Average Error: 14.7 → 0.4
Time: 24.0s
Precision: 64
Internal Precision: 128
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \log \left(e^{\sin b \cdot \sin a}\right)}\]

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 14.7

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]

  2. Initial simplification14.7

    \[\leadsto \frac{r \cdot \sin b}{\cos \left(b + a\right)}\]
  3. Using strategy rm

  4. Applied cos-sum0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}}\]
  5. Using strategy rm

  6. Applied add-log-exp0.4

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

  7. Final simplification0.4

    \[\leadsto \frac{r \cdot \sin b}{\cos a \cdot \cos b - \log \left(e^{\sin b \cdot \sin a}\right)}\]

Reproduce

herbie shell --seed 2018365 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))

Details

Time bar (total: 12.0s)Debug log

start441.0ms

Algorithm
intervals

setup89.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 13.6b

localize37.0ms

Local error

Found 3 expressions with local error:

2.0b
(cos (+ b a))
0.3b
(/ (* r (sin b)) (cos (+ b a)))
0.1b
(* r (sin b))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
3 → 38
Calls

3 calls. Slowest were:

5.0ms
(/ (* r (sin b)) (cos (+ b a)))
3.0ms
(cos (+ b a))
1.0ms
(* r (sin b))

series236.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

154.0ms
(/ (* r (sin b)) (cos (+ b a)))
57.0ms
(* r (sin b))
25.0ms
(cos (+ b a))

simplify887.0ms

Counts
24 → 47
Calls

24 calls. Slowest were:

511.0ms
(/ (* (* (* r (sin b)) (* r (sin b))) (* r (sin b))) (* (* (cos (+ b a)) (cos (+ b a))) (cos (+ b a))))
191.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
82.0ms
(- 1 (+ (* 1/2 (pow b 2)) (* a b)))

prune563.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.1b

localize43.0ms

Local error

Found 4 expressions with local error:

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

rewrite40.0ms

Algorithm
rewrite-expression-head
Counts
4 → 70
Calls

4 calls. Slowest were:

21.0ms
(/ (* r (sin b)) (- (* (cos b) (cos a)) (* (sin b) (sin a))))
7.0ms
(* (sin b) (sin a))
7.0ms
(* (cos b) (cos a))

series422.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

254.0ms
(/ (* r (sin b)) (- (* (cos b) (cos a)) (* (sin b) (sin a))))
80.0ms
(* (sin b) (sin a))
47.0ms
(* (cos b) (cos a))
42.0ms
(* r (sin b))

simplify1.1s

Counts
47 → 82
Calls

47 calls. Slowest were:

211.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
163.0ms
(+ (* (* (cos b) (cos a)) (* (cos b) (cos a))) (+ (* (* (sin b) (sin a)) (* (sin b) (sin a))) (* (* (cos b) (cos a)) (* (sin b) (sin a)))))
154.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* 1/2 (pow b 2))))

prune1.3s

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

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

rewrite61.0ms

Algorithm
rewrite-expression-head
Counts
4 → 93
Calls

4 calls. Slowest were:

24.0ms
(* r (/ (sin b) (- (* (cos b) (cos a)) (* (sin b) (sin a)))))
21.0ms
(/ (sin b) (- (* (cos b) (cos a)) (* (sin b) (sin a))))
7.0ms
(* (cos b) (cos a))

series466.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

237.0ms
(* r (/ (sin b) (- (* (cos b) (cos a)) (* (sin b) (sin a)))))
116.0ms
(/ (sin b) (- (* (cos b) (cos a)) (* (sin b) (sin a))))
76.0ms
(* (sin b) (sin a))
36.0ms
(* (cos b) (cos a))

simplify1.6s

Counts
75 → 105
Calls

75 calls. Slowest were:

188.0ms
(/ (* (* (sin b) (sin b)) (sin b)) (* (* (- (* (cos b) (cos a)) (* (sin b) (sin a))) (- (* (cos b) (cos a)) (* (sin b) (sin a)))) (- (* (cos b) (cos a)) (* (sin b) (sin a)))))
148.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
143.0ms
(+ (* (* (cos b) (cos a)) (* (cos b) (cos a))) (+ (* (* (sin b) (sin a)) (* (sin b) (sin a))) (* (* (cos b) (cos a)) (* (sin b) (sin a)))))

prune1.6s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

5.5b
(log (exp (* (sin b) (sin a))))
0.2b
(* (sin b) (sin a))
0.2b
(/ (* r (sin b)) (- (* (cos b) (cos a)) (log (exp (* (sin b) (sin a))))))
0.1b
(* r (sin b))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

19.0ms
(/ (* r (sin b)) (- (* (cos b) (cos a)) (log (exp (* (sin b) (sin a))))))
4.0ms
(* (sin b) (sin a))
2.0ms
(* r (sin b))

series405.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

252.0ms
(/ (* r (sin b)) (- (* (cos b) (cos a)) (log (exp (* (sin b) (sin a))))))
67.0ms
(log (exp (* (sin b) (sin a))))
43.0ms
(* (sin b) (sin a))
41.0ms
(* r (sin b))

simplify1.1s

Counts
43 → 76
Calls

43 calls. Slowest were:

284.0ms
(+ (* (* (cos b) (cos a)) (* (cos b) (cos a))) (+ (* (log (exp (* (sin b) (sin a)))) (log (exp (* (sin b) (sin a))))) (* (* (cos b) (cos a)) (log (exp (* (sin b) (sin a)))))))
197.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
116.0ms
(/ (* (* (* r (sin b)) (* r (sin b))) (* r (sin b))) (* (* (- (* (cos b) (cos a)) (log (exp (* (sin b) (sin a))))) (- (* (cos b) (cos a)) (log (exp (* (sin b) (sin a)))))) (- (* (cos b) (cos a)) (log (exp (* (sin b) (sin a)))))))

prune1.2s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0b

regimes295.0ms

Accuracy

0% (0.3b remaining)

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

bsearch2.0ms