Average Error: 0.1 → 0.1
Time: 16.3s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{\sin v}{1 + \cos v \cdot e} \cdot e\]

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 *-un-lft-identity0.1

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

  4. Applied times-frac0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019008 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))

Details

Time bar (total: 15.5s)Debug log

sample249.0ms

Algorithm
intervals

simplify36.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

35.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))

prune12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize71.0ms

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))

series221.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

112.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))
66.0ms
(* e (sin v))
42.0ms
(* e (cos v))

simplify1.1s

Counts
28 → 52
Calls

28 calls. Slowest were:

418.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
270.0ms
(+ (* 1 1) (- (* (* e (cos v)) (* e (cos v))) (* 1 (* e (cos v)))))
160.0ms
(/ (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))) (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))))

prune412.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 3 expressions with local error:

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

series203.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

106.0ms
(* e (/ (sin v) (+ 1 (* e (cos v)))))
64.0ms
(/ (sin v) (+ 1 (* e (cos v))))
33.0ms
(* e (cos v))

simplify1.9s

Counts
55 → 73
Calls

55 calls. Slowest were:

380.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
291.0ms
(/ (* (* (sin v) (sin v)) (sin v)) (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))))
279.0ms
(+ (* 1 1) (- (* (* e (cos v)) (* e (cos v))) (* 1 (* e (cos v)))))

prune654.0ms

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))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
3 → 64
Calls

3 calls. Slowest were:

8.0ms
(/ (+ 1 (* e (cos v))) (sin v))
1.0ms
(* e (cos v))
1.0ms
(/ e (/ (+ 1 (* e (cos v))) (sin v)))

series191.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

85.0ms
(/ (+ 1 (* e (cos v))) (sin v))
77.0ms
(/ e (/ (+ 1 (* e (cos v))) (sin v)))
28.0ms
(* e (cos v))

simplify1.3s

Counts
53 → 73
Calls

53 calls. Slowest were:

425.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
299.0ms
(/ (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))) (* (* (sin v) (sin v)) (sin v)))
90.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune788.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (sin v))
0.6b
(cbrt (sin v))
0.6b
(cbrt (sin v))
0.5b
(* (cbrt (sin v)) (cbrt (sin v)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

4.0ms
(* (cbrt (sin v)) (cbrt (sin v)))
1.0ms
(cbrt (sin v))
1.0ms
(cbrt (sin v))

series644.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

183.0ms
(* (cbrt (sin v)) (cbrt (sin v)))
166.0ms
(cbrt (sin v))
151.0ms
(cbrt (sin v))
144.0ms
(cbrt (sin v))

simplify316.0ms

Counts
41 → 73
Calls

41 calls. Slowest were:

68.0ms
(- (pow v 1/3) (+ (* 1/18 (pow (pow v 7) 1/3)) (* 1/3240 (pow (pow v 13) 1/3))))
67.0ms
(- (pow v 1/3) (+ (* 1/18 (pow (pow v 7) 1/3)) (* 1/3240 (pow (pow v 13) 1/3))))
63.0ms
(- (pow v 1/3) (+ (* 1/18 (pow (pow v 7) 1/3)) (* 1/3240 (pow (pow v 13) 1/3))))

prune1.0s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

regimes129.0ms

Accuracy

0% (0.1b remaining)

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

bsearch3.0ms

end0.0ms

sample6.2s

Algorithm
intervals