Average Error: 26.7 → 5.9
Time: 24.4s
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 26.7

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

  3. Applied unpow226.7

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

  4. Applied associate-*r*20.9

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

  6. Applied unpow220.9

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

  7. Applied associate-*l*15.6

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

  9. Applied associate-*l*12.8

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

  11. Applied associate-*r*5.9

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

  12. Final simplification5.9

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

Reproduce

herbie shell --seed 2019016 +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: 22.7s)Debug log

sample153.0ms

Algorithm
intervals
Results
363×(pre true 80)
282×(body real 80)
29×(body real 640)
24×(body real 1280)
19×(body real 320)
9×(body real 160)

simplify181.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
181.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))

prune15.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 26.5b

localize28.0ms

Local error

Found 4 expressions with local error:

16.8b
(* (pow cos 2) (* (* x (pow sin 2)) x))
7.9b
(* x (pow sin 2))
5.2b
(* (* x (pow sin 2)) x)
0.5b
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))

rewrite43.0ms

Algorithm
rewrite-expression-head
Rules
14×add-exp-log
11×add-cbrt-cube
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
5×associate-*l*
5×associate-*r*
4×add-log-exp
4×log1p-expm1-u
4×prod-exp
4×pow1
4×expm1-log1p-u
3×associate-/l*
3×*-commutative
3×div-exp
3×times-frac
2×unpow2
2×cbrt-unprod
2×cbrt-undiv
2×pow-to-exp
1×div-inv
1×cos-2
1×div-sub
1×frac-2neg
1×associate-/r*
1×clear-num
Counts
4 → 68
Calls
4 calls:
Slowest
21.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))
12.0ms
(* (pow cos 2) (* (* x (pow sin 2)) x))
7.0ms
(* (* x (pow sin 2)) x)
2.0ms
(* x (pow sin 2))

series174.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
81.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))
42.0ms
(* (pow cos 2) (* (* x (pow sin 2)) x))
27.0ms
(* (* x (pow sin 2)) x)
24.0ms
(* x (pow sin 2))

simplify3.5s

Counts
35 → 80
Calls
35 calls:
Slowest
577.0ms
(* (* (* (pow cos 2) (pow cos 2)) (pow cos 2)) (* (* (* (* x (pow sin 2)) x) (* (* x (pow sin 2)) x)) (* (* x (pow sin 2)) x)))
271.0ms
(- (log (cos (* 2 x))) (+ (log (pow cos 2)) (log (* (* x (pow sin 2)) x))))
267.0ms
(- (log (cos (* 2 x))) (log (* (pow cos 2) (* (* x (pow sin 2)) x))))
260.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))
259.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))

prune988.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 8.9b

localize29.0ms

Local error

Found 4 expressions with local error:

16.8b
(* (pow cos 2) (* (* (* x sin) sin) x))
5.2b
(* (* (* x sin) sin) x)
0.5b
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
0.1b
(* (* x sin) sin)

rewrite66.0ms

Algorithm
rewrite-expression-head
Rules
14×add-exp-log
11×add-cbrt-cube
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×associate-*l*
4×add-log-exp
4×log1p-expm1-u
4×prod-exp
4×pow1
4×expm1-log1p-u
3×associate-/l*
3×*-commutative
3×div-exp
3×times-frac
2×cbrt-unprod
2×cbrt-undiv
2×pow-to-exp
1×unpow2
1×div-inv
1×cos-2
1×associate-*r*
1×div-sub
1×frac-2neg
1×associate-/r*
1×clear-num
Counts
4 → 65
Calls
4 calls:
Slowest
27.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
23.0ms
(* (pow cos 2) (* (* (* x sin) sin) x))
11.0ms
(* (* (* x sin) sin) x)
4.0ms
(* (* x sin) sin)

series198.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
82.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
62.0ms
(* (pow cos 2) (* (* (* x sin) sin) x))
27.0ms
(* (* x sin) sin)
27.0ms
(* (* (* x sin) sin) x)

simplify3.4s

Counts
33 → 77
Calls
33 calls:
Slowest
677.0ms
(- (+ (/ 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)))))
428.0ms
(- (log (cos (* 2 x))) (+ (log (pow cos 2)) (log (* (* (* x sin) sin) x))))
288.0ms
(/ (* (cos x) (cos x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
277.0ms
(- (log (cos (* 2 x))) (+ (* (log cos) 2) (log (* (* (* x sin) sin) x))))
258.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))

prune843.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 6.7b

localize14.0ms

Local error

Found 4 expressions with local error:

