Average Error: 15.2 → 1.5
Time: 23.3s
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 1.5

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

  3. Final simplification1.5

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

Reproduce

herbie shell --seed 2019008 
(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: 21.6s)Debug log

sample158.0ms

Algorithm
intervals

simplify145.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

145.0ms
(* (cos (- (/ (* K (+ m n)) 2) M)) (exp (- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n))))))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 17.4b

localize64.0ms

Local error

Found 4 expressions with local error:

35.7b
(cos (- (/ (* K (+ m n)) 2) M))
0.0b
(* K (+ m n))
0.0b
(pow (- (/ (+ m n) 2) M) 2)
0.0b
(- (/ (* K (+ m n)) 2) M)

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 50
Calls

4 calls. Slowest were:

6.0ms
(cos (- (/ (* K (+ m n)) 2) M))
5.0ms
(pow (- (/ (+ m n) 2) M) 2)
3.0ms
(- (/ (* K (+ m n)) 2) M)

series207.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

64.0ms
(pow (- (/ (+ m n) 2) M) 2)
58.0ms
(- (/ (* K (+ m n)) 2) M)
55.0ms
(cos (- (/ (* K (+ m n)) 2) M))
30.0ms
(* K (+ m n))

simplify1.0s

Counts
27 → 62
Calls

27 calls. Slowest were:

103.0ms
(cos (- (+ (* 1/2 (* m K)) (* 1/2 (* K n))) M))
92.0ms
(- (pow M 2) (+ (* m M) (* M n)))
90.0ms
(- (pow M 2) (+ (* m M) (* M n)))

prune1.3s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 1.4b

localize3.0ms

Local error

Found 1 expressions with local error:

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

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
1 → 14
Calls

1 calls. Slowest were:

5.0ms
(pow (- (/ (+ m n) 2) M) 2)

series64.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

64.0ms
(pow (- (/ (+ m n) 2) M) 2)

simplify338.0ms

Counts
8 → 17
Calls

8 calls. Slowest were:

109.0ms
(- (pow M 2) (+ (* m M) (* M n)))
88.0ms
(+ (* 1/2 (* m n)) (+ (* 1/4 (pow m 2)) (* 1/4 (pow n 2))))
84.0ms
(- (pow M 2) (+ (* m M) (* M n)))

prune358.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 1.4b

localize27.0ms

Local error

Found 4 expressions with local error:

44.2b
(cos (* (* (cbrt (- (/ (* K (+ m n)) 2) M)) (cbrt (- (/ (* K (+ m n)) 2) M))) (cbrt (- (/ (* K (+ m n)) 2) M))))
15.7b
(cbrt (- (/ (* K (+ m n)) 2) M))
15.7b
(cbrt (- (/ (* K (+ m n)) 2) M))
15.7b
(cbrt (- (/ (* K (+ m n)) 2) M))

rewrite38.0ms

Algorithm
rewrite-expression-head
Counts
4 → 46
Calls

4 calls. Slowest were:

28.0ms
(cos (* (* (cbrt (- (/ (* K (+ m n)) 2) M)) (cbrt (- (/ (* K (+ m n)) 2) M))) (cbrt (- (/ (* K (+ m n)) 2) M))))
4.0ms
(cbrt (- (/ (* K (+ m n)) 2) M))
3.0ms
(cbrt (- (/ (* K (+ m n)) 2) M))

series809.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

262.0ms
(cbrt (- (/ (* K (+ m n)) 2) M))
249.0ms
(cbrt (- (/ (* K (+ m n)) 2) M))
240.0ms
(cbrt (- (/ (* K (+ m n)) 2) M))
58.0ms
(cos (* (* (cbrt (- (/ (* K (+ m n)) 2) M)) (cbrt (- (/ (* K (+ m n)) 2) M))) (cbrt (- (/ (* K (+ m n)) 2) M))))

simplify3.7s

Counts
27 → 58
Calls

27 calls. Slowest were:

394.0ms
(exp (* 1/3 (- (log 1/2) (+ (log (/ 1 m)) (log (/ 1 K))))))
382.0ms
(exp (* 1/3 (- (log 1/2) (+ (log (/ 1 m)) (log (/ 1 K))))))
344.0ms
(exp (* 1/3 (- (log 1/2) (+ (log (/ 1 m)) (log (/ 1 K))))))

prune1.2s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 1.4b

localize25.0ms

Local error

Found 4 expressions with local error:

43.6b
(cos (* (* (cbrt (- (/ (* K (+ m n)) 2) M)) (cbrt (* 1/2 (* K m)))) (cbrt (- (/ (* K (+ m n)) 2) M))))
15.7b
(cbrt (- (/ (* K (+ m n)) 2) M))
15.7b
(cbrt (- (/ (* K (+ m n)) 2) M))
14.9b
(cbrt (* 1/2 (* K m)))

rewrite38.0ms

Algorithm
rewrite-expression-head
Counts
4 → 42
Calls

4 calls. Slowest were:

29.0ms
(cos (* (* (cbrt (- (/ (* K (+ m n)) 2) M)) (cbrt (* 1/2 (* K m)))) (cbrt (- (/ (* K (+ m n)) 2) M))))
5.0ms
(cbrt (- (/ (* K (+ m n)) 2) M))
2.0ms
(cbrt (- (/ (* K (+ m n)) 2) M))

series2.2s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.6s
(cos (* (* (cbrt (- (/ (* K (+ m n)) 2) M)) (cbrt (* 1/2 (* K m)))) (cbrt (- (/ (* K (+ m n)) 2) M))))
278.0ms
(cbrt (- (/ (* K (+ m n)) 2) M))
260.0ms
(cbrt (- (/ (* K (+ m n)) 2) M))
126.0ms
(cbrt (* 1/2 (* K m)))

simplify3.9s

Counts
23 → 54
Calls

23 calls. Slowest were:

470.0ms
(cos (* (cbrt 1/2) (exp (* 1/3 (- (log 1/4) (+ (* 3 (log (/ -1 K))) (* 3 (log (/ -1 m)))))))))
395.0ms
(cos (* (cbrt 1/2) (exp (* 1/3 (- (log 1/4) (+ (* 3 (log (/ 1 m))) (* 3 (log (/ 1 K)))))))))
385.0ms
(exp (* 1/3 (- (log 1/2) (+ (log (/ 1 m)) (log (/ 1 K))))))

prune1.4s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 1.4b

regimes491.0ms

Accuracy

0% (0.3b remaining)

Error of 1.5b against oracle of 1.2b and baseline of 1.5b

bsearch2.0ms

end0.0ms

sample3.9s

Algorithm
intervals