Average Error: 0.1 → 0.1
Time: 21.1s
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 2019007 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))

Details

Time bar (total: 19.8s)Debug log

sample194.0ms

Algorithm
intervals

simplify52.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

51.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))

prune19.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize31.0ms

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))

series176.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

116.0ms
(/ (* e (sin v)) (fma (cos v) e 1))
60.0ms
(* e (sin v))

simplify1.6s

Counts
20 → 40
Calls

20 calls. Slowest were:

860.0ms
(/ (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))) (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)))
501.0ms
(- (+ (* (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))))

prune394.0ms

Pruning

7 alts after pruning (6 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:

7.0ms
(/ (fma (cos v) e 1) (sin v))
2.0ms
(/ e (/ (fma (cos v) e 1) (sin v)))

series185.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

110.0ms
(/ e (/ (fma (cos v) e 1) (sin v)))
75.0ms
(/ (fma (cos v) e 1) (sin v))

simplify1.3s

Counts
45 → 61
Calls

45 calls. Slowest were:

528.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
171.0ms
(/ (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)) (* (* (sin v) (sin v)) (sin v)))
80.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune678.0ms

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 3 expressions with local error:

0.3b
(/ 1 (/ (fma (cos v) e 1) (sin v)))
0.1b
(/ (fma (cos v) e 1) (sin v))
0.1b
(* e (/ 1 (/ (fma (cos v) e 1) (sin v))))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
3 → 76
Calls

3 calls. Slowest were:

4.0ms
(/ 1 (/ (fma (cos v) e 1) (sin v)))
3.0ms
(/ (fma (cos v) e 1) (sin v))
1.0ms
(* e (/ 1 (/ (fma (cos v) e 1) (sin v))))

series205.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

84.0ms
(/ (fma (cos v) e 1) (sin v))
80.0ms
(* e (/ 1 (/ (fma (cos v) e 1) (sin v))))
41.0ms
(/ 1 (/ (fma (cos v) e 1) (sin v)))

simplify2.0s

Counts
56 → 85
Calls

56 calls. Slowest were:

520.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
253.0ms
(- v (+ (* 1/6 (pow v 3)) (* e v)))
164.0ms
(- (+ (/ (sin v) (* (pow e 3) (pow (cos v) 3))) (/ (sin v) (* e (cos v)))) (/ (sin v) (* (pow e 2) (pow (cos v) 2))))

prune818.0ms

Pruning

11 alts after pruning (8 fresh and 3 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 4 expressions with local error:

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

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 125
Calls

4 calls. Slowest were:

13.0ms
(* (cbrt (/ (fma (cos v) e 1) (sin v))) (cbrt (/ (fma (cos v) e 1) (sin v))))
2.0ms
(cbrt (/ (fma (cos v) e 1) (sin v)))
2.0ms
(cbrt (/ (fma (cos v) e 1) (sin v)))

series379.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

114.0ms
(* (cbrt (/ (fma (cos v) e 1) (sin v))) (cbrt (/ (fma (cos v) e 1) (sin v))))
93.0ms
(cbrt (/ (fma (cos v) e 1) (sin v)))
90.0ms
(cbrt (/ (fma (cos v) e 1) (sin v)))
81.0ms
(cbrt (/ (fma (cos v) e 1) (sin v)))

simplify3.8s

Counts
98 → 137
Calls

98 calls. Slowest were:

338.0ms
(* (/ (fma (cos v) e 1) (sin v)) (/ (fma (cos v) e 1) (sin v)))
299.0ms
(* (/ (fma (cos v) e 1) (sin v)) (/ (fma (cos v) e 1) (sin v)))
281.0ms
(- (+ (exp (* 1/3 (- (log (/ (pow (cos v) 2) (pow (sin v) 2))) (* 2 (log (/ 1 e)))))) (* 2/3 (/ (exp (* 1/3 (- (log (/ (pow (cos v) 2) (pow (sin v) 2))) (* 2 (log (/ 1 e)))))) (* e (cos v))))) (* 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)))))

prune1.9s

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 0.0b

regimes204.0ms

Accuracy

0% (0.1b remaining)

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

bsearch2.0ms

end0.0ms

sample5.7s

Algorithm
intervals