Average Error: 15.1 → 0.3
Time: 26.6s
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 15.1

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 2018362 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))

Details

Time bar (total: 14.7s)Debug log

start394.0ms

Algorithm
intervals

setup405.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 12.4b

localize20.0ms

Local error

Found 3 expressions with local error:

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

rewrite19.0ms

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))

series216.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

133.0ms
(/ (* r (sin b)) (cos (+ a b)))
54.0ms
(* r (sin b))
29.0ms
(cos (+ a b))

simplify873.0ms

Counts
24 → 47
Calls

24 calls. Slowest were:

545.0ms
(/ (* (* (* r (sin b)) (* r (sin b))) (* r (sin b))) (* (* (cos (+ a b)) (cos (+ a b))) (cos (+ a b))))
162.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
90.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))

prune638.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.1b

localize42.0ms

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

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 70
Calls

4 calls. Slowest were:

13.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
3.0ms
(* (sin a) (sin b))
3.0ms
(* (cos a) (cos b))

series434.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

274.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
57.0ms
(* r (sin b))
56.0ms
(* (sin a) (sin b))
46.0ms
(* (cos a) (cos b))

simplify1.1s

Counts
47 → 82
Calls

47 calls. Slowest were:

186.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
154.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* 1/2 (pow b 2))))
153.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))

prune1.1s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

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))

rewrite61.0ms

Algorithm
rewrite-expression-head
Counts
4 → 93
Calls

4 calls. Slowest were:

23.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
22.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
7.0ms
(* (cos a) (cos b))

series417.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

207.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
109.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
51.0ms
(* (sin a) (sin b))
50.0ms
(* (cos a) (cos b))

simplify1.7s

Counts
75 → 105
Calls

75 calls. Slowest were:

233.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
184.0ms
(/ (* (* (sin b) (sin b)) (sin b)) (* (* (- (* (cos a) (cos b)) (* (sin a) (sin b))) (- (* (cos a) (cos b)) (* (sin a) (sin b)))) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
147.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))

prune1.4s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.0b

localize7.0ms

Local error

Found 4 expressions with local error:

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

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 92
Calls

4 calls. Slowest were:

12.0ms
(/ (- (* (cos a) (cos b)) (* (sin a) (sin b))) (sin b))
3.0ms
(* (sin a) (sin b))
3.0ms
(* (cos a) (cos b))

series542.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

279.0ms
(/ r (/ (- (* (cos a) (cos b)) (* (sin a) (sin b))) (sin b)))
143.0ms
(/ (- (* (cos a) (cos b)) (* (sin a) (sin b))) (sin b))
61.0ms
(* (sin a) (sin b))
59.0ms
(* (cos a) (cos b))

simplify3.6s

Counts
74 → 104
Calls

74 calls. Slowest were:

369.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
332.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
306.0ms
(- (pow (* (cos a) (cos b)) 3) (pow (* (sin a) (sin b)) 3))

prune1.5s

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 0.0b

regimes282.0ms

Accuracy

0% (0.3b remaining)

Error of 0.3b against oracle of 0.0b and baseline of 0.3b

bsearch3.0ms