Average Error: 15.3 → 0.3
Time: 26.4s
Precision: 64
Internal Precision: 128
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a}\]

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

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

  3. Applied cos-sum0.3

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

  5. Applied associate-*r/0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019007 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))

Details

Time bar (total: 24.6s)Debug log

sample251.0ms

Algorithm
intervals

simplify72.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

71.0ms
(* r (/ (sin b) (cos (+ a b))))

prune15.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 14.9b

localize22.0ms

Local error

Found 3 expressions with local error:

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

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
3 → 57
Calls

3 calls. Slowest were:

4.0ms
(/ (sin b) (cos (+ a b)))
3.0ms
(cos (+ a b))
2.0ms
(* r (/ (sin b) (cos (+ a b))))

series148.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

84.0ms
(* r (/ (sin b) (cos (+ a b))))
43.0ms
(/ (sin b) (cos (+ a b)))
20.0ms
(cos (+ a b))

simplify554.0ms

Counts
49 → 66
Calls

49 calls. Slowest were:

127.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
93.0ms
(+ b (+ (* 1/3 (pow b 3)) (* a (pow b 2))))
79.0ms
(/ (* (* (sin b) (sin b)) (sin b)) (* (* (cos (+ a b)) (cos (+ a b))) (cos (+ a b))))

prune584.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.2b

localize28.0ms

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)))

rewrite48.0ms

Algorithm
rewrite-expression-head
Counts
4 → 89
Calls

4 calls. Slowest were:

24.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
10.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
9.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))

series502.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

237.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
118.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
85.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
62.0ms
(* (sin a) (sin b))

simplify2.2s

Counts
70 → 101
Calls

70 calls. Slowest were:

353.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
301.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
163.0ms
(/ (* (* (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

localize12.0ms

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))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

11.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
8.0ms
(* (sin a) (sin b))
6.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))

series429.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

262.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
87.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
47.0ms
(* (sin a) (sin b))
33.0ms
(* r (sin b))

simplify1.7s

Counts
42 → 78
Calls

42 calls. Slowest were:

326.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
323.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
190.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))

prune1.1s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.0b

localize36.0ms

Local error

Found 4 expressions with local error:

0.6b
(pow (* (cos a) (cos b)) 3)
0.4b
(* (* (cos a) (cos b)) (* (cos a) (cos b)))
0.3b
(pow (* (sin a) (sin b)) 3)
0.3b
(* (* (cos a) (cos b)) (* (sin a) (sin b)))

rewrite55.0ms

Algorithm
rewrite-expression-head
Counts
4 → 95
Calls

4 calls. Slowest were:

29.0ms
(* (* (cos a) (cos b)) (* (cos a) (cos b)))
15.0ms
(* (* (cos a) (cos b)) (* (sin a) (sin b)))
6.0ms
(pow (* (cos a) (cos b)) 3)

series359.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

139.0ms
(* (* (cos a) (cos b)) (* (sin a) (sin b)))
76.0ms
(* (* (cos a) (cos b)) (* (cos a) (cos b)))
75.0ms
(pow (* (sin a) (sin b)) 3)
68.0ms
(pow (* (cos a) (cos b)) 3)

simplify5.1s

Counts
74 → 107
Calls

74 calls. Slowest were:

685.0ms
(* (* (* (* (cos a) (cos a)) (cos a)) (* (* (cos b) (cos b)) (cos b))) (* (* (* (cos a) (cos b)) (* (cos a) (cos b))) (* (cos a) (cos b))))
648.0ms
(* (* (* (* (cos a) (cos b)) (* (cos a) (cos b))) (* (cos a) (cos b))) (* (* (* (cos a) (cos b)) (* (cos a) (cos b))) (* (cos a) (cos b))))
595.0ms
(* (* (* (* (cos a) (cos b)) (* (cos a) (cos b))) (* (cos a) (cos b))) (* (* (* (cos a) (cos a)) (cos a)) (* (* (cos b) (cos b)) (cos b))))

prune2.0s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.0b

regimes324.0ms

Accuracy

0% (0.3b remaining)

Error of 0.3b against oracle of 0.1b and baseline of 0.3b

bsearch1.0ms

end0.0ms

sample7.7s

Algorithm
intervals