Average Error: 14.7 → 0.4
Time: 24.3s
Precision: 64
Internal Precision: 128
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \log \left(e^{\sin b \cdot \sin a}\right)}\]

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

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

  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), B"
  (* r (/ (sin b) (cos (+ a b)))))

Details

Time bar (total: 11.6s)Debug log

start349.0ms

Algorithm
intervals

setup88.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 13.5b

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))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
3 → 38
Calls

3 calls. Slowest were:

8.0ms
(/ (* r (sin b)) (cos (+ b a)))
6.0ms
(cos (+ b a))
2.0ms
(* r (sin b))

series226.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

134.0ms
(/ (* r (sin b)) (cos (+ b a)))
57.0ms
(* r (sin b))
35.0ms
(cos (+ b a))

simplify1.1s

Counts
24 → 47
Calls

24 calls. Slowest were:

683.0ms
(/ (* (* (* r (sin b)) (* r (sin b))) (* r (sin b))) (* (* (cos (+ b a)) (cos (+ b a))) (cos (+ b a))))
157.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
104.0ms
(- 1 (+ (* 1/2 (pow b 2)) (* a b)))

prune515.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.1b

localize33.0ms

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))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
4 → 70
Calls

4 calls. Slowest were:

9.0ms
(/ (* r (sin b)) (- (* (cos b) (cos a)) (* (sin b) (sin a))))
4.0ms
(* (sin b) (sin a))
3.0ms
(* (cos b) (cos a))

series412.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

240.0ms
(/ (* r (sin b)) (- (* (cos b) (cos a)) (* (sin b) (sin a))))
69.0ms
(* (sin b) (sin a))
57.0ms
(* r (sin b))
46.0ms
(* (cos b) (cos a))

simplify927.0ms

Counts
47 → 82
Calls

47 calls. Slowest were:

208.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
146.0ms
(+ (* (* (cos b) (cos a)) (* (cos b) (cos a))) (+ (* (* (sin b) (sin a)) (* (sin b) (sin a))) (* (* (cos b) (cos a)) (* (sin b) (sin a)))))
100.0ms
(* (* (* (cos b) (cos b)) (cos b)) (* (* (cos a) (cos a)) (cos a)))

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))

rewrite56.0ms

Algorithm
rewrite-expression-head
Counts
4 → 93
Calls

4 calls. Slowest were:

22.0ms
(* r (/ (sin b) (- (* (cos b) (cos a)) (* (sin b) (sin a)))))
18.0ms
(/ (sin b) (- (* (cos b) (cos a)) (* (sin b) (sin a))))
7.0ms
(* (sin b) (sin a))

series487.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

232.0ms
(* r (/ (sin b) (- (* (cos b) (cos a)) (* (sin b) (sin a)))))
142.0ms
(/ (sin b) (- (* (cos b) (cos a)) (* (sin b) (sin a))))
78.0ms
(* (sin b) (sin a))
35.0ms
(* (cos b) (cos a))

simplify1.7s

Counts
75 → 105
Calls

75 calls. Slowest were:

217.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
166.0ms
(/ (* (* (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)))))
164.0ms
(+ (* (* (cos b) (cos a)) (* (cos b) (cos a))) (+ (* (* (sin b) (sin a)) (* (sin b) (sin a))) (* (* (cos b) (cos a)) (* (sin b) (sin a)))))

prune1.4s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize11.0ms

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))

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

11.0ms
(/ (* r (sin b)) (- (* (cos b) (cos a)) (log (exp (* (sin b) (sin a))))))
5.0ms
(log (exp (* (sin b) (sin a))))
3.0ms
(* (sin b) (sin a))

series401.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

218.0ms
(/ (* r (sin b)) (- (* (cos b) (cos a)) (log (exp (* (sin b) (sin a))))))
75.0ms
(log (exp (* (sin b) (sin a))))
55.0ms
(* r (sin b))
53.0ms
(* (sin b) (sin a))

simplify1.1s

Counts
43 → 76
Calls

43 calls. Slowest were:

245.0ms
(+ (* (* (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)))))))
148.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
110.0ms
(/ (* (* (* 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)))))))

prune1.1s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0b

regimes246.0ms

Accuracy

0% (0.3b remaining)

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

bsearch2.0ms