Average Error: 0.1 → 0.1
Time: 15.2s
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 2019010 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))

Details

Time bar (total: 14.2s)Debug log

sample388.0ms

Algorithm
intervals

simplify81.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
80.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize22.0ms

Local error

Found 3 expressions with local error:

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

rewrite15.0ms

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
6.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))
5.0ms
(* e (sin v))
2.0ms
(* e (cos v))

series177.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
81.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))
65.0ms
(* e (sin v))
31.0ms
(* e (cos v))

simplify944.0ms

Counts
28 → 52
Calls
28 calls:
Slowest
340.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
285.0ms
(+ (* 1 1) (- (* (* e (cos v)) (* e (cos v))) (* 1 (* e (cos v)))))
102.0ms
(/ (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))) (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))))
57.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))
55.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))

prune506.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 4 expressions with local error:

0.1b
(* (* e (cos v)) (* e (cos v)))
0.1b
(* e (sin v))
0.1b
(pow (* e (cos v)) 3)
0.1b
(* e (cos v))

rewrite21.0ms

Algorithm
rewrite-expression-head
Rules
10×pow1
7×add-exp-log
7×associate-*r*
7×add-cbrt-cube
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
4×add-log-exp
3×*-commutative
1×rem-cube-cbrt
1×cbrt-unprod
1×pow-prod-up
1×prod-exp
1×pow-exp
1×pow-prod-down
1×associate-*l*
1×cube-prod
1×unpow3
1×pow-plus
1×pow-pow
1×unpow-prod-down
1×pow-to-exp
1×cube-mult
1×pow2
Counts
4 → 53
Calls
4 calls:
Slowest
13.0ms
(* (* e (cos v)) (* e (cos v)))
2.0ms
(pow (* e (cos v)) 3)
2.0ms
(* e (sin v))
2.0ms
(* e (cos v))

series203.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
58.0ms
(pow (* e (cos v)) 3)
58.0ms
(* (* e (cos v)) (* e (cos v)))
48.0ms
(* e (cos v))
38.0ms
(* e (sin v))

simplify752.0ms

Counts
30 → 65
Calls
30 calls:
Slowest
342.0ms
(- (pow e 3) (* 3/2 (* (pow e 3) (pow v 2))))
187.0ms
(* (* (* (* e (cos v)) (* e (cos v))) (* e (cos v))) (* (* (* e (cos v)) (* e (cos v))) (* e (cos v))))
55.0ms
(- (pow e 2) (* (pow e 2) (pow v 2)))
42.0ms
(* (pow e 3) (pow (cos v) 3))
23.0ms
(* (pow e 3) (pow (cos v) 3))

prune864.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 3 expressions with local error:

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

rewrite13.0ms

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
9.0ms
(/ (+ 1 (* e (cos v))) (sin v))
2.0ms
(* e (cos v))
1.0ms
(/ e (/ (+ 1 (* e (cos v))) (sin v)))

series193.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
88.0ms
(/ e (/ (+ 1 (* e (cos v))) (sin v)))
60.0ms
(/ (+ 1 (* e (cos v))) (sin v))
45.0ms
(* e (cos v))

simplify1.1s

Counts
53 → 73
Calls
53 calls:
Slowest
365.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
223.0ms
(/ (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))) (* (* (sin v) (sin v)) (sin v)))
70.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))
70.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))
44.0ms
(- (* 1 1) (* (* e (cos v)) (* e (cos v))))

prune695.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

0.7b
(/ 1 (/ (+ 1 (* e (cos v))) (* e (sin v))))
0.2b
(/ (+ 1 (* e (cos v))) (* e (sin v)))
0.1b
(* e (sin v))
0.1b
(* e (cos v))

rewrite22.0ms

Algorithm
rewrite-expression-head
Rules
10×add-cube-cbrt
10×*-un-lft-identity
10×add-sqr-sqrt
9×add-exp-log
8×associate-/r*
6×associate-*r*
6×add-cbrt-cube
6×times-frac
5×pow1
4×add-log-exp
3×associate-/l*
3×div-inv
2×associate-/l/
2×*-commutative
2×div-exp
2×frac-2neg
2×clear-num
2×rec-exp
1×inv-pow
1×pow-flip
1×flip-+
1×associate-/r/
1×flip3-+
1×cbrt-undiv
Counts
4 → 65
Calls
4 calls:
Slowest
13.0ms
(/ (+ 1 (* e (cos v))) (* e (sin v)))
3.0ms
(/ 1 (/ (+ 1 (* e (cos v))) (* e (sin v))))
2.0ms
(* e (sin v))
2.0ms
(* e (cos v))

series310.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
129.0ms
(/ (+ 1 (* e (cos v))) (* e (sin v)))
90.0ms
(/ 1 (/ (+ 1 (* e (cos v))) (* e (sin v))))
54.0ms
(* e (sin v))
37.0ms
(* e (cos v))

simplify1.0s

Counts
40 → 77
Calls
40 calls:
Slowest
372.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
151.0ms
(/ (* (* (+ 1 (* e (cos v))) (+ 1 (* e (cos v)))) (+ 1 (* e (cos v)))) (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))))
80.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))
78.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))
73.0ms
(+ (pow 1 3) (pow (* e (cos v)) 3))

prune669.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

regimes135.0ms

Accuracy

0% (0.1b remaining)

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

bsearch4.0ms

end0.0ms

sample6.1s

Algorithm
intervals