Average Error: 15.3 → 0.3
Time: 22.8s
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 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))

Details

Time bar (total: 21.1s)Debug log

sample346.0ms

Algorithm
intervals

simplify76.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

76.0ms
(/ (* r (sin b)) (cos (+ a b)))

prune15.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 13.4b

localize38.0ms

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))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
3 → 38
Calls

3 calls. Slowest were:

9.0ms
(/ (* r (sin b)) (cos (+ a b)))
5.0ms
(cos (+ a b))
2.0ms
(* r (sin b))

series197.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

114.0ms
(/ (* r (sin b)) (cos (+ a b)))
49.0ms
(* r (sin b))
33.0ms
(cos (+ a b))

simplify919.0ms

Counts
24 → 47
Calls

24 calls. Slowest were:

526.0ms
(/ (* (* (* r (sin b)) (* r (sin b))) (* r (sin b))) (* (* (cos (+ a b)) (cos (+ a b))) (cos (+ a b))))
216.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
74.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))

prune695.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.2b

localize19.0ms

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))

rewrite39.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

14.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
14.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
8.0ms
(* (sin a) (sin b))

series427.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

252.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
82.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
48.0ms
(* (sin a) (sin b))
46.0ms
(* r (sin b))

simplify1.6s

Counts
42 → 78
Calls

42 calls. Slowest were:

398.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
312.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
177.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))

prune1.2s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.0b

localize6.0ms

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)))

rewrite59.0ms

Algorithm
rewrite-expression-head
Counts
4 → 89
Calls

4 calls. Slowest were:

23.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
17.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
10.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))

series577.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

257.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
141.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
102.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
75.0ms
(* (sin a) (sin b))

simplify2.3s

Counts
70 → 101
Calls

70 calls. Slowest were:

360.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
328.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
174.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))

prune1.7s

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)))))

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

14.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b))))))
7.0ms
(- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b)))))
4.0ms
(log (exp (* (sin a) (sin b))))

series462.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

240.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b))))))
99.0ms
(- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b)))))
67.0ms
(log (exp (* (sin a) (sin b))))
56.0ms
(* (sin a) (sin b))

simplify1.2s

Counts
44 → 80
Calls

44 calls. Slowest were:

252.0ms
(+ (* (* (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)))))))
179.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
95.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))

prune1.2s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.0b

regimes287.0ms

Accuracy

0% (0.3b remaining)

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

bsearch3.0ms

end0.0ms

sample7.6s

Algorithm
intervals