Average Error: 0.1 → 0.2
Time: 18.7s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot (e^{\log_* (1 + \frac{\sin v}{(\left(\cos v\right) \cdot e + 1)_*})} - 1)^*\]

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. 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 expm1-log1p-u0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))

Details

Time bar (total: 17.7s)Debug log

sample317.0ms

Algorithm
intervals

simplify64.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
64.0ms
(/ (* e (sin v)) (+ 1 (* e (cos v))))

prune17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize37.0ms

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

series149.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
82.0ms
(/ (* e (sin v)) (fma (cos v) e 1))
66.0ms
(* e (sin v))

simplify1.5s

Counts
20 → 40
Calls
20 calls:
Slowest
846.0ms
(/ (* (* (* e (sin v)) (* e (sin v))) (* e (sin v))) (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)))
418.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
110.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))
75.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))
27.0ms
(- (log (* e (sin v))) (log (fma (cos v) e 1)))

prune513.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

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
Rules
18×times-frac
17×add-cube-cbrt
17×*-un-lft-identity
17×add-sqr-sqrt
13×associate-*r*
4×add-exp-log
4×add-cbrt-cube
3×associate-/l*
3×associate-/r*
2×add-log-exp
2×div-inv
2×log1p-expm1-u
2×pow1
2×expm1-log1p-u
1×*-commutative
1×associate-*r/
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
2 → 53
Calls
2 calls:
Slowest
7.0ms
(/ (sin v) (fma (cos v) e 1))
4.0ms
(* e (/ (sin v) (fma (cos v) e 1)))

series160.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
101.0ms
(* e (/ (sin v) (fma (cos v) e 1)))
59.0ms
(/ (sin v) (fma (cos v) e 1))

simplify2.0s

Counts
45 → 59
Calls
45 calls:
Slowest
478.0ms
(/ (* (* (sin v) (sin v)) (sin v)) (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)))
425.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
228.0ms
(- v (+ (* 1/6 (pow v 3)) (* e v)))
150.0ms
(- (+ (/ (sin v) (* (pow e 3) (pow (cos v) 3))) (/ (sin v) (* e (cos v)))) (/ (sin v) (* (pow e 2) (pow (cos v) 2))))
139.0ms
(- (+ (/ (sin v) (* (pow e 3) (pow (cos v) 3))) (/ (sin v) (* e (cos v)))) (/ (sin v) (* (pow e 2) (pow (cos v) 2))))

prune610.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

localize20.0ms

Local error

Found 4 expressions with local error:

0.3b
(log1p (/ (sin v) (fma (cos v) e 1)))
0.1b
(expm1 (log1p (/ (sin v) (fma (cos v) e 1))))
0.1b
(* e (expm1 (log1p (/ (sin v) (fma (cos v) e 1)))))
0.0b
(/ (sin v) (fma (cos v) e 1))

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
13×add-cube-cbrt
13×*-un-lft-identity
13×add-sqr-sqrt
9×times-frac
6×add-exp-log
6×add-cbrt-cube
5×expm1-log1p-u
4×add-log-exp
4×log1p-expm1-u
4×pow1
3×associate-/l*
3×associate-*r*
3×associate-/r*
1×expm1-udef
1×div-inv
1×*-commutative
1×log1p-expm1
1×div-exp
1×frac-2neg
1×clear-num
1×expm1-log1p
1×log1p-udef
1×cbrt-undiv
Counts
4 → 64
Calls
4 calls:
Slowest
7.0ms
(/ (sin v) (fma (cos v) e 1))
2.0ms
(* e (expm1 (log1p (/ (sin v) (fma (cos v) e 1)))))
1.0ms
(log1p (/ (sin v) (fma (cos v) e 1)))
0.0ms
(expm1 (log1p (/ (sin v) (fma (cos v) e 1))))

series400.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
134.0ms
(* e (expm1 (log1p (/ (sin v) (fma (cos v) e 1)))))
113.0ms
(expm1 (log1p (/ (sin v) (fma (cos v) e 1))))
108.0ms
(log1p (/ (sin v) (fma (cos v) e 1)))
45.0ms
(/ (sin v) (fma (cos v) e 1))

simplify2.7s

Counts
42 → 76
Calls
42 calls:
Slowest
510.0ms
(/ (* (* (sin v) (sin v)) (sin v)) (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)))
396.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
195.0ms
(- v (+ (* 1/6 (pow v 3)) (* e v)))
190.0ms
(- v (+ (* 1/6 (pow v 3)) (* e v)))
190.0ms
(- v (+ (* 1/2 (pow v 2)) (* e v)))

prune962.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize6.0ms

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))

rewrite5.0ms

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
3.0ms
(/ (fma (cos v) e 1) (sin v))
1.0ms
(/ e (/ (fma (cos v) e 1) (sin v)))

series155.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
93.0ms
(/ e (/ (fma (cos v) e 1) (sin v)))
62.0ms
(/ (fma (cos v) e 1) (sin v))

simplify1.2s

Counts
45 → 61
Calls
45 calls:
Slowest
489.0ms
(- (+ (* (pow e 3) v) (* e v)) (* (pow e 2) v))
131.0ms
(/ (* (* (fma (cos v) e 1) (fma (cos v) e 1)) (fma (cos v) e 1)) (* (* (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))))
76.0ms
(- (+ (/ (sin v) (* (pow e 2) (pow (cos v) 3))) (/ (sin v) (cos v))) (/ (sin v) (* e (pow (cos v) 2))))
38.0ms
(/ (* (cbrt (fma (cos v) e 1)) (cbrt (fma (cos v) e 1))) (* (cbrt (sin v)) (cbrt (sin v))))

prune639.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

regimes129.0ms

Accuracy

0% (0.2b remaining)

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

bsearch2.0ms

end0.0ms

sample6.0s

Algorithm
intervals