Average Error: 0.1 → 0.1
Time: 14.2s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\left(\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right) + \left(1 - \cos v \cdot e\right)\right) \cdot \frac{e \cdot \sin v}{1 + {\left(\cos v \cdot e\right)}^{3}}\]

Error

Bits error versus e

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
  2. Using strategy rm

  3. Applied flip3-+0.1

    \[\leadsto \frac{e \cdot \sin v}{\color{blue}{\frac{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}{1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)}}}\]

  4. Applied associate-/r/0.1

    \[\leadsto \color{blue}{\frac{e \cdot \sin v}{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)\right)}\]

  5. Simplified0.1

    \[\leadsto \frac{e \cdot \sin v}{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}} \cdot \color{blue}{\left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) + \left(1 - e \cdot \cos v\right)\right)}\]

  6. Final simplification0.1

    \[\leadsto \left(\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right) + \left(1 - \cos v \cdot e\right)\right) \cdot \frac{e \cdot \sin v}{1 + {\left(\cos v \cdot e\right)}^{3}}\]

Reproduce

herbie shell --seed 2019016 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))

Details

Time bar (total: 13.4s)Debug log

sample319.0ms

Algorithm
intervals
Results
256×(pre true 80)
134×(body real 80)
60×(body real 1280)
43×(body real 640)
11×(body real 320)
8×(body real 160)

simplify41.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
40.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize41.0ms

Local error

Found 3 expressions with local error:

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

rewrite18.0ms

Algorithm
rewrite-expression-head
Rules
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×associate-*r*
5×add-exp-log
5×add-cbrt-cube
3×add-log-exp
3×times-frac
3×associate-/r*
3×pow1
2×*-commutative
2×associate-/r/
1×associate-/l*
1×div-inv
1×flip-+
1×div-exp
1×flip3-+
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
3 → 43
Calls
3 calls:
Slowest
9.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))
5.0ms
(* e (cos v))
2.0ms
(* e (sin v))

series183.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
85.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))
54.0ms
(* e (cos v))
43.0ms
(* e (sin v))

simplify912.0ms

Counts
28 → 52
Calls
28 calls:
Slowest
347.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
249.0ms
(+ (* 1 1) (- (* (* e (cos v)) (* e (cos v))) (* 1 (* e (cos v)))))
90.0ms
(/ (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))) (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))))
78.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))
67.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune561.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize27.0ms

Local error

Found 4 expressions with local error:

0.1b
(* e (cos v))
0.1b
(* e (cos v))
0.1b
(* e (cos v))
0.1b
(* e (cos v))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
12×associate-*r*
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
4×add-log-exp
4×*-commutative
4×add-exp-log
4×add-cbrt-cube
4×pow1
Counts
4 → 44
Calls
4 calls:
Slowest
2.0ms
(* e (cos v))
1.0ms
(* e (cos v))
1.0ms
(* e (cos v))
1.0ms
(* e (cos v))

series177.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
52.0ms
(* e (cos v))
49.0ms
(* e (cos v))
47.0ms
(* e (cos v))
29.0ms
(* e (cos v))

simplify44.0ms

Counts
24 → 56
Calls
24 calls:
Slowest
3.0ms
(* e (cos v))
3.0ms
(* e (cos v))
3.0ms
(* e (cos v))
3.0ms
(* e (cos v))
3.0ms
(* e (cos v))

prune865.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize5.0ms

Local error

Found 3 expressions with local error:

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

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
19×add-cube-cbrt
19×*-un-lft-identity
19×add-sqr-sqrt
18×times-frac
16×associate-/r*
5×add-exp-log
5×add-cbrt-cube
3×add-log-exp
3×associate-/l*
3×div-inv
3×associate-*r*
3×pow1
2×associate-/l/
2×frac-2neg
2×clear-num
1×flip-+
1×*-commutative
1×associate-/r/
1×div-exp
1×flip3-+
1×cbrt-undiv
Counts
3 → 64
Calls
3 calls:
Slowest
6.0ms
(/ (+ 1 (* e (cos v))) (sin v))
1.0ms
(* e (cos v))
1.0ms
(/ e (/ (+ 1 (* e (cos v))) (sin v)))

series151.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
65.0ms
(/ e (/ (+ 1 (* e (cos v))) (sin v)))
56.0ms
(/ (+ 1 (* e (cos v))) (sin v))
29.0ms
(* e (cos v))

simplify1.6s

Counts
53 → 73
Calls
53 calls:
Slowest
386.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
354.0ms
(/ (* (cbrt (+ 1 (* e (cos v)))) (cbrt (+ 1 (* e (cos v))))) (sqrt (sin v)))
249.0ms
(/ (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))) (* (* (sin v) (sin v)) (sin v)))
109.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))
77.0ms
(+ (pow 1 3) (pow (* e (cos v)) 3))

prune651.0ms

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (sin v))
0.6b
(cbrt (sin v))
0.6b
(cbrt (sin v))
0.4b
(* (cbrt (sin v)) (cbrt (sin v)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
15×cbrt-prod
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
9×pow1
7×pow1/3
6×add-exp-log
6×associate-*l*
6×associate-*r*
4×add-log-exp
4×add-cbrt-cube
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 61
Calls
4 calls:
Slowest
5.0ms
(* (cbrt (sin v)) (cbrt (sin v)))
1.0ms
(cbrt (sin v))
0.0ms
(cbrt (sin v))
0.0ms
(cbrt (sin v))

series727.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
193.0ms
(cbrt (sin v))
191.0ms
(cbrt (sin v))
172.0ms
(cbrt (sin v))
171.0ms
(* (cbrt (sin v)) (cbrt (sin v)))

simplify365.0ms

Counts
41 → 73
Calls
41 calls:
Slowest
85.0ms
(- (pow v 1/3) (+ (* 1/18 (pow (pow v 7) 1/3)) (* 1/3240 (pow (pow v 13) 1/3))))
76.0ms
(- (pow v 1/3) (+ (* 1/18 (pow (pow v 7) 1/3)) (* 1/3240 (pow (pow v 13) 1/3))))
70.0ms
(- (pow v 1/3) (+ (* 1/18 (pow (pow v 7) 1/3)) (* 1/3240 (pow (pow v 13) 1/3))))
61.0ms
(- (+ (pow v 2/3) (* 1/405 (pow (pow v 14) 1/3))) (* 1/9 (pow (pow v 8) 1/3)))
4.0ms
(pow (pow (sin v) 2) 1/3)

prune855.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

regimes191.0ms

Accuracy

0% (0.1b remaining)

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

bsearch3.0ms

end0.0ms

sample5.7s

Algorithm
intervals
Results
8000×(pre true 80)
4322×(body real 80)
1530×(body real 1280)
1225×(body real 640)
642×(body real 320)
281×(body real 160)