Average Error: 0.1 → 0.1
Time: 19.3s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{\log \left(e^{\cos v \cdot e}\right) + 1}\]

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-log-exp0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2018360 
(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

start488.0ms

Algorithm
intervals

setup576.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize22.0ms

Local error

Found 3 expressions with local error:

0.2b
(* 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))

series156.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

67.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))
46.0ms
(* e (sin v))
42.0ms
(* e (cos v))

simplify724.0ms

Counts
28 → 52
Calls

28 calls. Slowest were:

256.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
146.0ms
(+ (* 1 1) (- (* (* e (cos v)) (* e (cos v))) (* 1 (* e (cos v)))))
97.0ms
(/ (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))) (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))))

prune576.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

6.0b
(log (exp (* e (cos v))))
0.2b
(* e (sin v))
0.1b
(* e (cos v))
0.0b
(/ (* e (sin v)) (+ 1 (log (exp (* e (cos v))))))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

9.0ms
(/ (* e (sin v)) (+ 1 (log (exp (* e (cos v))))))
3.0ms
(log (exp (* e (cos v))))
2.0ms
(* e (sin v))

series196.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

87.0ms
(/ (* e (sin v)) (+ 1 (log (exp (* e (cos v))))))
38.0ms
(* e (cos v))
37.0ms
(* e (sin v))
34.0ms
(log (exp (* e (cos v))))

simplify833.0ms

Counts
35 → 68
Calls

35 calls. Slowest were:

247.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
155.0ms
(+ (* 1 1) (- (* (log (exp (* e (cos v)))) (log (exp (* e (cos v))))) (* 1 (log (exp (* e (cos v)))))))
112.0ms
(/ (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))) (* (* (+ 1 (log (exp (* e (cos v))))) (+ 1 (log (exp (* e (cos v)))))) (+ 1 (log (exp (* e (cos v)))))))

prune916.0ms

Pruning

8 alts after pruning (7 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)))
0.2b
(/ (+ 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)))

series218.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

105.0ms
(/ e (/ (+ 1 (* e (cos v))) (sin v)))
72.0ms
(/ (+ 1 (* e (cos v))) (sin v))
41.0ms
(* e (cos v))

simplify1.1s

Counts
53 → 73
Calls

53 calls. Slowest were:

263.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
262.0ms
(/ (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))) (* (* (sin v) (sin v)) (sin v)))
100.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune813.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.6b
(/ 1 (/ (/ (+ 1 (* e (cos v))) (sin v)) e))
0.2b
(/ (/ (+ 1 (* e (cos v))) (sin v)) e)
0.2b
(/ (+ 1 (* e (cos v))) (sin v))
0.1b
(* e (cos v))

rewrite36.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

16.0ms
(/ (/ (+ 1 (* e (cos v))) (sin v)) e)
12.0ms
(/ (+ 1 (* e (cos v))) (sin v))
4.0ms
(/ 1 (/ (/ (+ 1 (* e (cos v))) (sin v)) e))

series307.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

113.0ms
(/ (/ (+ 1 (* e (cos v))) (sin v)) e)
88.0ms
(/ 1 (/ (/ (+ 1 (* e (cos v))) (sin v)) e))
77.0ms
(/ (+ 1 (* e (cos v))) (sin v))
29.0ms
(* e (cos v))

simplify1.4s

Counts
60 → 94
Calls

60 calls. Slowest were:

372.0ms
(/ (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))) (* (* (sin v) (sin v)) (sin v)))
254.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
106.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune1.1s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

regimes139.0ms

Accuracy

0% (0.1b remaining)

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

bsearch1.0ms