Average Error: 0.1 → 0.1
Time: 18.4s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\left(\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right) + \left(1 - \cos v \cdot e\right)\right) \cdot \frac{e \cdot \sin v}{1 + {\left(\cos v \cdot e\right)}^{3}}\]

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 flip3-+0.1

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

  4. Applied associate-/r/0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019007 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))

Details

Time bar (total: 17.3s)Debug log

sample241.0ms

Algorithm
intervals

simplify65.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

64.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))

prune17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize36.0ms

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))))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
3 → 43
Calls

3 calls. Slowest were:

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

series200.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

84.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))
64.0ms
(* e (sin v))
52.0ms
(* e (cos v))

simplify992.0ms

Counts
28 → 52
Calls

28 calls. Slowest were:

351.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
295.0ms
(+ (* 1 1) (- (* (* e (cos v)) (* e (cos v))) (* 1 (* e (cos v)))))
93.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune545.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize29.0ms

Local error

Found 4 expressions with local error:

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

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 44
Calls

4 calls. Slowest were:

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

series140.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

42.0ms
(* e (cos v))
37.0ms
(* e (cos v))
30.0ms
(* e (cos v))
30.0ms
(* e (cos v))

simplify27.0ms

Counts
24 → 56
Calls

24 calls. Slowest were:

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

prune817.0ms

Pruning

6 alts after pruning (4 fresh and 2 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))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
3 → 64
Calls

3 calls. Slowest were:

13.0ms
(/ (+ 1 (* e (cos v))) (sin v))
2.0ms
(* e (cos v))
2.0ms
(/ e (/ (+ 1 (* e (cos v))) (sin v)))

series203.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

88.0ms
(/ (+ 1 (* e (cos v))) (sin v))
86.0ms
(/ e (/ (+ 1 (* e (cos v))) (sin v)))
29.0ms
(* e (cos v))

simplify1.4s

Counts
53 → 73
Calls

53 calls. Slowest were:

485.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
292.0ms
(/ (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))) (* (* (sin v) (sin v)) (sin v)))
109.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune771.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

localize29.0ms

Local error

Found 4 expressions with local error:

0.5b
(* (cbrt (/ (+ 1 (* e (cos v))) (sin v))) (cbrt (/ (+ 1 (* e (cos v))) (sin v))))
0.5b
(cbrt (/ (+ 1 (* e (cos v))) (sin v)))
0.5b
(cbrt (/ (+ 1 (* e (cos v))) (sin v)))
0.5b
(cbrt (/ (+ 1 (* e (cos v))) (sin v)))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 117
Calls

4 calls. Slowest were:

12.0ms
(* (cbrt (/ (+ 1 (* e (cos v))) (sin v))) (cbrt (/ (+ 1 (* e (cos v))) (sin v))))
2.0ms
(cbrt (/ (+ 1 (* e (cos v))) (sin v)))
2.0ms
(cbrt (/ (+ 1 (* e (cos v))) (sin v)))

series456.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

131.0ms
(* (cbrt (/ (+ 1 (* e (cos v))) (sin v))) (cbrt (/ (+ 1 (* e (cos v))) (sin v))))
120.0ms
(cbrt (/ (+ 1 (* e (cos v))) (sin v)))
105.0ms
(cbrt (/ (+ 1 (* e (cos v))) (sin v)))
99.0ms
(cbrt (/ (+ 1 (* e (cos v))) (sin v)))

simplify3.5s

Counts
98 → 129
Calls

98 calls. Slowest were:

263.0ms
(- (+ (* 2/3 (/ (exp (* 1/3 (- (log (/ (pow (cos v) 2) (pow (sin v) 2))) (* 2 (log (/ -1 e)))))) (* e (cos v)))) (exp (* 1/3 (- (log (/ (pow (cos v) 2) (pow (sin v) 2))) (* 2 (log (/ -1 e))))))) (* 1/9 (/ (exp (* 1/3 (- (log (/ (pow (cos v) 2) (pow (sin v) 2))) (* 2 (log (/ -1 e)))))) (* (pow e 2) (pow (cos v) 2)))))
258.0ms
(- (+ (* 1/3 (/ (exp (* 1/3 (- (log (* -1 (/ (cos v) (sin v)))) (log (/ -1 e))))) (* e (cos v)))) (exp (* 1/3 (- (log (* -1 (/ (cos v) (sin v)))) (log (/ -1 e)))))) (* 1/9 (/ (exp (* 1/3 (- (log (* -1 (/ (cos v) (sin v)))) (log (/ -1 e))))) (* (pow e 2) (pow (cos v) 2)))))
237.0ms
(- (+ (exp (* 1/3 (- (log (/ (pow (cos v) 2) (pow (sin v) 2))) (* 2 (log (/ 1 e)))))) (* 2/3 (/ (exp (* 1/3 (- (log (/ (pow (cos v) 2) (pow (sin v) 2))) (* 2 (log (/ 1 e)))))) (* e (cos v))))) (* 1/9 (/ (exp (* 1/3 (- (log (/ (pow (cos v) 2) (pow (sin v) 2))) (* 2 (log (/ 1 e)))))) (* (pow e 2) (pow (cos v) 2)))))

prune1.7s

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 0.0b

regimes220.0ms

Accuracy

0% (0.1b remaining)

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

bsearch5.0ms

end0.0ms

sample5.8s

Algorithm
intervals