Average Error: 26.8 → 5.8
Time: 20.5s
Precision: 64
Internal Precision: 128
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(\left(\left(sin \cdot x\right) \cdot cos\right) \cdot \left(sin \cdot x\right)\right)}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 26.8

    \[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
  2. Using strategy rm

  3. Applied unpow226.8

    \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot \color{blue}{\left(sin \cdot sin\right)}\right) \cdot x\right)}\]

  4. Applied associate-*r*20.7

    \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\color{blue}{\left(\left(x \cdot sin\right) \cdot sin\right)} \cdot x\right)}\]
  5. Using strategy rm

  6. Applied unpow220.7

    \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(cos \cdot cos\right)} \cdot \left(\left(\left(x \cdot sin\right) \cdot sin\right) \cdot x\right)}\]

  7. Applied associate-*l*15.7

    \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{cos \cdot \left(cos \cdot \left(\left(\left(x \cdot sin\right) \cdot sin\right) \cdot x\right)\right)}}\]
  8. Using strategy rm

  9. Applied associate-*l*13.3

    \[\leadsto \frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(cos \cdot \color{blue}{\left(\left(x \cdot sin\right) \cdot \left(sin \cdot x\right)\right)}\right)}\]
  10. Using strategy rm

  11. Applied associate-*r*5.8

    \[\leadsto \frac{\cos \left(2 \cdot x\right)}{cos \cdot \color{blue}{\left(\left(cos \cdot \left(x \cdot sin\right)\right) \cdot \left(sin \cdot x\right)\right)}}\]

  12. Final simplification5.8

    \[\leadsto \frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(\left(\left(sin \cdot x\right) \cdot cos\right) \cdot \left(sin \cdot x\right)\right)}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))))

Details

Time bar (total: 19.2s)Debug log

sample106.0ms

Algorithm
intervals

simplify157.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

157.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 28.5b

localize23.0ms

Local error

Found 4 expressions with local error:

18.3b
(* (pow cos 2) (* (* x (pow sin 2)) x))
10.1b
(* x (pow sin 2))
4.6b
(* (* x (pow sin 2)) x)
0.2b
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

11.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))
9.0ms
(* (pow cos 2) (* (* x (pow sin 2)) x))
4.0ms
(* (* x (pow sin 2)) x)

series121.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

51.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))
37.0ms
(* (pow cos 2) (* (* x (pow sin 2)) x))
17.0ms
(* (* x (pow sin 2)) x)
16.0ms
(* x (pow sin 2))

simplify2.2s

Counts
35 → 72
Calls

35 calls. Slowest were:

372.0ms
(* (* (* (pow cos 2) (pow cos 2)) (pow cos 2)) (* (* (* (* x (pow sin 2)) x) (* (* x (pow sin 2)) x)) (* (* x (pow sin 2)) x)))
211.0ms
(- (log (cos (* 2 x))) (log (* (pow cos 2) (* (* x (pow sin 2)) x))))
175.0ms
(- (log (cos (* 2 x))) (+ (log (pow cos 2)) (log (* (* x (pow sin 2)) x))))

prune612.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 10.5b

localize15.0ms

Local error

Found 4 expressions with local error:

18.3b
(* (pow cos 2) (* (* (* x sin) sin) x))
4.6b
(* (* (* x sin) sin) x)
0.2b
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
0.1b
(* (* x sin) sin)

rewrite36.0ms

Algorithm
rewrite-expression-head
Counts
4 → 57
Calls

4 calls. Slowest were:

14.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
13.0ms
(* (pow cos 2) (* (* (* x sin) sin) x))
6.0ms
(* (* (* x sin) sin) x)

series119.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

50.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
34.0ms
(* (pow cos 2) (* (* (* x sin) sin) x))
18.0ms
(* (* (* x sin) sin) x)
16.0ms
(* (* x sin) sin)

simplify2.0s

Counts
33 → 69
Calls

33 calls. Slowest were:

269.0ms
(- (log (cos (* 2 x))) (+ (log (pow cos 2)) (log (* (* (* x sin) sin) x))))
205.0ms
(/ (* (cos x) (cos x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
172.0ms
(- (+ (/ 1 (* (pow x 2) (* (pow sin 2) (pow cos 2)))) (* 2/3 (/ (pow x 2) (* (pow sin 2) (pow cos 2))))) (* 2 (/ 1 (* (pow sin 2) (pow cos 2)))))

prune666.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 7.4b

localize21.0ms

Local error

Found 4 expressions with local error:

8.0b
(* cos (* (* (* x sin) sin) x))
4.6b
(* (* (* x sin) sin) x)
4.1b
(* cos (* cos (* (* (* x sin) sin) x)))
0.2b
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))

rewrite80.0ms

Algorithm
rewrite-expression-head
Counts
4 → 47
Calls

4 calls. Slowest were:

27.0ms
(* cos (* cos (* (* (* x sin) sin) x)))
26.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))
14.0ms
(* cos (* (* (* x sin) sin) x))

series180.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

62.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))
58.0ms
(* cos (* (* (* x sin) sin) x))
34.0ms
(* cos (* cos (* (* (* x sin) sin) x)))
26.0ms
(* (* (* x sin) sin) x)

simplify2.3s

Counts
26 → 59
Calls

26 calls. Slowest were:

299.0ms
(- (+ (/ 1 (* (pow x 2) (* (pow sin 2) (pow cos 2)))) (* 2/3 (/ (pow x 2) (* (pow sin 2) (pow cos 2))))) (* 2 (/ 1 (* (pow sin 2) (pow cos 2)))))
273.0ms
(/ (* (cos x) (cos x)) (* cos (* cos (* (* (* x sin) sin) x))))
271.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))

prune657.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 3.5b

localize12.0ms

Local error

Found 4 expressions with local error:

8.0b
(* cos (* (* x sin) (* sin x)))
4.1b
(* cos (* cos (* (* x sin) (* sin x))))
0.3b
(* (* x sin) (* sin x))
0.2b
(/ (cos (* 2 x)) (* cos (* cos (* (* x sin) (* sin x)))))

rewrite59.0ms

Algorithm
rewrite-expression-head
Counts
4 → 51
Calls

4 calls. Slowest were:

19.0ms
(* cos (* cos (* (* x sin) (* sin x))))
18.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* x sin) (* sin x)))))
14.0ms
(* cos (* (* x sin) (* sin x)))

series221.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

69.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* x sin) (* sin x)))))
63.0ms
(* cos (* (* x sin) (* sin x)))
61.0ms
(* cos (* cos (* (* x sin) (* sin x))))
28.0ms
(* (* x sin) (* sin x))

simplify2.5s

Counts
29 → 63
Calls

29 calls. Slowest were:

280.0ms
(/ (* (cos x) (cos x)) (* cos (* cos (* (* x sin) (* sin x)))))
260.0ms
(- (+ (/ 1 (* (pow x 2) (* (pow sin 2) (pow cos 2)))) (* 2/3 (/ (pow x 2) (* (pow sin 2) (pow cos 2))))) (* 2 (/ 1 (* (pow sin 2) (pow cos 2)))))
238.0ms
(* (* (* (* x sin) (* x sin)) (* x sin)) (* (* (* sin x) (* sin x)) (* sin x)))

prune655.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 2.6b

regimes655.0ms

Accuracy

0% (2.2b remaining)

Error of 5.8b against oracle of 3.6b and baseline of 5.8b

bsearch3.0ms

end0.0ms

sample5.8s

Algorithm
intervals