Average Error: 0.1 → 0.1
Time: 19.5s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{\cos v \cdot e + 1}\]

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. Final simplification0.1

    \[\leadsto \frac{e \cdot \sin v}{\cos v \cdot e + 1}\]

Reproduce

herbie shell --seed 2019004 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))

Details

Time bar (total: 18.6s)Debug log

sample214.0ms

Algorithm
intervals

simplify21.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune41.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize25.0ms

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))))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
3 → 43
Calls

3 calls. Slowest were:

8.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))
4.0ms
(* e (cos v))
2.0ms
(* e (sin v))

series217.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

87.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))
68.0ms
(* e (sin v))
62.0ms
(* e (cos v))

simplify1.1s

Counts
28 → 52
Calls

28 calls. Slowest were:

452.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
280.0ms
(+ (* 1 1) (- (* (* e (cos v)) (* e (cos v))) (* 1 (* e (cos v)))))
108.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune494.0ms

Pruning

7 alts after pruning (6 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
(* 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)))

series229.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

95.0ms
(/ (+ 1 (* e (cos v))) (sin v))
86.0ms
(/ e (/ (+ 1 (* e (cos v))) (sin v)))
48.0ms
(* e (cos v))

simplify1.3s

Counts
53 → 73
Calls

53 calls. Slowest were:

419.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
234.0ms
(/ (* (* (+ 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))))

prune719.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.0b

localize11.0ms

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))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

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

series309.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

127.0ms
(/ (* e (cos v)) (sin v))
122.0ms
(/ e (+ (/ (* e (cos v)) (sin v)) (/ 1 (sin v))))
34.0ms
(* e (cos v))
26.0ms
(/ 1 (sin v))

simplify1.6s

Counts
41 → 79
Calls

41 calls. Slowest were:

498.0ms
(/ (* (* (* e (cos v)) (* e (cos v))) (* e (cos v))) (* (* (sin v) (sin v)) (sin v)))
414.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
247.0ms
(+ (* (/ (* e (cos v)) (sin v)) (/ (* e (cos v)) (sin v))) (- (* (/ 1 (sin v)) (/ 1 (sin v))) (* (/ (* e (cos v)) (sin v)) (/ 1 (sin v)))))

prune835.0ms

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)))
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))) (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)))

series467.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

145.0ms
(* (cbrt (/ (+ 1 (* e (cos v))) (sin v))) (cbrt (/ (+ 1 (* e (cos v))) (sin v))))
119.0ms
(cbrt (/ (+ 1 (* e (cos v))) (sin v)))
108.0ms
(cbrt (/ (+ 1 (* e (cos v))) (sin v)))
95.0ms
(cbrt (/ (+ 1 (* e (cos v))) (sin v)))

simplify3.5s

Counts
98 → 129
Calls

98 calls. Slowest were:

317.0ms
(* (/ (+ 1 (* e (cos v))) (sin v)) (/ (+ 1 (* e (cos v))) (sin v)))
260.0ms
(- (+ (* 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)))))
225.0ms
(- (+ (* 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)))))

prune1.7s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0b

regimes199.0ms

Accuracy

0% (0.1b remaining)

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

bsearch1.0ms

end0.0ms

sample5.6s

Algorithm
intervals