Average Error: 26.8 → 5.8
Time: 17.6s
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 +o rules:numerics
(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: 16.4s)Debug log

sample166.0ms

Algorithm
intervals

simplify270.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

270.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 28.5b

localize31.0ms

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)))

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

15.0ms
(* (pow cos 2) (* (* x (pow sin 2)) x))
10.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))
3.0ms
(* (* x (pow sin 2)) x)

series140.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

61.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))
46.0ms
(* (pow cos 2) (* (* x (pow sin 2)) x))
16.0ms
(* (* x (pow sin 2)) x)
16.0ms
(* x (pow sin 2))

simplify2.7s

Counts
35 → 80
Calls

35 calls. Slowest were:

478.0ms
(* (* (* (pow cos 2) (pow cos 2)) (pow cos 2)) (* (* (* (* x (pow sin 2)) x) (* (* x (pow sin 2)) x)) (* (* x (pow sin 2)) x)))
262.0ms
(- (+ (/ 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)))))
229.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))

prune689.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 9.5b

localize12.0ms

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 → 65
Calls

4 calls. Slowest were:

15.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
12.0ms
(* (pow cos 2) (* (* (* x sin) sin) x))
6.0ms
(* (* (* x sin) sin) x)

series113.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

49.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
32.0ms
(* (pow cos 2) (* (* (* x sin) sin) x))
16.0ms
(* (* (* x sin) sin) x)
15.0ms
(* (* x sin) sin)

simplify2.1s

Counts
33 → 77
Calls

33 calls. Slowest were:

289.0ms
(- (+ (/ 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)))))
267.0ms
(- (log (cos (* 2 x))) (+ (log (pow cos 2)) (log (* (* (* x sin) sin) x))))
203.0ms
(/ (* (cos x) (cos x)) (* (pow cos 2) (* (* (* x sin) sin) x)))

prune601.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 7.0b

localize8.0ms

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))))

rewrite39.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

13.0ms
(* cos (* cos (* (* (* x sin) sin) x)))
12.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))
7.0ms
(* (* (* x sin) sin) x)

series131.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

48.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))
34.0ms
(* cos (* (* (* x sin) sin) x))
33.0ms
(* cos (* cos (* (* (* x sin) sin) x)))
16.0ms
(* (* (* x sin) sin) x)

simplify1.5s

Counts
26 → 67
Calls

26 calls. Slowest were:

204.0ms
(- (+ (/ 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)))))
167.0ms
(/ (* (cos x) (cos x)) (* cos (* cos (* (* (* x sin) sin) x))))
149.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))

prune510.0ms

Pruning

15 alts after pruning (15 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)))))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

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

series132.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

49.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* x sin) (* sin x)))))
34.0ms
(* cos (* (* x sin) (* sin x)))
32.0ms
(* cos (* cos (* (* x sin) (* sin x))))
16.0ms
(* (* x sin) (* sin x))

simplify1.6s

Counts
29 → 71
Calls

29 calls. Slowest were:

202.0ms
(- (+ (/ 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)))))
167.0ms
(/ (* (cos x) (cos x)) (* cos (* cos (* (* x sin) (* sin x)))))
166.0ms
(* (* (* (* x sin) (* x sin)) (* x sin)) (* (* (* sin x) (* sin x)) (* sin x)))

prune680.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 2.6b

regimes440.0ms

Accuracy

0% (2.4b remaining)

Error of 5.8b against oracle of 3.4b and baseline of 5.8b

bsearch2.0ms

end0.0ms

sample4.4s

Algorithm
intervals