Average Error: 0.1 → 0.1
Time: 15.7s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \frac{\sin v}{(\left(\cos v\right) \cdot e + 1)_*}\]

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. Using strategy rm

  4. Applied *-un-lft-identity0.1

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

  5. Applied times-frac0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

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

Details

Time bar (total: 14.8s)Debug log

sample368.0ms

Algorithm
intervals

simplify80.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

79.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))

prune23.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize33.0ms

Local error

Found 2 expressions with local error:

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

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
2 → 34
Calls

2 calls. Slowest were:

7.0ms
(/ (* e (sin v)) (fma (cos v) e 1))
5.0ms
(* e (sin v))

series161.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

84.0ms
(/ (* e (sin v)) (fma (cos v) e 1))
77.0ms
(* e (sin v))

simplify1.5s

Counts
20 → 40
Calls

20 calls. Slowest were:

784.0ms
(/ (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))) (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)))
432.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
110.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune418.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 2 expressions with local error:

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

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
2 → 53
Calls

2 calls. Slowest were:

3.0ms
(/ (sin v) (fma (cos v) e 1))
2.0ms
(* e (/ (sin v) (fma (cos v) e 1)))

series137.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

73.0ms
(* e (/ (sin v) (fma (cos v) e 1)))
63.0ms
(/ (sin v) (fma (cos v) e 1))

simplify1.9s

Counts
45 → 59
Calls

45 calls. Slowest were:

474.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
467.0ms
(/ (* (* (sin v) (sin v)) (sin v)) (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)))
223.0ms
(- v (+ (* 1/6 (pow v 3)) (* e v)))

prune653.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.0b

localize6.0ms

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))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
2 → 55
Calls

2 calls. Slowest were:

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

series145.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

83.0ms
(/ e (/ (fma (cos v) e 1) (sin v)))
62.0ms
(/ (fma (cos v) e 1) (sin v))

simplify1.2s

Counts
45 → 61
Calls

45 calls. Slowest were:

516.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
120.0ms
(/ (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)) (* (* (sin v) (sin v)) (sin v)))
101.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune621.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 3 expressions with local error:

0.6b
(/ 1 (/ (fma (cos v) e 1) (* e (sin v))))
0.2b
(/ (fma (cos v) e 1) (* e (sin v)))
0.1b
(* e (sin v))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
3 → 58
Calls

3 calls. Slowest were:

7.0ms
(/ (fma (cos v) e 1) (* e (sin v)))
4.0ms
(/ 1 (/ (fma (cos v) e 1) (* e (sin v))))
2.0ms
(* e (sin v))

series187.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

92.0ms
(/ 1 (/ (fma (cos v) e 1) (* e (sin v))))
60.0ms
(/ (fma (cos v) e 1) (* e (sin v)))
35.0ms
(* e (sin v))

simplify916.0ms

Counts
32 → 67
Calls

32 calls. Slowest were:

423.0ms
(- (* e v) (+ (* 1/6 (* e (pow v 3))) (* (pow e 2) v)))
108.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))
74.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune630.0ms

Pruning

7 alts after pruning (4 fresh and 3 done)

Merged error: 0.0b

regimes120.0ms

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