Average Error: 0.1 → 0.1
Time: 14.2s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 + \sqrt[3]{\left(\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right)\right) \cdot \left(\cos v \cdot e\right)}}\]

Error

Bits error versus e

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
  2. Using strategy rm

  3. Applied add-cbrt-cube0.1

    \[\leadsto \frac{e \cdot \sin v}{1 + \color{blue}{\sqrt[3]{\left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)\right) \cdot \left(e \cdot \cos v\right)}}}\]

  4. Final simplification0.1

    \[\leadsto \frac{e \cdot \sin v}{1 + \sqrt[3]{\left(\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right)\right) \cdot \left(\cos v \cdot e\right)}}\]

Reproduce

herbie shell --seed 2019002 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))

Details

Time bar (total: 13.2s)Debug log

sample377.0ms

Algorithm
intervals

simplify26.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

26.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))

prune33.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize42.0ms

Local error

Found 3 expressions with local error:

0.1b
(* e (cos v))
0.1b
(* e (sin v))
0.0b
(/ (* e (sin v)) (+ 1 (* e (cos v))))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
3 → 43
Calls

3 calls. Slowest were:

8.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))
5.0ms
(* e (cos v))
2.0ms
(* e (sin v))

series228.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

93.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))
70.0ms
(* e (sin v))
65.0ms
(* e (cos v))

simplify1.0s

Counts
28 → 52
Calls

28 calls. Slowest were:

376.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
198.0ms
(+ (* 1 1) (- (* (* e (cos v)) (* e (cos v))) (* 1 (* e (cos v)))))
120.0ms
(/ (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))) (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))))

prune526.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

37.6b
(cbrt (* (* (* e (cos v)) (* e (cos v))) (* e (cos v))))
0.1b
(* e (cos v))
0.1b
(* e (cos v))
0.1b
(* e (cos v))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 43
Calls

4 calls. Slowest were:

9.0ms
(cbrt (* (* (* e (cos v)) (* e (cos v))) (* e (cos v))))
3.0ms
(* e (cos v))
1.0ms
(* e (cos v))

series184.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

49.0ms
(* e (cos v))
48.0ms
(* e (cos v))
48.0ms
(cbrt (* (* (* e (cos v)) (* e (cos v))) (* e (cos v))))
38.0ms
(* e (cos v))

simplify96.0ms

Counts
22 → 55
Calls

22 calls. Slowest were:

57.0ms
(* -1 (* e (* (cbrt -1) (cos v))))
13.0ms
(cbrt (* (* e (cos v)) (* e (cos v))))
3.0ms
(* e (cos v))

prune575.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize5.0ms

Local error

Found 3 expressions with local error:

0.2b
(/ e (/ (+ 1 (* e (cos v))) (sin v)))
0.1b
(* e (cos v))
0.1b
(/ (+ 1 (* e (cos v))) (sin v))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
3 → 64
Calls

3 calls. Slowest were:

6.0ms
(/ (+ 1 (* e (cos v))) (sin v))
1.0ms
(* e (cos v))
1.0ms
(/ e (/ (+ 1 (* e (cos v))) (sin v)))

series212.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

95.0ms
(/ (+ 1 (* e (cos v))) (sin v))
78.0ms
(/ e (/ (+ 1 (* e (cos v))) (sin v)))
39.0ms
(* e (cos v))

simplify1.2s

Counts
53 → 73
Calls

53 calls. Slowest were:

380.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
269.0ms
(/ (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))) (* (* (sin v) (sin v)) (sin v)))
68.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune792.0ms

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (sin v))
0.6b
(cbrt (sin v))
0.6b
(cbrt (sin v))
0.5b
(* (cbrt (sin v)) (cbrt (sin v)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

8.0ms
(* (cbrt (sin v)) (cbrt (sin v)))
1.0ms
(cbrt (sin v))
0.0ms
(cbrt (sin v))

series614.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

173.0ms
(* (cbrt (sin v)) (cbrt (sin v)))
167.0ms
(cbrt (sin v))
142.0ms
(cbrt (sin v))
132.0ms
(cbrt (sin v))

simplify309.0ms

Counts
41 → 73
Calls

41 calls. Slowest were:

90.0ms
(- (pow v 1/3) (+ (* 1/18 (pow (pow v 7) 1/3)) (* 1/3240 (pow (pow v 13) 1/3))))
64.0ms
(- (pow v 1/3) (+ (* 1/18 (pow (pow v 7) 1/3)) (* 1/3240 (pow (pow v 13) 1/3))))
61.0ms
(- (pow v 1/3) (+ (* 1/18 (pow (pow v 7) 1/3)) (* 1/3240 (pow (pow v 13) 1/3))))

prune877.0ms

Pruning

10 alts after pruning (7 fresh and 3 done)

Merged error: 0.0b

regimes172.0ms

Accuracy

0% (0.1b remaining)

Error of 0.1b against oracle of 0.0b and baseline of 0.1b

bsearch3.0ms

end0.0ms

sample5.8s

Algorithm
intervals