Average Error: 0.1 → 0.1
Time: 20.0s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{(\left(\cos v\right) \cdot e + 1)_*}\]

Error

Bits error versus e

Bits error versus v

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))

Details

Time bar (total: 18.7s)Debug log

sample218.0ms

Algorithm
intervals

simplify23.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

22.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))

prune32.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize21.0ms

Local error

Found 2 expressions with local error:

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

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
2 → 34
Calls

2 calls. Slowest were:

4.0ms
(/ (* e (sin v)) (fma (cos v) e 1))
3.0ms
(* e (sin v))

series202.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

132.0ms
(/ (* e (sin v)) (fma (cos v) e 1))
70.0ms
(* e (sin v))

simplify1.7s

Counts
20 → 40
Calls

20 calls. Slowest were:

895.0ms
(/ (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))) (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)))
500.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
101.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune476.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 2 expressions with local error:

0.2b
(/ e (/ (fma (cos v) e 1) (sin v)))
0.1b
(/ (fma (cos v) e 1) (sin v))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
2 → 55
Calls

2 calls. Slowest were:

6.0ms
(/ (fma (cos v) e 1) (sin v))
2.0ms
(/ e (/ (fma (cos v) e 1) (sin v)))

series164.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

83.0ms
(/ e (/ (fma (cos v) e 1) (sin v)))
81.0ms
(/ (fma (cos v) e 1) (sin v))

simplify1.3s

Counts
45 → 61
Calls

45 calls. Slowest were:

503.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
160.0ms
(/ (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)) (* (* (sin v) (sin v)) (sin v)))
85.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune659.0ms

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 0.0b

localize17.0ms

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)))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

7.0ms
(* (cbrt (sin v)) (cbrt (sin v)))
1.0ms
(cbrt (sin v))
1.0ms
(cbrt (sin v))

series638.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

178.0ms
(cbrt (sin v))
159.0ms
(cbrt (sin v))
153.0ms
(* (cbrt (sin v)) (cbrt (sin v)))
148.0ms
(cbrt (sin v))

simplify449.0ms

Counts
41 → 81
Calls

41 calls. Slowest were:

104.0ms
(- (pow v 1/3) (+ (* 1/18 (pow (pow v 7) 1/3)) (* 1/3240 (pow (pow v 13) 1/3))))
94.0ms
(- (pow v 1/3) (+ (* 1/18 (pow (pow v 7) 1/3)) (* 1/3240 (pow (pow v 13) 1/3))))
93.0ms
(- (pow v 1/3) (+ (* 1/18 (pow (pow v 7) 1/3)) (* 1/3240 (pow (pow v 13) 1/3))))

prune1.1s

Pruning

10 alts after pruning (7 fresh and 3 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 4 expressions with local error:

15.2b
(cbrt (/ e (/ (fma (cos v) e 1) (sin v))))
15.2b
(cbrt (/ e (/ (fma (cos v) e 1) (sin v))))
15.2b
(cbrt (/ e (/ (fma (cos v) e 1) (sin v))))
0.4b
(* (cbrt (/ e (/ (fma (cos v) e 1) (sin v)))) (cbrt (/ e (/ (fma (cos v) e 1) (sin v)))))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
4 → 85
Calls

4 calls. Slowest were:

6.0ms
(* (cbrt (/ e (/ (fma (cos v) e 1) (sin v)))) (cbrt (/ e (/ (fma (cos v) e 1) (sin v)))))
1.0ms
(cbrt (/ e (/ (fma (cos v) e 1) (sin v))))
0.0ms
(cbrt (/ e (/ (fma (cos v) e 1) (sin v))))

series557.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

153.0ms
(cbrt (/ e (/ (fma (cos v) e 1) (sin v))))
150.0ms
(* (cbrt (/ e (/ (fma (cos v) e 1) (sin v)))) (cbrt (/ e (/ (fma (cos v) e 1) (sin v)))))
132.0ms
(cbrt (/ e (/ (fma (cos v) e 1) (sin v))))
122.0ms
(cbrt (/ e (/ (fma (cos v) e 1) (sin v))))

simplify4.0s

Counts
58 → 97
Calls

58 calls. Slowest were:

421.0ms
(* (/ e (/ (fma (cos v) e 1) (sin v))) (/ e (/ (fma (cos v) e 1) (sin v))))
357.0ms
(* (/ e (/ (fma (cos v) e 1) (sin v))) (/ e (/ (fma (cos v) e 1) (sin v))))
309.0ms
(- (+ (* (cbrt -1) (pow (/ (* (sin v) -1) (cos v)) 1/3)) (* 2/9 (* (/ (cbrt -1) (pow e 2)) (pow (/ (* (sin v) -1) (pow (cos v) 7)) 1/3)))) (* 1/3 (* (/ (cbrt -1) e) (pow (/ (* (sin v) -1) (pow (cos v) 4)) 1/3))))

prune1.3s

Pruning

10 alts after pruning (7 fresh and 3 done)

Merged error: 0.0b

regimes147.0ms

Accuracy

0% (0.1b remaining)

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

bsearch1.0ms

end0.0ms

sample5.8s

Algorithm
intervals