Average Error: 0.1 → 0.1
Time: 19.7s
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 2018362 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))

Details

Time bar (total: 9.6s)Debug log

start376.0ms

Algorithm
intervals

setup59.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize449.0ms

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))))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
3 → 43
Calls

3 calls. Slowest were:

4.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))
2.0ms
(* e (cos v))
1.0ms
(* e (sin v))

series192.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

92.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))
53.0ms
(* e (sin v))
47.0ms
(* e (cos v))

simplify1.1s

Counts
28 → 52
Calls

28 calls. Slowest were:

416.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
257.0ms
(+ (* 1 1) (- (* (* e (cos v)) (* e (cos v))) (* 1 (* e (cos v)))))
138.0ms
(/ (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))) (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))))

prune523.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 4 expressions with local error:

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

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
4 → 49
Calls

4 calls. Slowest were:

11.0ms
(* (* e (cos v)) (* e (cos v)))
2.0ms
(* e (cos v))
2.0ms
(* e (cos v))

series213.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

72.0ms
(* (* e (cos v)) (* e (cos v)))
54.0ms
(* e (cos v))
49.0ms
(* e (cos v))
39.0ms
(* e (cos v))

simplify299.0ms

Counts
28 → 61
Calls

28 calls. Slowest were:

181.0ms
(* (* (* (* e (cos v)) (* e (cos v))) (* e (cos v))) (* (* (* e (cos v)) (* e (cos v))) (* e (cos v))))
39.0ms
(- (pow e 2) (* (pow e 2) (pow v 2)))
21.0ms
(+ (log (* e (cos v))) (log (* e (cos v))))

prune829.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 3 expressions with local error:

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

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
3 → 64
Calls

3 calls. Slowest were:

6.0ms
(/ (sin v) (+ 1 (* e (cos v))))
4.0ms
(* e (/ (sin v) (+ 1 (* e (cos v)))))
2.0ms
(* e (cos v))

series168.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

77.0ms
(* e (/ (sin v) (+ 1 (* e (cos v)))))
48.0ms
(/ (sin v) (+ 1 (* e (cos v))))
43.0ms
(* e (cos v))

simplify2.2s

Counts
55 → 73
Calls

55 calls. Slowest were:

512.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
325.0ms
(/ (* (* (sin v) (sin v)) (sin v)) (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))))
272.0ms
(+ (* 1 1) (- (* (* e (cos v)) (* e (cos v))) (* 1 (* e (cos v)))))

prune716.0ms

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

series181.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

83.0ms
(/ e (/ (+ 1 (* e (cos v))) (sin v)))
60.0ms
(/ (+ 1 (* e (cos v))) (sin v))
38.0ms
(* e (cos v))

simplify1.4s

Counts
53 → 73
Calls

53 calls. Slowest were:

460.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
274.0ms
(/ (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))) (* (* (sin v) (sin v)) (sin v)))
84.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune765.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.0b

regimes131.0ms

Accuracy

0% (0.1b remaining)

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

bsearch3.0ms