Average Error: 15.3 → 0.4
Time: 25.5s
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 15.3

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 2019007 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))

Details

Time bar (total: 24.0s)Debug log

sample408.0ms

Algorithm
intervals

simplify68.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

67.0ms
(/ (* r (sin b)) (cos (+ a b)))

prune19.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 14.9b

localize34.0ms

Local error

Found 3 expressions with local error:

4.2b
(cos (+ a b))
0.2b
(/ (* r (sin b)) (cos (+ a b)))
0.1b
(* r (sin b))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
3 → 38
Calls

3 calls. Slowest were:

9.0ms
(/ (* r (sin b)) (cos (+ a b)))
5.0ms
(cos (+ a b))
2.0ms
(* r (sin b))

series226.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

141.0ms
(/ (* r (sin b)) (cos (+ a b)))
53.0ms
(* r (sin b))
32.0ms
(cos (+ a b))

simplify1.1s

Counts
24 → 47
Calls

24 calls. Slowest were:

580.0ms
(/ (* (* (* r (sin b)) (* r (sin b))) (* r (sin b))) (* (* (cos (+ a b)) (cos (+ a b))) (cos (+ a b))))
194.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
136.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))

prune526.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.2b

localize41.0ms

Local error

Found 4 expressions with local error:

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)))
0.1b
(* r (sin b))

rewrite43.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

19.0ms
(/ (* 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))

series523.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

279.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
110.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
78.0ms
(* (sin a) (sin b))
55.0ms
(* r (sin b))

simplify1.5s

Counts
42 → 78
Calls

42 calls. Slowest were:

312.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
267.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
171.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))

prune1.0s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

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

rewrite37.0ms

Algorithm
rewrite-expression-head
Counts
4 → 89
Calls

4 calls. Slowest were:

11.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
10.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
7.0ms
(* (sin a) (sin b))

series599.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

252.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
147.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
117.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
82.0ms
(* (sin a) (sin b))

simplify2.4s

Counts
70 → 101
Calls

70 calls. Slowest were:

380.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
328.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
222.0ms
(/ (* (* (sin b) (sin b)) (sin b)) (* (* (- (* (cos a) (cos b)) (* (sin a) (sin b))) (- (* (cos a) (cos b)) (* (sin a) (sin b)))) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))

prune1.4s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 4 expressions with local error:

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

rewrite45.0ms

Algorithm
rewrite-expression-head
Counts
4 → 88
Calls

4 calls. Slowest were:

25.0ms
(/ (- (* (cos a) (cos b)) (* (sin a) (sin b))) (sin b))
8.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
7.0ms
(* (sin a) (sin b))

series489.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

223.0ms
(/ r (/ (- (* (cos a) (cos b)) (* (sin a) (sin b))) (sin b)))
121.0ms
(/ (- (* (cos a) (cos b)) (* (sin a) (sin b))) (sin b))
88.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
56.0ms
(* (sin a) (sin b))

simplify4.2s

Counts
69 → 100
Calls

69 calls. Slowest were:

389.0ms
(- (pow (* (cos a) (cos b)) 3) (pow (* (sin a) (sin b)) 3))
340.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
336.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))

prune1.2s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.0b

regimes321.0ms

Accuracy

0% (0.3b remaining)

Error of 0.4b against oracle of 0.1b and baseline of 0.4b

bsearch3.0ms

end0.0ms

sample7.8s

Algorithm
intervals