Average Error: 0.1 → 0.1
Time: 5.5m
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{(\left(\cos v\right) \cdot e + 1)_*}\]

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))

Details

Time bar (total: 5.5m)Debug log

sample194.0ms

Algorithm
intervals
Results
55.0ms139×body80valid
42.0ms47×body1280valid
25.0ms40×body640valid
9.0ms20×body320valid
8.0ms256×pre80true
3.0ms10×body160valid

simplify22.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
21.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))

prune32.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize20.0ms

Local error

Found 2 expressions with local error:

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

rewrite12.0ms

Algorithm
rewrite-expression-head
Rules
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-exp-log
4×add-cbrt-cube
3×associate-*r*
3×times-frac
3×associate-/r*
2×add-log-exp
2×log1p-expm1-u
2×pow1
2×expm1-log1p-u
1×associate-/l*
1×div-inv
1×*-commutative
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
2 → 34
Calls
2 calls:
Slowest
6.0ms
(/ (* e (sin v)) (fma (cos v) e 1))
3.0ms
(* e (sin v))

series541.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
481.0ms
(/ (* e (sin v)) (fma (cos v) e 1))
60.0ms
(* e (sin v))

simplify1.6s

Counts
20 → 40
Calls
20 calls:
Slowest
866.0ms
(/ (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))) (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)))
472.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
70.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))
66.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))
47.0ms
(- (log (* e (sin v))) (log (fma (cos v) e 1)))

prune404.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.1b

localize11.0ms

Local error

Found 2 expressions with local error:

0.3b
(/ e (/ (fma (cos v) e 1) (sin v)))
0.2b
(/ (fma (cos v) e 1) (sin v))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
18×times-frac
17×add-cube-cbrt
17×*-un-lft-identity
17×add-sqr-sqrt
16×associate-/r*
4×add-exp-log
4×add-cbrt-cube
3×associate-/l*
3×div-inv
2×add-log-exp
2×log1p-expm1-u
2×frac-2neg
2×clear-num
2×pow1
2×expm1-log1p-u
1×associate-/r/
1×div-exp
1×cbrt-undiv
Counts
2 → 55
Calls
2 calls:
Slowest
4.0ms
(/ (fma (cos v) e 1) (sin v))
2.0ms
(/ e (/ (fma (cos v) e 1) (sin v)))

series137.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
78.0ms
(/ e (/ (fma (cos v) e 1) (sin v)))
60.0ms
(/ (fma (cos v) e 1) (sin v))

simplify1.2s

Counts
45 → 61
Calls
45 calls:
Slowest
488.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
110.0ms
(/ (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)) (* (* (sin v) (sin v)) (sin v)))
75.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))
69.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))
28.0ms
(+ (/ (* e (cos v)) (sin v)) (/ 1 (sin v)))

prune691.0ms

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 0.0b

localize27.0ms

Local error

Found 4 expressions with local error:

0.3b
(/ (/ e (/ (sqrt (fma (cos v) e 1)) 1)) (/ (sqrt (fma (cos v) e 1)) (sin v)))
0.1b
(/ (sqrt (fma (cos v) e 1)) (sin v))
0.0b
(/ e (/ (sqrt (fma (cos v) e 1)) 1))
0.0b
(sqrt (fma (cos v) e 1))

rewrite31.0ms

Algorithm
rewrite-expression-head
Rules
236×times-frac
123×add-sqr-sqrt
122×add-cube-cbrt
122×*-un-lft-identity
69×sqrt-prod
33×div-inv
29×associate-/r*
25×associate-/r/
11×associate-/l*
11×add-exp-log
11×add-cbrt-cube
4×add-log-exp
4×log1p-expm1-u
4×div-exp
4×cbrt-undiv
4×pow1
4×expm1-log1p-u
3×frac-2neg
3×clear-num
1×associate-/l/
1×pow1/2
1×rem-sqrt-square
Counts
4 → 227
Calls
4 calls:
Slowest
10.0ms
(/ (/ e (/ (sqrt (fma (cos v) e 1)) 1)) (/ (sqrt (fma (cos v) e 1)) (sin v)))
7.0ms
(/ (sqrt (fma (cos v) e 1)) (sin v))
5.0ms
(sqrt (fma (cos v) e 1))
1.0ms
(/ e (/ (sqrt (fma (cos v) e 1)) 1))

series321.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
108.0ms
(/ (sqrt (fma (cos v) e 1)) (sin v))
94.0ms
(/ (/ e (/ (sqrt (fma (cos v) e 1)) 1)) (/ (sqrt (fma (cos v) e 1)) (sin v)))
61.0ms
(/ e (/ (sqrt (fma (cos v) e 1)) 1))
57.0ms
(sqrt (fma (cos v) e 1))

simplify17.5s

Counts
316 → 239
Calls
316 calls:
Slowest
475.0ms
(/ (/ e (sqrt (fma (cos v) e 1))) (/ 1 (* (cbrt (sin v)) (cbrt (sin v)))))
475.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
392.0ms
(/ (/ e (sqrt (fma (cos v) e 1))) (/ (sqrt 1) (* (cbrt (sin v)) (cbrt (sin v)))))
387.0ms
(/ (/ e (sqrt (fma (cos v) e 1))) (/ (* (cbrt (sqrt (fma (cos v) e 1))) (cbrt (sqrt (fma (cos v) e 1)))) (* (cbrt (sin v)) (cbrt (sin v)))))
367.0ms
(/ (* (cbrt (/ e (/ (sqrt (fma (cos v) e 1)) 1))) (cbrt (/ e (/ (sqrt (fma (cos v) e 1)) 1)))) (/ (* (cbrt (sqrt (fma (cos v) e 1))) (cbrt (sqrt (fma (cos v) e 1)))) (* (cbrt (sin v)) (cbrt (sin v)))))

prune2.8s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 4 expressions with local error:

