Average Error: 15.1 → 0.4
Time: 35.3s
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 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))

Details

Time bar (total: 22.6s)Debug log

start1.2s

Algorithm
intervals

setup62.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 12.3b

localize21.0ms

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))))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
3 → 63
Calls

3 calls. Slowest were:

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

series179.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

111.0ms
(* r (/ (sin b) (cos (+ a b))))
45.0ms
(/ (sin b) (cos (+ a b)))
22.0ms
(cos (+ a b))

simplify898.0ms

Counts
49 → 72
Calls

49 calls. Slowest were:

240.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
171.0ms
(+ b (+ (* 1/3 (pow b 3)) (* a (pow b 2))))
163.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))

prune983.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.2b

localize25.0ms

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))

rewrite49.0ms

Algorithm
rewrite-expression-head
Counts
4 → 101
Calls

4 calls. Slowest were:

23.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
13.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
7.0ms
(* (cos a) (cos b))

series528.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

268.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
138.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
80.0ms
(* (sin a) (sin b))
42.0ms
(* (cos a) (cos b))

simplify2.4s

Counts
75 → 113
Calls

75 calls. Slowest were:

263.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
262.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))
246.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))

prune1.8s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize19.0ms

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))

rewrite51.0ms

Algorithm
rewrite-expression-head
Counts
4 → 80
Calls

4 calls. Slowest were:

25.0ms
(* (* r (sin b)) (/ 1 (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
16.0ms
(/ 1 (- (* (cos a) (cos b)) (* (sin a) (sin b))))
6.0ms
(* (sin a) (sin b))

series456.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

237.0ms
(* (* r (sin b)) (/ 1 (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
128.0ms
(/ 1 (- (* (cos a) (cos b)) (* (sin a) (sin b))))
56.0ms
(* (sin a) (sin b))
35.0ms
(* r (sin b))

simplify1.8s

Counts
44 → 92
Calls

44 calls. Slowest were:

314.0ms
(* (* (* (* 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))))))
291.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
199.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))

prune1.5s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.0b

localize39.0ms

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) (- (pow (* (cos a) (cos b)) 3) (pow (* (sin a) (sin b)) 3)))) (fma (fma (sin a) (sin b) (* (cos b) (cos a))) (* (sin b) (sin a)) (* (* (cos b) (cos a)) (* (cos b) (cos a)))))

rewrite62.0ms

Algorithm
rewrite-expression-head
Counts
4 → 96
Calls

4 calls. Slowest were:

32.0ms
(* (* r (/ (sin b) (- (pow (* (cos a) (cos b)) 3) (pow (* (sin a) (sin b)) 3)))) (fma (fma (sin a) (sin b) (* (cos b) (cos a))) (* (sin b) (sin a)) (* (* (cos b) (cos a)) (* (cos b) (cos a)))))
22.0ms
(* (* (cos b) (cos a)) (* (cos b) (cos a)))
3.0ms
(pow (* (cos a) (cos b)) 3)

series959.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

763.0ms
(* (* r (/ (sin b) (- (pow (* (cos a) (cos b)) 3) (pow (* (sin a) (sin b)) 3)))) (fma (fma (sin a) (sin b) (* (cos b) (cos a))) (* (sin b) (sin a)) (* (* (cos b) (cos a)) (* (cos b) (cos a)))))
75.0ms
(pow (* (sin a) (sin b)) 3)
62.0ms
(pow (* (cos a) (cos b)) 3)
59.0ms
(* (* (cos b) (cos a)) (* (cos b) (cos a)))

simplify6.4s

Counts
66 → 108
Calls

66 calls. Slowest were:

670.0ms
(* (* (* (* (cos b) (cos a)) (* (cos b) (cos a))) (* (cos b) (cos a))) (* (* (* (cos b) (cos a)) (* (cos b) (cos a))) (* (cos b) (cos a))))
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))))
621.0ms
(* (* (* (* (cos b) (cos b)) (cos b)) (* (* (cos a) (cos a)) (cos a))) (* (* (* (cos b) (cos a)) (* (cos b) (cos a))) (* (cos b) (cos a))))

prune2.8s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

regimes366.0ms

Accuracy

0% (0.3b remaining)

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

bsearch4.0ms