6.3b
(* cos (* cos (* (* (* x sin) sin) x)))
6.2b
(* cos (* (* (* x sin) sin) x))
5.2b
(* (* (* x sin) sin) x)
0.5b
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))

rewrite57.0ms

Algorithm
rewrite-expression-head
Rules
6×add-cube-cbrt
6×add-exp-log
6×add-cbrt-cube
6×*-un-lft-identity
6×add-sqr-sqrt
4×add-log-exp
4×log1p-expm1-u
4×pow1
4×expm1-log1p-u
3×associate-/l*
3×*-commutative
3×times-frac
2×associate-*r*
1×div-inv
1×cos-2
1×associate-*l*
1×div-exp
1×div-sub
1×frac-2neg
1×associate-/r*
1×clear-num
1×cbrt-undiv
Counts
4 → 55
Calls
4 calls:
Slowest
30.0ms
(* cos (* cos (* (* (* x sin) sin) x)))
13.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))
7.0ms
(* cos (* (* (* x sin) sin) x))
6.0ms
(* (* (* x sin) sin) x)

series196.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
85.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))
52.0ms
(* cos (* (* (* x sin) sin) x))
41.0ms
(* cos (* cos (* (* (* x sin) sin) x)))
17.0ms
(* (* (* x sin) sin) x)

simplify2.1s

Counts
26 → 67
Calls
26 calls:
Slowest
276.0ms
(/ (* (cos x) (cos x)) (* cos (* cos (* (* (* x sin) sin) x))))
256.0ms
(- (+ (/ 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)))))
250.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))
244.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))
147.0ms
(/ (* (* (cos (* 2 x)) (cos (* 2 x))) (cos (* 2 x))) (* (* (* cos (* cos (* (* (* x sin) sin) x))) (* cos (* cos (* (* (* x sin) sin) x)))) (* cos (* cos (* (* (* x sin) sin) x)))))

prune1.0s

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 4.8b

localize20.0ms

Local error

Found 4 expressions with local error:

6.3b
(* cos (* cos (* (* x sin) (* sin x))))
6.2b
(* cos (* (* x sin) (* sin x)))
0.5b
(/ (cos (* 2 x)) (* cos (* cos (* (* x sin) (* sin x)))))
0.2b
(* (* x sin) (* sin x))

rewrite57.0ms

Algorithm
rewrite-expression-head
Rules
8×add-exp-log
8×add-cbrt-cube
6×add-cube-cbrt
6×*-un-lft-identity
6×pow1
6×add-sqr-sqrt
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×associate-/l*
3×*-commutative
3×associate-*r*
3×times-frac
1×div-inv
1×cos-2
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×associate-*l*
1×div-exp
1×div-sub
1×frac-2neg
1×associate-/r*
1×clear-num
1×cbrt-undiv
Counts
4 → 59
Calls
4 calls:
Slowest
18.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* x sin) (* sin x)))))
18.0ms
(* cos (* cos (* (* x sin) (* sin x))))
13.0ms
(* cos (* (* x sin) (* sin x)))
7.0ms
(* (* x sin) (* sin x))

series197.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
78.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* x sin) (* sin x)))))
52.0ms
(* cos (* (* x sin) (* sin x)))
37.0ms
(* cos (* cos (* (* x sin) (* sin x))))
28.0ms
(* (* x sin) (* sin x))

simplify2.5s

Counts
29 → 71
Calls
29 calls:
Slowest
275.0ms
(/ (* (cos x) (cos x)) (* cos (* cos (* (* x sin) (* sin x)))))
270.0ms
(- (+ (/ 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)))))
239.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))
224.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))
206.0ms
(* (* (* (* x sin) (* x sin)) (* x sin)) (* (* (* sin x) (* sin x)) (* sin x)))

prune815.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 3.1b

regimes691.0ms

Accuracy

0% (2.5b remaining)

Error of 5.9b against oracle of 3.4b and baseline of 5.9b

bsearch2.0ms

end0.0ms

sample5.4s

Algorithm
intervals
Results
11760×(pre true 80)
9181×(body real 80)
898×(body real 640)
820×(body real 1280)
569×(body real 320)
292×(body real 160)