Average Error: 15.2 → 0.4
Time: 26.9s
Precision: 64
Internal Precision: 128
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\frac{\cos b \cdot \cos a - \sin b \cdot \sin a}{\sin b}}\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.2

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  2. Using strategy rm

  3. Applied cos-sum0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]
  4. Using strategy rm

  5. Applied associate-/l*0.4

    \[\leadsto \color{blue}{\frac{r}{\frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\sin b}}}\]

  6. Final simplification0.4

    \[\leadsto \frac{r}{\frac{\cos b \cdot \cos a - \sin b \cdot \sin a}{\sin b}}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))

Details

Time bar (total: 25.0s)Debug log

sample1.2s

Algorithm
intervals

simplify68.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

68.0ms
(/ (* r (sin b)) (cos (+ a b)))

prune51.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 13.2b

localize45.0ms

Local error

Found 3 expressions with local error:

5.7b
(cos (+ a b))
0.2b
(/ (* r (sin b)) (cos (+ a b)))
0.1b
(* r (sin b))

rewrite62.0ms

Algorithm
rewrite-expression-head
Counts
3 → 38
Calls

3 calls. Slowest were:

41.0ms
(cos (+ a b))
13.0ms
(/ (* r (sin b)) (cos (+ a b)))
3.0ms
(* r (sin b))

series285.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

152.0ms
(/ (* r (sin b)) (cos (+ a b)))
78.0ms
(cos (+ a b))
54.0ms
(* r (sin b))

simplify1.0s

Counts
24 → 47
Calls

24 calls. Slowest were:

611.0ms
(/ (* (* (* r (sin b)) (* r (sin b))) (* r (sin b))) (* (* (cos (+ a b)) (cos (+ a b))) (cos (+ a b))))
158.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
137.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))

prune548.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.2b

localize21.0ms

Local error

Found 4 expressions with local error:

0.2b
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
0.2b
(* (sin a) (sin b))
0.1b
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
0.1b
(* r (sin b))

rewrite40.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

16.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
13.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
7.0ms
(* (sin a) (sin b))

series513.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

317.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
84.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
69.0ms
(* (sin a) (sin b))
42.0ms
(* r (sin b))

simplify1.6s

Counts
42 → 78
Calls

42 calls. Slowest were:

379.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
362.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
170.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))

prune1.0s

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 4 expressions with local error:

0.2b
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
0.2b
(* (sin a) (sin b))
0.2b
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
0.1b
(- (* (cos a) (cos b)) (* (sin a) (sin b)))

rewrite42.0ms

Algorithm
rewrite-expression-head
Counts
4 → 89
Calls

4 calls. Slowest were:

13.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
13.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
11.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))

series541.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

266.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
122.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
99.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
54.0ms
(* (sin a) (sin b))

simplify2.2s

Counts
70 → 101
Calls

70 calls. Slowest were:

365.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
350.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
185.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))

prune1.4s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.3b
(/ r (/ (- (* (cos a) (cos b)) (* (sin a) (sin b))) (sin b)))
0.2b
(/ (- (* (cos a) (cos b)) (* (sin a) (sin b))) (sin b))
0.2b
(* (sin a) (sin b))
0.1b
(- (* (cos a) (cos b)) (* (sin a) (sin b)))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 88
Calls

4 calls. Slowest were:

14.0ms
(/ (- (* (cos a) (cos b)) (* (sin a) (sin b))) (sin b))
7.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
4.0ms
(* (sin a) (sin b))

series491.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

244.0ms
(/ r (/ (- (* (cos a) (cos b)) (* (sin a) (sin b))) (sin b)))
102.0ms
(/ (- (* (cos a) (cos b)) (* (sin a) (sin b))) (sin b))
76.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
70.0ms
(* (sin a) (sin b))

simplify4.2s

Counts
69 → 100
Calls

69 calls. Slowest were:

370.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
363.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
358.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))

prune1.3s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.0b

regimes413.0ms

Accuracy

0% (0.3b remaining)

Error of 0.4b against oracle of 0.0b and baseline of 0.4b

bsearch3.0ms

end0.0ms

sample7.7s

Algorithm
intervals