Average Error: 14.7 → 0.4
Time: 27.6s
Precision: 64
Internal Precision: 128
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{\left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)\right) \cdot \left(\sin a \cdot \sin b\right)}} \cdot r\]

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 14.7

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 add-cbrt-cube0.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \color{blue}{\sqrt[3]{\left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)\right) \cdot \left(\sin a \cdot \sin b\right)}}}\]

  6. Final simplification0.4

    \[\leadsto \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{\left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)\right) \cdot \left(\sin a \cdot \sin b\right)}} \cdot r\]

Reproduce

herbie shell --seed 2019004 +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: 26.0s)Debug log

sample265.0ms

Algorithm
intervals

simplify22.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 16.0b

localize19.0ms

Local error

Found 3 expressions with local error:

4.1b
(cos (+ a b))
0.2b
(/ (sin b) (cos (+ a b)))
0.2b
(* r (/ (sin b) (cos (+ a b))))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
3 → 63
Calls

3 calls. Slowest were:

7.0ms
(/ (sin b) (cos (+ a b)))
5.0ms
(cos (+ a b))
4.0ms
(* r (/ (sin b) (cos (+ a b))))

series175.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

102.0ms
(* r (/ (sin b) (cos (+ a b))))
38.0ms
(cos (+ a b))
36.0ms
(/ (sin b) (cos (+ a b)))

simplify927.0ms

Counts
49 → 72
Calls

49 calls. Slowest were:

232.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
222.0ms
(+ b (+ (* 1/3 (pow b 3)) (* a (pow b 2))))
149.0ms
(/ (* (* (sin b) (sin b)) (sin b)) (* (* (cos (+ a b)) (cos (+ a b))) (cos (+ a b))))

prune871.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.2b

localize42.0ms

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)))

rewrite41.0ms

Algorithm
rewrite-expression-head
Counts
4 → 99
Calls

4 calls. Slowest were:

13.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
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))))

series517.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

248.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
135.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
80.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
54.0ms
(* (sin a) (sin b))

simplify2.9s

Counts
73 → 111
Calls

73 calls. Slowest were:

369.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
263.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
254.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))

prune1.6s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize34.0ms

Local error

Found 4 expressions with local error:

28.0b
(cbrt (* (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (sin a) (sin b))))
0.3b
(* (* (sin a) (sin b)) (* (sin a) (sin b)))
0.2b
(* (sin a) (sin b))
0.2b
(* (sin a) (sin b))

rewrite89.0ms

Algorithm
rewrite-expression-head
Counts
4 → 93
Calls

4 calls. Slowest were:

52.0ms
(cbrt (* (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (sin a) (sin b))))
20.0ms
(* (* (sin a) (sin b)) (* (sin a) (sin b)))
7.0ms
(* (sin a) (sin b))

series255.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

71.0ms
(* (sin a) (sin b))
64.0ms
(* (* (sin a) (sin b)) (* (sin a) (sin b)))
62.0ms
(* (sin a) (sin b))
57.0ms
(cbrt (* (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (sin a) (sin b))))

simplify5.0s

Counts
66 → 105
Calls

66 calls. Slowest were:

604.0ms
(* (* (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (sin a) (sin b))) (* (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (sin a) (sin b))))
598.0ms
(* (* (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (sin a) (sin b))) (* (* (* (sin a) (sin a)) (sin a)) (* (* (sin b) (sin b)) (sin b))))
595.0ms
(* (* (* (* (sin a) (sin a)) (sin a)) (* (* (sin b) (sin b)) (sin b))) (* (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (sin a) (sin b))))

prune1.6s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 4 expressions with local error:

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

rewrite46.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

16.0ms
(/ (* (sin b) r) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
14.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
11.0ms
(* (sin a) (sin b))

series449.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

273.0ms
(/ (* (sin b) r) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
92.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
55.0ms
(* (sin a) (sin b))
28.0ms
(* (sin b) r)

simplify1.9s

Counts
45 → 88
Calls

45 calls. Slowest were:

364.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
298.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
268.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))

prune1.2s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.0b

regimes331.0ms

Accuracy

0% (0.3b remaining)

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

bsearch4.0ms

end0.0ms

sample7.7s

Algorithm
intervals