0.3b
(/ (/ (/ e (/ (sqrt (fma (cos v) e 1)) 1)) (/ (sqrt (sqrt (fma (cos v) e 1))) 1)) (/ (sqrt (sqrt (fma (cos v) e 1))) (sin v)))
0.2b
(/ (sqrt (sqrt (fma (cos v) e 1))) (sin v))
0.0b
(/ (/ e (/ (sqrt (fma (cos v) e 1)) 1)) (/ (sqrt (sqrt (fma (cos v) e 1))) 1))
0.0b
(sqrt (sqrt (fma (cos v) e 1)))

rewrite78.0ms

Algorithm
rewrite-expression-head
Rules
2164×times-frac
855×add-sqr-sqrt
853×add-cube-cbrt
853×*-un-lft-identity
748×sqrt-prod
360×div-inv
167×associate-/r/
39×associate-/l*
38×associate-/r*
20×add-exp-log
20×add-cbrt-cube
10×div-exp
10×cbrt-undiv
4×add-log-exp
4×log1p-expm1-u
4×pow1
4×expm1-log1p-u
3×frac-2neg
3×clear-num
2×associate-/l/
2×rem-sqrt-square
1×pow1/2
Counts
4 → 969
Calls
4 calls:
Slowest
16.0ms
(/ (/ (/ e (/ (sqrt (fma (cos v) e 1)) 1)) (/ (sqrt (sqrt (fma (cos v) e 1))) 1)) (/ (sqrt (sqrt (fma (cos v) e 1))) (sin v)))
6.0ms
(/ (sqrt (sqrt (fma (cos v) e 1))) (sin v))
4.0ms
(/ (/ e (/ (sqrt (fma (cos v) e 1)) 1)) (/ (sqrt (sqrt (fma (cos v) e 1))) 1))
2.0ms
(sqrt (sqrt (fma (cos v) e 1)))

series357.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
103.0ms
(/ (/ (/ e (/ (sqrt (fma (cos v) e 1)) 1)) (/ (sqrt (sqrt (fma (cos v) e 1))) 1)) (/ (sqrt (sqrt (fma (cos v) e 1))) (sin v)))
95.0ms
(/ (sqrt (sqrt (fma (cos v) e 1))) (sin v))
79.0ms
(sqrt (sqrt (fma (cos v) e 1)))
79.0ms
(/ (/ e (/ (sqrt (fma (cos v) e 1)) 1)) (/ (sqrt (sqrt (fma (cos v) e 1))) 1))

simplify4.7m

Counts
1751 → 981
Calls
1751 calls:
Slowest
1.2s
(/ (* (* (/ e (/ (sqrt (fma (cos v) e 1)) 1)) (/ e (/ (sqrt (fma (cos v) e 1)) 1))) (/ e (/ (sqrt (fma (cos v) e 1)) 1))) (* (* (/ (sqrt (sqrt (fma (cos v) e 1))) 1) (/ (sqrt (sqrt (fma (cos v) e 1))) 1)) (/ (sqrt (sqrt (fma (cos v) e 1))) 1)))
937.0ms
(/ (/ (/ 1 (/ (sqrt (fma (cos v) e 1)) 1)) (sqrt (/ (sqrt (sqrt (fma (cos v) e 1))) 1))) (/ (sqrt (sqrt (cbrt (fma (cos v) e 1)))) (sin v)))
902.0ms
(/ (/ e (sqrt (sqrt (fma (cos v) e 1)))) (/ (sqrt 1) (* (cbrt (sin v)) (cbrt (sin v)))))
897.0ms
(/ (/ (/ e (/ (sqrt (fma (cos v) e 1)) 1)) (/ (sqrt (sqrt (fma (cos v) e 1))) 1)) (/ (sqrt (sqrt (sqrt (fma (cos v) e 1)))) (cbrt (sin v))))
882.0ms
(/ (/ (/ e (/ (sqrt (fma (cos v) e 1)) 1)) (cbrt (/ (sqrt (sqrt (fma (cos v) e 1))) 1))) (/ (sqrt (sqrt (fma (cos v) e 1))) (sin v)))

prune16.1s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0b

regimes158.0ms

Accuracy

0% (0.1b remaining)

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

bsearch1.0ms

end0.0ms

sample5.4s

Algorithm
intervals
Results
2.4s1473×body1280valid
1.1s1224×body640valid
839.0ms4346×body80valid
383.0ms633×body320valid
313.0ms8000×pre80true
180.0ms324×body160valid