Average Error: 0.1 → 0.1
Time: 19.8s
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 2019004 +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

sample303.0ms

Algorithm
intervals

simplify78.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune21.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize19.0ms

Local error

Found 2 expressions with local error:

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

rewrite8.0ms

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))

series184.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

105.0ms
(/ (* e (sin v)) (fma (cos v) e 1))
78.0ms
(* e (sin v))

simplify1.6s

Counts
20 → 40
Calls

20 calls. Slowest were:

807.0ms
(/ (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))) (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)))
473.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
126.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune438.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.0b

localize5.0ms

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))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 55
Calls

2 calls. Slowest were:

3.0ms
(/ (fma (cos v) e 1) (sin v))
1.0ms
(/ e (/ (fma (cos v) e 1) (sin v)))

series182.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

92.0ms
(/ (fma (cos v) e 1) (sin v))
90.0ms
(/ e (/ (fma (cos v) e 1) (sin v)))

simplify1.0s

Counts
45 → 61
Calls

45 calls. Slowest were:

434.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
128.0ms
(/ (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)) (* (* (sin v) (sin v)) (sin v)))
73.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune577.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.0b

localize24.0ms

Local error

Found 4 expressions with local error:

0.2b
(/ e (fma (/ e (sin v)) (cos v) (/ 1 (sin v))))
0.1b
(/ e (sin v))
0.1b
(/ 1 (sin v))
0.0b
(fma (/ e (sin v)) (cos v) (/ 1 (sin v)))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 58
Calls

4 calls. Slowest were:

2.0ms
(/ 1 (sin v))
1.0ms
(/ e (fma (/ e (sin v)) (cos v) (/ 1 (sin v))))
1.0ms
(/ e (sin v))

series352.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

130.0ms
(fma (/ e (sin v)) (cos v) (/ 1 (sin v)))
125.0ms
(/ e (fma (/ e (sin v)) (cos v) (/ 1 (sin v))))
73.0ms
(/ e (sin v))
23.0ms
(/ 1 (sin v))

simplify930.0ms

Counts
22 → 70
Calls

22 calls. Slowest were:

555.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
83.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))
79.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune680.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

0.5b
(cbrt (/ (fma (cos v) e 1) (sin v)))
0.5b
(cbrt (/ (fma (cos v) e 1) (sin v)))
0.5b
(cbrt (/ (fma (cos v) e 1) (sin v)))
0.5b
(* (cbrt (/ (fma (cos v) e 1) (sin v))) (cbrt (/ (fma (cos v) e 1) (sin v))))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 125
Calls

4 calls. Slowest were:

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

series452.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

118.0ms
(cbrt (/ (fma (cos v) e 1) (sin v)))
116.0ms
(cbrt (/ (fma (cos v) e 1) (sin v)))
114.0ms
(cbrt (/ (fma (cos v) e 1) (sin v)))
104.0ms
(* (cbrt (/ (fma (cos v) e 1) (sin v))) (cbrt (/ (fma (cos v) e 1) (sin v))))

simplify3.7s

Counts
98 → 137
Calls

98 calls. Slowest were:

358.0ms
(- (+ (* 2/3 (/ (exp (* 1/3 (- (log (/ (pow (cos v) 2) (pow (sin v) 2))) (* 2 (log (/ -1 e)))))) (* e (cos v)))) (exp (* 1/3 (- (log (/ (pow (cos v) 2) (pow (sin v) 2))) (* 2 (log (/ -1 e))))))) (* 1/9 (/ (exp (* 1/3 (- (log (/ (pow (cos v) 2) (pow (sin v) 2))) (* 2 (log (/ -1 e)))))) (* (pow e 2) (pow (cos v) 2)))))
340.0ms
(* (/ (fma (cos v) e 1) (sin v)) (/ (fma (cos v) e 1) (sin v)))
303.0ms
(* (/ (fma (cos v) e 1) (sin v)) (/ (fma (cos v) e 1) (sin v)))

prune1.8s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0b

regimes177.0ms

Accuracy

0% (0.1b remaining)

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

bsearch3.0ms

end0.0ms

sample6.1s

Algorithm
intervals