Average Error: 0.1 → 0.1
Time: 21.3s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\left(e \cdot \sin v\right) \cdot \frac{1}{(\left(\cos v\right) \cdot e + 1)_*}\]

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 *-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. Using strategy rm

  8. Applied div-inv0.1

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

  9. Applied associate-*r*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2018362 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))

Details

Time bar (total: 11.5s)Debug log

start354.0ms

Algorithm
intervals

setup57.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize420.0ms

Local error

Found 2 expressions with local error:

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

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
2 → 34
Calls

2 calls. Slowest were:

8.0ms
(/ (* e (sin v)) (fma (cos v) e 1))
5.0ms
(* e (sin v))

series206.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

132.0ms
(/ (* e (sin v)) (fma (cos v) e 1))
73.0ms
(* e (sin v))

simplify1.7s

Counts
20 → 40
Calls

20 calls. Slowest were:

833.0ms
(/ (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))) (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)))
575.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
99.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune426.0ms

Pruning

5 alts after pruning (5 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))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
2 → 53
Calls

2 calls. Slowest were:

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

series126.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

75.0ms
(* e (/ (sin v) (fma (cos v) e 1)))
51.0ms
(/ (sin v) (fma (cos v) e 1))

simplify2.3s

Counts
45 → 59
Calls

45 calls. Slowest were:

587.0ms
(/ (* (* (sin v) (sin v)) (sin v)) (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)))
564.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
266.0ms
(- v (+ (* 1/6 (pow v 3)) (* e v)))

prune689.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 3 expressions with local error:

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

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
3 → 52
Calls

3 calls. Slowest were:

10.0ms
(* (* e (sin v)) (/ 1 (fma (cos v) e 1)))
2.0ms
(* e (sin v))
2.0ms
(/ 1 (fma (cos v) e 1))

series148.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

87.0ms
(* (* e (sin v)) (/ 1 (fma (cos v) e 1)))
34.0ms
(* e (sin v))
27.0ms
(/ 1 (fma (cos v) e 1))

simplify1.8s

Counts
26 → 61
Calls

26 calls. Slowest were:

575.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
477.0ms
(* (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))) (* (* (/ 1 (fma (cos v) e 1)) (/ 1 (fma (cos v) e 1))) (/ 1 (fma (cos v) e 1))))
146.0ms
(- (+ (/ 1 (* e (cos v))) (/ 1 (* (pow e 3) (pow (cos v) 3)))) (/ 1 (* (pow e 2) (pow (cos v) 2))))

prune809.0ms

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))

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)))

series160.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

80.0ms
(/ e (/ (fma (cos v) e 1) (sin v)))
79.0ms
(/ (fma (cos v) e 1) (sin v))

simplify1.4s

Counts
45 → 61
Calls

45 calls. Slowest were:

635.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
165.0ms
(/ (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)) (* (* (sin v) (sin v)) (sin v)))
83.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune711.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0b

regimes162.0ms

Accuracy

0% (0.1b remaining)

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

bsearch2.0ms