Average Error: 17.4 → 0.4
Time: 37.2s
Precision: 64
Internal Precision: 128
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + \cos \left(\frac{K}{2}\right) \cdot \left(\left(\frac{1}{60} \cdot {\ell}^{5} + \ell \cdot (\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right) \cdot J\right)\]

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.4

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

  2. Taylor expanded around 0 0.4

    \[\leadsto \left(J \cdot \color{blue}{\left(2 \cdot \ell + \left(\frac{1}{3} \cdot {\ell}^{3} + \frac{1}{60} \cdot {\ell}^{5}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

  3. Simplified0.4

    \[\leadsto \left(J \cdot \color{blue}{(\left((\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right) \cdot \ell + \left({\ell}^{5} \cdot \frac{1}{60}\right))_*}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
  4. Using strategy rm

  5. Applied fma-udef0.4

    \[\leadsto \left(J \cdot \color{blue}{\left((\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_* \cdot \ell + {\ell}^{5} \cdot \frac{1}{60}\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

  6. Final simplification0.4

    \[\leadsto U + \cos \left(\frac{K}{2}\right) \cdot \left(\left(\frac{1}{60} \cdot {\ell}^{5} + \ell \cdot (\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right) \cdot J\right)\]

Reproduce

herbie shell --seed 2018365 +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: 20.9s)Debug log

start388.0ms

Algorithm
intervals

setup288.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 17.0b

localize42.0ms

Local error

Found 3 expressions with local error:

5.9b
(- (exp l) (exp (- l)))
0.0b
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
0.0b
(* J (- (exp l) (exp (- l))))

rewrite36.0ms

Algorithm
rewrite-expression-head
Counts
3 → 86
Calls

3 calls. Slowest were:

19.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
8.0ms
(* J (- (exp l) (exp (- l))))
7.0ms
(- (exp l) (exp (- l)))

series273.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

131.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
95.0ms
(* J (- (exp l) (exp (- l))))
47.0ms
(- (exp l) (exp (- l)))

simplify4.1s

Counts
70 → 95
Calls

70 calls. Slowest were:

369.0ms
(* (* J (- (pow (exp l) 3) (pow (exp (- l)) 3))) (cos (/ K 2)))
227.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (- l)))))
203.0ms
(+ (* (exp l) (exp l)) (+ (* (exp (- l)) (exp (- l))) (* (exp l) (exp (- l)))))

prune1.5s

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize31.0ms

Local error

Found 4 expressions with local error:

0.1b
(* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))
0.0b
(* (pow l 5) 1/60)
0.0b
(* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)))
0.0b
(fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

6.0ms
(* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))
2.0ms
(* (pow l 5) 1/60)
1.0ms
(* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)))

series323.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

139.0ms
(* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))
111.0ms
(* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)))
42.0ms
(fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))
31.0ms
(* (pow l 5) 1/60)

simplify2.2s

Counts
25 → 67
Calls

25 calls. Slowest were:

338.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
328.0ms
(+ (* 1/3 (* J (pow l 3))) (+ (* 2 (* J l)) (* 1/60 (* J (pow l 5)))))
304.0ms
(+ (* 1/3 (* J (pow l 3))) (+ (* 2 (* J l)) (* 1/60 (* J (pow l 5)))))

prune1.4s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

localize18.0ms

Local error

Found 4 expressions with local error:

0.1b
(* (* J (+ (* (fma 1/3 (* l l) 2) l) (* (pow l 5) 1/60))) (cos (/ K 2)))
0.0b
(* (pow l 5) 1/60)
0.0b
(* J (+ (* (fma 1/3 (* l l) 2) l) (* (pow l 5) 1/60)))
0.0b
(* (fma 1/3 (* l l) 2) l)

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 62
Calls

4 calls. Slowest were:

16.0ms
(* (* J (+ (* (fma 1/3 (* l l) 2) l) (* (pow l 5) 1/60))) (cos (/ K 2)))
6.0ms
(* J (+ (* (fma 1/3 (* l l) 2) l) (* (pow l 5) 1/60)))
5.0ms
(* (fma 1/3 (* l l) 2) l)

series313.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

141.0ms
(* (* J (+ (* (fma 1/3 (* l l) 2) l) (* (pow l 5) 1/60))) (cos (/ K 2)))
77.0ms
(* J (+ (* (fma 1/3 (* l l) 2) l) (* (pow l 5) 1/60)))
59.0ms
(* (fma 1/3 (* l l) 2) l)
36.0ms
(* (pow l 5) 1/60)

simplify3.0s

Counts
31 → 74
Calls

31 calls. Slowest were:

507.0ms
(* (* J (- (* (* (fma 1/3 (* l l) 2) l) (* (fma 1/3 (* l l) 2) l)) (* (* (pow l 5) 1/60) (* (pow l 5) 1/60)))) (cos (/ K 2)))
394.0ms
(+ (* 1/3 (* J (pow l 3))) (+ (* 2 (* J l)) (* 1/60 (* J (pow l 5)))))
314.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))

prune1.5s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0b

localize17.0ms

Local error

Found 4 expressions with local error:

0.1b
(* J (* (+ (* (fma 1/3 (* l l) 2) l) (* (pow l 5) 1/60)) (cos (/ K 2))))
0.1b
(* (+ (* (fma 1/3 (* l l) 2) l) (* (pow l 5) 1/60)) (cos (/ K 2)))
0.0b
(* (pow l 5) 1/60)
0.0b
(* (fma 1/3 (* l l) 2) l)

rewrite39.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

21.0ms
(* J (* (+ (* (fma 1/3 (* l l) 2) l) (* (pow l 5) 1/60)) (cos (/ K 2))))
13.0ms
(* (+ (* (fma 1/3 (* l l) 2) l) (* (pow l 5) 1/60)) (cos (/ K 2)))
2.0ms
(* (fma 1/3 (* l l) 2) l)

series284.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

151.0ms
(* J (* (+ (* (fma 1/3 (* l l) 2) l) (* (pow l 5) 1/60)) (cos (/ K 2))))
62.0ms
(* (+ (* (fma 1/3 (* l l) 2) l) (* (pow l 5) 1/60)) (cos (/ K 2)))
41.0ms
(* (fma 1/3 (* l l) 2) l)
30.0ms
(* (pow l 5) 1/60)

simplify2.7s

Counts
29 → 72
Calls

29 calls. Slowest were:

378.0ms
(* (+ (pow (* (fma 1/3 (* l l) 2) l) 3) (pow (* (pow l 5) 1/60) 3)) (cos (/ K 2)))
348.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
330.0ms
(* (- (* (* (fma 1/3 (* l l) 2) l) (* (fma 1/3 (* l l) 2) l)) (* (* (pow l 5) 1/60) (* (pow l 5) 1/60))) (cos (/ K 2)))

prune1.4s

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0b

regimes756.0ms

Accuracy

0% (0.3b remaining)

Error of 0.4b against oracle of 0.0b and baseline of 0.4b

bsearch4.0ms