Average Error: 15.3 → 0.3
Time: 25.3s
Precision: 64
Internal Precision: 128
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

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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 *-un-lft-identity0.3

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

  6. Applied times-frac0.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))

Details

Time bar (total: 23.5s)Debug log

sample333.0ms

Algorithm
intervals

simplify68.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune18.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 13.4b

localize36.0ms

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))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
3 → 44
Calls

3 calls. Slowest were:

9.0ms
(/ (* r (sin b)) (cos (+ a b)))
6.0ms
(cos (+ a b))
3.0ms
(* r (sin b))

series196.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

108.0ms
(/ (* r (sin b)) (cos (+ a b)))
56.0ms
(* r (sin b))
33.0ms
(cos (+ a b))

simplify1.0s

Counts
24 → 53
Calls

24 calls. Slowest were:

496.0ms
(/ (* (* (* r (sin b)) (* r (sin b))) (* r (sin b))) (* (* (cos (+ a b)) (cos (+ a b))) (cos (+ a b))))
265.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
137.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))

prune751.0ms

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

rewrite49.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

25.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
14.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
7.0ms
(* (sin a) (sin b))

series435.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

218.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
92.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
70.0ms
(* (sin a) (sin b))
55.0ms
(* r (sin b))

simplify1.9s

Counts
45 → 88
Calls

45 calls. Slowest were:

353.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
275.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
264.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))

prune1.3s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

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

rewrite47.0ms

Algorithm
rewrite-expression-head
Counts
4 → 99
Calls

4 calls. Slowest were:

19.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
12.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
11.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))

series610.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

295.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
144.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
97.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
74.0ms
(* (sin a) (sin b))

simplify3.3s

Counts
73 → 111
Calls

73 calls. Slowest were:

374.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
365.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
284.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))

prune1.4s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 4 expressions with local error:

3.3b
(log (exp (* (sin a) (sin b))))
0.2b
(/ (* r (sin b)) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b))))))
0.2b
(* (sin a) (sin b))
0.1b
(- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b)))))

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
4 → 83
Calls

4 calls. Slowest were:

12.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b))))))
8.0ms
(- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b)))))
4.0ms
(log (exp (* (sin a) (sin b))))

series466.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

249.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b))))))
93.0ms
(- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b)))))
73.0ms
(* (sin a) (sin b))
51.0ms
(log (exp (* (sin a) (sin b))))

simplify2.0s

Counts
57 → 95
Calls

57 calls. Slowest were:

379.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (log (exp (* (sin a) (sin b)))) (log (exp (* (sin a) (sin b))))) (* (* (cos a) (cos b)) (log (exp (* (sin a) (sin b)))))))
252.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
169.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))

prune1.5s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.0b

regimes300.0ms

Accuracy

0% (0.3b remaining)

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

bsearch3.0ms

end0.0ms

sample7.5s

Algorithm
intervals