Average Error: 0.1 → 0.1
Time: 19.7s
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 2018365 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))

Details

Time bar (total: 9.9s)Debug log

start893.0ms

Algorithm
intervals

setup96.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize44.0ms

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))))

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

series201.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

89.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))
69.0ms
(* e (sin v))
42.0ms
(* e (cos v))

simplify865.0ms

Counts
28 → 52
Calls

28 calls. Slowest were:

397.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
198.0ms
(+ (* 1 1) (- (* (* e (cos v)) (* e (cos v))) (* 1 (* e (cos v)))))
76.0ms
(/ (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))) (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))))

prune497.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 4 expressions with local error:

0.1b
(* (* e (cos v)) (* e (cos v)))
0.1b
(* e (sin v))
0.1b
(pow (* e (cos v)) 3)
0.1b
(* e (cos v))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 53
Calls

4 calls. Slowest were:

6.0ms
(* (* e (cos v)) (* e (cos v)))
1.0ms
(* e (cos v))
1.0ms
(pow (* e (cos v)) 3)

series218.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

64.0ms
(pow (* e (cos v)) 3)
56.0ms
(* e (sin v))
51.0ms
(* (* e (cos v)) (* e (cos v)))
47.0ms
(* e (cos v))

simplify718.0ms

Counts
30 → 65
Calls

30 calls. Slowest were:

301.0ms
(- (pow e 3) (* 3/2 (* (pow e 3) (pow v 2))))
165.0ms
(* (* (* (* e (cos v)) (* e (cos v))) (* e (cos v))) (* (* (* e (cos v)) (* e (cos v))) (* e (cos v))))
49.0ms
(- (pow e 2) (* (pow e 2) (pow v 2)))

prune936.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize7.0ms

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)))

series189.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

94.0ms
(/ e (/ (+ 1 (* e (cos v))) (sin v)))
57.0ms
(/ (+ 1 (* e (cos v))) (sin v))
37.0ms
(* e (cos v))

simplify1.2s

Counts
53 → 73
Calls

53 calls. Slowest were:

359.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
329.0ms
(/ (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))) (* (* (sin v) (sin v)) (sin v)))
86.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune918.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 4 expressions with local error:

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

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

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

series338.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

140.0ms
(/ (* e (cos v)) (sin v))
124.0ms
(/ e (+ (/ (* e (cos v)) (sin v)) (/ 1 (sin v))))
49.0ms
(* e (cos v))
24.0ms
(/ 1 (sin v))

simplify1.6s

Counts
41 → 79
Calls

41 calls. Slowest were:

596.0ms
(/ (* (* (* e (cos v)) (* e (cos v))) (* e (cos v))) (* (* (sin v) (sin v)) (sin v)))
397.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
209.0ms
(+ (* (/ (* e (cos v)) (sin v)) (/ (* e (cos v)) (sin v))) (- (* (/ 1 (sin v)) (/ 1 (sin v))) (* (/ (* e (cos v)) (sin v)) (/ 1 (sin v)))))

prune899.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.0b

regimes122.0ms

Accuracy

0% (0.1b remaining)

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

bsearch5.0ms