Average Error: 15.1 → 0.4
Time: 33.1s
Precision: 64
Internal Precision: 128
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{1}{\cos b \cdot \cos a - \sin b \cdot \sin a} \cdot \left(r \cdot \sin b\right)\]

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 15.1

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 div-inv0.4

    \[\leadsto r \cdot \color{blue}{\left(\sin b \cdot \frac{1}{\cos a \cdot \cos b - \sin a \cdot \sin b}\right)}\]

  6. Applied associate-*r*0.4

    \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]

  7. Final simplification0.4

    \[\leadsto \frac{1}{\cos b \cdot \cos a - \sin b \cdot \sin a} \cdot \left(r \cdot \sin b\right)\]

Reproduce

herbie shell --seed 2018362 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))

Details

Time bar (total: 20.9s)Debug log

start357.0ms

Algorithm
intervals

setup381.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 12.3b

localize22.0ms

Local error

Found 3 expressions with local error:

3.0b
(cos (+ a b))
0.2b
(/ (sin b) (cos (+ a b)))
0.2b
(* r (/ (sin b) (cos (+ a b))))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
3 → 57
Calls

3 calls. Slowest were:

6.0ms
(/ (sin b) (cos (+ a b)))
5.0ms
(cos (+ a b))
4.0ms
(* r (/ (sin b) (cos (+ a b))))

series176.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

93.0ms
(* r (/ (sin b) (cos (+ a b))))
60.0ms
(/ (sin b) (cos (+ a b)))
21.0ms
(cos (+ a b))

simplify784.0ms

Counts
49 → 66
Calls

49 calls. Slowest were:

186.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
133.0ms
(/ (* (* (sin b) (sin b)) (sin b)) (* (* (cos (+ a b)) (cos (+ a b))) (cos (+ a b))))
89.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))

prune812.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.2b

localize47.0ms

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

rewrite64.0ms

Algorithm
rewrite-expression-head
Counts
4 → 93
Calls

4 calls. Slowest were:

24.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
19.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
12.0ms
(* (sin a) (sin b))

series558.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

281.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
162.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
74.0ms
(* (sin a) (sin b))
41.0ms
(* (cos a) (cos b))

simplify1.7s

Counts
75 → 105
Calls

75 calls. Slowest were:

172.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))
150.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))
149.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))

prune1.4s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (* r (sin b)) (/ 1 (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
0.2b
(/ 1 (- (* (cos a) (cos b)) (* (sin a) (sin b))))
0.2b
(* (sin a) (sin b))
0.2b
(* r (sin b))

rewrite36.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

15.0ms
(/ 1 (- (* (cos a) (cos b)) (* (sin a) (sin b))))
11.0ms
(* (* r (sin b)) (/ 1 (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
6.0ms
(* (sin a) (sin b))

series461.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

279.0ms
(* (* r (sin b)) (/ 1 (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
92.0ms
(/ 1 (- (* (cos a) (cos b)) (* (sin a) (sin b))))
48.0ms
(* (sin a) (sin b))
42.0ms
(* r (sin b))

simplify1.7s

Counts
44 → 84
Calls

44 calls. Slowest were:

436.0ms
(* (* (* (* r (sin b)) (* r (sin b))) (* r (sin b))) (* (* (/ 1 (- (* (cos a) (cos b)) (* (sin a) (sin b)))) (/ 1 (- (* (cos a) (cos b)) (* (sin a) (sin b))))) (/ 1 (- (* (cos a) (cos b)) (* (sin a) (sin b))))))
229.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
166.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))

prune1.2s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize38.0ms

Local error

Found 4 expressions with local error:

0.5b
(pow (* (cos a) (cos b)) 3)
0.3b
(* (* (cos b) (cos a)) (* (cos b) (cos a)))
0.3b
(pow (* (sin a) (sin b)) 3)
0.2b
(* (* r (sin b)) (/ 1 (- (pow (* (cos a) (cos b)) 3) (pow (* (sin a) (sin b)) 3))))

rewrite44.0ms

Algorithm
rewrite-expression-head
Counts
4 → 93
Calls

4 calls. Slowest were:

19.0ms
(* (* (cos b) (cos a)) (* (cos b) (cos a)))
15.0ms
(* (* r (sin b)) (/ 1 (- (pow (* (cos a) (cos b)) 3) (pow (* (sin a) (sin b)) 3))))
5.0ms
(pow (* (cos a) (cos b)) 3)

series472.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

307.0ms
(* (* r (sin b)) (/ 1 (- (pow (* (cos a) (cos b)) 3) (pow (* (sin a) (sin b)) 3))))
69.0ms
(pow (* (sin a) (sin b)) 3)
49.0ms
(pow (* (cos a) (cos b)) 3)
48.0ms
(* (* (cos b) (cos a)) (* (cos b) (cos a)))

simplify7.8s

Counts
71 → 105
Calls

71 calls. Slowest were:

653.0ms
(* (* (* (* (cos b) (cos b)) (cos b)) (* (* (cos a) (cos a)) (cos a))) (* (* (* (cos b) (cos a)) (* (cos b) (cos a))) (* (cos b) (cos a))))
626.0ms
(* (* (* (* (cos b) (cos a)) (* (cos b) (cos a))) (* (cos b) (cos a))) (* (* (* (cos b) (cos b)) (cos b)) (* (* (cos a) (cos a)) (cos a))))
548.0ms
(* (* (* (* (cos b) (cos a)) (* (cos b) (cos a))) (* (cos b) (cos a))) (* (* (* (cos b) (cos a)) (* (cos b) (cos a))) (* (cos b) (cos a))))

prune2.4s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

regimes332.0ms

Accuracy

0% (0.3b remaining)

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

bsearch4.0ms