Average Error: 0.1 → 0.1
Time: 15.1s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)\]

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 flip-+0.1

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

  4. Applied associate-/r/0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019005 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))

Details

Time bar (total: 14.2s)Debug log

sample273.0ms

Algorithm
intervals

simplify62.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

61.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))

prune17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize23.0ms

Local error

Found 3 expressions with local error:

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

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
3 → 43
Calls

3 calls. Slowest were:

4.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))
3.0ms
(* e (sin v))
1.0ms
(* e (cos v))

series185.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

87.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))
60.0ms
(* e (sin v))
38.0ms
(* e (cos v))

simplify984.0ms

Counts
28 → 52
Calls

28 calls. Slowest were:

358.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
242.0ms
(+ (* 1 1) (- (* (* e (cos v)) (* e (cos v))) (* 1 (* e (cos v)))))
124.0ms
(/ (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))) (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))))

prune516.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 4 expressions with local error:

0.1b
(* e (sin v))
0.1b
(* e (cos v))
0.1b
(* e (cos v))
0.1b
(* e (cos v))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 44
Calls

4 calls. Slowest were:

3.0ms
(* e (cos v))
1.0ms
(* e (sin v))
1.0ms
(* e (cos v))

series169.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

51.0ms
(* e (sin v))
48.0ms
(* e (cos v))
39.0ms
(* e (cos v))
30.0ms
(* e (cos v))

simplify36.0ms

Counts
24 → 56
Calls

24 calls. Slowest were:

3.0ms
(* e (sin v))
3.0ms
(* e (sin v))
3.0ms
(* e (cos v))

prune662.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 3 expressions with local error:

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

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
3 → 64
Calls

3 calls. Slowest were:

8.0ms
(/ (+ 1 (* e (cos v))) (sin v))
2.0ms
(/ e (/ (+ 1 (* e (cos v))) (sin v)))
1.0ms
(* e (cos v))

series216.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

100.0ms
(/ e (/ (+ 1 (* e (cos v))) (sin v)))
84.0ms
(/ (+ 1 (* e (cos v))) (sin v))
32.0ms
(* e (cos v))

simplify1.2s

Counts
53 → 73
Calls

53 calls. Slowest were:

386.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
260.0ms
(/ (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))) (* (* (sin v) (sin v)) (sin v)))
82.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune729.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

5.4b
(log (exp (* e (cos v))))
0.2b
(/ e (/ (+ 1 (log (exp (* e (cos v))))) (sin v)))
0.1b
(/ (+ 1 (log (exp (* e (cos v))))) (sin v))
0.1b
(* e (cos v))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 77
Calls

4 calls. Slowest were:

21.0ms
(/ (+ 1 (log (exp (* e (cos v))))) (sin v))
2.0ms
(* e (cos v))
2.0ms
(log (exp (* e (cos v))))

series249.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

90.0ms
(/ (+ 1 (log (exp (* e (cos v))))) (sin v))
73.0ms
(/ e (/ (+ 1 (log (exp (* e (cos v))))) (sin v)))
48.0ms
(* e (cos v))
39.0ms
(log (exp (* e (cos v))))

simplify1.6s

Counts
60 → 89
Calls

60 calls. Slowest were:

326.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
305.0ms
(/ (* (* (+ 1 (log (exp (* e (cos v))))) (+ 1 (log (exp (* e (cos v)))))) (+ 1 (log (exp (* e (cos v)))))) (* (* (sin v) (sin v)) (sin v)))
114.0ms
(+ (pow 1 3) (pow (log (exp (* e (cos v)))) 3))

prune1.1s

Pruning

9 alts after pruning (7 fresh and 2 done)

Merged error: 0.0b

regimes237.0ms

Accuracy

0% (0.1b remaining)

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

bsearch2.0ms

end0.0ms

sample5.9s

Algorithm
intervals