Average Error: 15.2 → 1.3
Time: 22.9s
Precision: 64
Internal Precision: 128
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]

Error

Bits error versus K

Bits error versus m

Bits error versus n

Bits error versus M

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.2

    \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]

  2. Taylor expanded around 0 1.3

    \[\leadsto \color{blue}{1} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]

  3. Final simplification1.3

    \[\leadsto e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]

Reproduce

herbie shell --seed 2018365 
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  (* (cos (- (/ (* K (+ m n)) 2) M)) (exp (- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n)))))))

Details

Time bar (total: 16.5s)Debug log

start186.0ms

Algorithm
intervals

setup166.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 16.2b

localize56.0ms

Local error

Found 4 expressions with local error:

35.5b
(cos (- (/ (* K (+ m n)) 2) M))
0.0b
(pow (- (/ (+ m n) 2) M) 2)
0.0b
(- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n))))
0.0b
(* K (+ m n))

rewrite192.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

173.0ms
(- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n))))
10.0ms
(pow (- (/ (+ m n) 2) M) 2)
6.0ms
(cos (- (/ (* K (+ m n)) 2) M))

series249.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

128.0ms
(- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n))))
64.0ms
(cos (- (/ (* K (+ m n)) 2) M))
38.0ms
(pow (- (/ (+ m n) 2) M) 2)
19.0ms
(* K (+ m n))

simplify2.6s

Counts
36 → 71
Calls

36 calls. Slowest were:

401.0ms
(/ (/ 1 (exp (pow (- (/ (+ m n) 2) M) 2))) (exp (- l (fabs (- m n)))))
274.0ms
(/ (exp (- (pow (- (/ (+ m n) 2) M) 2))) (exp (- l (fabs (- m n)))))
267.0ms
(- (sqrt (- (pow (- (/ (+ m n) 2) M) 2))) (sqrt (- l (fabs (- m n)))))

prune1.5s

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.6b

localize4.0ms

Local error

Found 3 expressions with local error:

0.0b
(pow (- (/ (+ m n) 2) M) 2)
0.0b
(- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n))))
0.0b
(- (/ (+ m n) 2) M)

rewrite153.0ms

Algorithm
rewrite-expression-head
Counts
3 → 45
Calls

3 calls. Slowest were:

139.0ms
(- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n))))
8.0ms
(- (/ (+ m n) 2) M)
5.0ms
(pow (- (/ (+ m n) 2) M) 2)

series185.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

104.0ms
(- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n))))
60.0ms
(pow (- (/ (+ m n) 2) M) 2)
21.0ms
(- (/ (+ m n) 2) M)

simplify2.3s

Counts
23 → 54
Calls

23 calls. Slowest were:

349.0ms
(/ (exp (- (pow (- (/ (+ m n) 2) M) 2))) (exp (- l (fabs (- m n)))))
276.0ms
(/ (/ 1 (exp (pow (- (/ (+ m n) 2) M) 2))) (exp (- l (fabs (- m n)))))
269.0ms
(- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n))))

prune1.1s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.6b

localize39.0ms

Local error

Found 4 expressions with local error:

17.9b
(cos (- (/ (exp (log (* K (+ m n)))) 2) M))
9.2b
(log (* K (+ m n)))
2.0b
(exp (log (* K (+ m n))))
0.0b
(pow (- (/ (+ m n) 2) M) 2)

rewrite47.0ms

Algorithm
rewrite-expression-head
Counts
4 → 51
Calls

4 calls. Slowest were:

27.0ms
(cos (- (/ (exp (log (* K (+ m n)))) 2) M))
10.0ms
(pow (- (/ (+ m n) 2) M) 2)
6.0ms
(exp (log (* K (+ m n))))

series205.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

78.0ms
(cos (- (/ (exp (log (* K (+ m n)))) 2) M))
54.0ms
(log (* K (+ m n)))
46.0ms
(pow (- (/ (+ m n) 2) M) 2)
27.0ms
(exp (log (* K (+ m n))))

simplify1.3s

Counts
31 → 63
Calls

31 calls. Slowest were:

258.0ms
(- (+ (log (/ -1 m)) (log (/ -1 K))))
150.0ms
(exp (log (* K (- (* m m) (* n n)))))
125.0ms
(log (* K (- (* m m) (* n n))))

prune1.4s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.6b

localize3.0ms

Local error

Found 4 expressions with local error:

35.5b
(cos (- (/ (* K (+ m n)) 2) M))
0.0b
(pow (- (/ (+ m n) 2) M) 2)
0.0b
(- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n))))
0.0b
(* K (+ m n))

rewrite133.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

110.0ms
(- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n))))
11.0ms
(cos (- (/ (* K (+ m n)) 2) M))
10.0ms
(pow (- (/ (+ m n) 2) M) 2)

series247.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

119.0ms
(- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n))))
51.0ms
(pow (- (/ (+ m n) 2) M) 2)
50.0ms
(cos (- (/ (* K (+ m n)) 2) M))
27.0ms
(* K (+ m n))

simplify2.7s

Counts
36 → 71
Calls

36 calls. Slowest were:

315.0ms
(/ (exp (- (pow (- (/ (+ m n) 2) M) 2))) (exp (- l (fabs (- m n)))))
292.0ms
(- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n))))
285.0ms
(/ (/ 1 (exp (pow (- (/ (+ m n) 2) M) 2))) (exp (- l (fabs (- m n)))))

prune1.4s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.6b

regimes507.0ms

Accuracy

0% (0.2b remaining)

Error of 1.3b against oracle of 1.1b and baseline of 1.3b

bsearch2.0ms