Average Error: 0.1 → 0.2
Time: 58.7s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e}{\sqrt{(\left(\cos v\right) \cdot e + 1)_*}} \cdot \frac{\sin v}{\sqrt{(\left(\cos v\right) \cdot e + 1)_*}}\]

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. Initial simplification0.1

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied times-frac0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))

Details

Time bar (total: 49.4s)Debug log

start506.0ms

Algorithm
intervals

setup94.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize39.0ms

Local error

Found 2 expressions with local error:

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

rewrite14.0ms

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))

series152.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

93.0ms
(/ (* e (sin v)) (fma (cos v) e 1))
58.0ms
(* e (sin v))

simplify1.8s

Counts
20 → 40
Calls

20 calls. Slowest were:

836.0ms
(/ (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))) (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)))
581.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
114.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune473.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 4 expressions with local error:

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

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
4 → 113
Calls

4 calls. Slowest were:

12.0ms
(* (/ e (sqrt (fma (cos v) e 1))) (/ (sin v) (sqrt (fma (cos v) e 1))))
7.0ms
(/ (sin v) (sqrt (fma (cos v) e 1)))
2.0ms
(sqrt (fma (cos v) e 1))

series312.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

108.0ms
(/ (sin v) (sqrt (fma (cos v) e 1)))
93.0ms
(* (/ e (sqrt (fma (cos v) e 1))) (/ (sin v) (sqrt (fma (cos v) e 1))))
56.0ms
(sqrt (fma (cos v) e 1))
55.0ms
(sqrt (fma (cos v) e 1))

simplify3.4s

Counts
99 → 125
Calls

99 calls. Slowest were:

554.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
480.0ms
(* (* (* (/ e (sqrt (fma (cos v) e 1))) (/ e (sqrt (fma (cos v) e 1)))) (/ e (sqrt (fma (cos v) e 1)))) (* (* (/ (sin v) (sqrt (fma (cos v) e 1))) (/ (sin v) (sqrt (fma (cos v) e 1)))) (/ (sin v) (sqrt (fma (cos v) e 1)))))
374.0ms
(- v (+ (* 1/6 (pow v 3)) (* 1/2 (* e v))))

prune1.5s

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))

rewrite9.0ms

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)))

series156.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

100.0ms
(/ e (/ (fma (cos v) e 1) (sin v)))
56.0ms
(/ (fma (cos v) e 1) (sin v))

simplify1.1s

Counts
45 → 61
Calls

45 calls. Slowest were:

471.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
108.0ms
(/ (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)) (* (* (sin v) (sin v)) (sin v)))
100.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune746.0ms

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 0b

localize36.0ms

Local error

Found 4 expressions with local error:

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

rewrite35.0ms

Algorithm
rewrite-expression-head
Counts
4 → 243
Calls

4 calls. Slowest were:

10.0ms
(* (cbrt (fma (cos v) e 1)) (cbrt (fma (cos v) e 1)))
10.0ms
(/ (/ e (/ (* (cbrt (fma (cos v) e 1)) (cbrt (fma (cos v) e 1))) 1)) (/ (cbrt (fma (cos v) e 1)) (sin v)))
7.0ms
(/ (cbrt (fma (cos v) e 1)) (sin v))

series417.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

119.0ms
(/ (cbrt (fma (cos v) e 1)) (sin v))
106.0ms
(* (cbrt (fma (cos v) e 1)) (cbrt (fma (cos v) e 1)))
96.0ms
(/ (/ e (/ (* (cbrt (fma (cos v) e 1)) (cbrt (fma (cos v) e 1))) 1)) (/ (cbrt (fma (cos v) e 1)) (sin v)))
96.0ms
(/ e (/ (* (cbrt (fma (cos v) e 1)) (cbrt (fma (cos v) e 1))) 1))

simplify34.9s

Counts
332 → 255
Calls

332 calls. Slowest were:

755.0ms
(/ (/ e (* (cbrt (fma (cos v) e 1)) (cbrt (fma (cos v) e 1)))) (/ (* (cbrt (cbrt (fma (cos v) e 1))) (cbrt (cbrt (fma (cos v) e 1)))) 1))
650.0ms
(/ (/ 1 (/ (* (cbrt (fma (cos v) e 1)) (cbrt (fma (cos v) e 1))) 1)) (/ (cbrt (fma (cos v) e 1)) (cbrt (sin v))))
571.0ms
(/ (/ 1 (/ (* (cbrt (fma (cos v) e 1)) (cbrt (fma (cos v) e 1))) 1)) (/ (cbrt (cbrt (fma (cos v) e 1))) (sin v)))

prune3.5s

Pruning

7 alts after pruning (4 fresh and 3 done)

Merged error: 0b

regimes106.0ms

Accuracy

0% (0.1b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 0.2b

bsearch1.0ms