- (exp l) (exp (- l))))

rewrite43.0ms

Algorithm
rewrite-expression-head
Counts
3 → 86
Calls

3 calls. Slowest were:

20.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
16.0ms
(* J (- (exp l) (exp (- l))))
5.0ms
(- (exp l) (exp (- l)))

series251.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

96.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
91.0ms
(* J (- (exp l) (exp (- l))))
64.0ms
(- (exp l) (exp (- l)))

simplify3.9s

Counts
70 → 95
Calls

70 calls. Slowest were:

422.0ms
(* (* J (- (pow (exp l) 3) (pow (exp (- l)) 3))) (cos (/ K 2)))
206.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
203.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (- l)))))

prune1.5s

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 2 expressions with local error:

0.1b
(* (* 2 (* J l)) (cos (/ K 2)))
0.0b
(+ (* (* 2 (* J l)) (cos (/ K 2))) U)

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
2 → 30
Calls

2 calls. Slowest were:

7.0ms
(+ (* (* 2 (* J l)) (cos (/ K 2))) U)
5.0ms
(* (* 2 (* J l)) (cos (/ K 2)))

series165.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

116.0ms
(+ (* (* 2 (* J l)) (cos (/ K 2))) U)
49.0ms
(* (* 2 (* J l)) (cos (/ K 2)))

simplify1.1s

Counts
13 → 36
Calls

13 calls. Slowest were:

803.0ms
(* (* (* (* 2 (* J l)) (* 2 (* J l))) (* 2 (* J l))) (* (* (cos (/ K 2)) (cos (/ K 2))) (cos (/ K 2))))
57.0ms
(+ (* 2 (* (cos (* 1/2 K)) (* J l))) U)
51.0ms
(* 2 (* (cos (* 1/2 K)) (* J l)))

prune469.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize4.0ms

Local error

Found 1 expressions with local error:

0.0b
(fma (* 2 (* J l)) (cos (/ K 2)) U)

rewrite0.0ms

Algorithm
rewrite-expression-head
Counts
1 → 10
Calls

1 calls. Slowest were:

0.0ms
(fma (* 2 (* J l)) (cos (/ K 2)) U)

series99.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

99.0ms
(fma (* 2 (* J l)) (cos (/ K 2)) U)

simplify74.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

34.0ms
(+ (* 2 (* (cos (* 1/2 K)) (* J l))) U)
28.0ms
(+ (* 2 (* (cos (* 1/2 K)) (* J l))) U)
10.0ms
(+ U (* 2 (* J l)))

prune214.0ms

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

12.8b
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
12.8b
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
12.8b
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
0.5b
(* (cbrt (* (* 2 (* J l)) (cos (/ K 2)))) (cbrt (* (* 2 (* J l)) (cos (/ K 2)))))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

18.0ms
(* (cbrt (* (* 2 (* J l)) (cos (/ K 2)))) (cbrt (* (* 2 (* J l)) (cos (/ K 2)))))
3.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
3.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))

series767.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

246.0ms
(* (cbrt (* (* 2 (* J l)) (cos (/ K 2)))) (cbrt (* (* 2 (* J l)) (cos (/ K 2)))))
197.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
171.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
153.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))

simplify2.3s

Counts
31 → 71
Calls

31 calls. Slowest were:

208.0ms
(* (exp (* 1/3 (+ (* 2 (log J)) (* 2 (log l))))) (pow (cbrt 2) 2))
203.0ms
(* (pow (cbrt 2) 2) (exp (* 1/3 (- (log (pow (cos (* 1/2 K)) 2)) (+ (* 2 (log (/ 1 l))) (* 2 (log (/ 1 J))))))))
190.0ms
(* (exp (* 1/3 (- (log (cos (* 1/2 K))) (+ (log (/ 1 l)) (log (/ 1 J)))))) (cbrt 2))

prune1.0s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

regimes485.0ms

Accuracy

0% (0.3b remaining)

Error of 0.5b against oracle of 0.2b and baseline of 0.5b

bsearch2.0ms

end0.0ms

sample11.1s

Algorithm
intervals