Average Error: 15.3 → 0.4
Time: 26.6s
Precision: 64
Internal Precision: 128
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\sin b}{\cos a \cdot \cos b - \log \left(e^{\sin a \cdot \sin b}\right)} \cdot r\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.3

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

  3. Applied cos-sum0.3

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

  5. Applied add-log-exp0.4

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

  6. Final simplification0.4

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

Reproduce

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

Details

Time bar (total: 24.7s)Debug log

sample242.0ms

Algorithm
intervals

simplify34.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune14.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 13.4b

localize35.0ms

Local error

Found 3 expressions with local error:

4.2b
(cos (+ a b))
0.2b
(/ (sin b) (cos (+ a b)))
0.2b
(* r (/ (sin b) (cos (+ a b))))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
3 → 63
Calls

3 calls. Slowest were:

7.0ms
(/ (sin b) (cos (+ a b)))
5.0ms
(cos (+ a b))
5.0ms
(* r (/ (sin b) (cos (+ a b))))

series227.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

138.0ms
(* r (/ (sin b) (cos (+ a b))))
45.0ms
(cos (+ a b))
43.0ms
(/ (sin b) (cos (+ a b)))

simplify909.0ms

Counts
49 → 72
Calls

49 calls. Slowest were:

237.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
200.0ms
(+ b (+ (* 1/3 (pow b 3)) (* a (pow b 2))))
124.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))

prune738.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.2b

localize24.0ms

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)))

rewrite51.0ms

Algorithm
rewrite-expression-head
Counts
4 → 99
Calls

4 calls. Slowest were:

20.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
14.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
10.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))

series550.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

279.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
128.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
96.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
48.0ms
(* (sin a) (sin b))

simplify2.9s

Counts
73 → 111
Calls

73 calls. Slowest were:

365.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
313.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
245.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))

prune1.7s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 4 expressions with local error:

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

rewrite40.0ms

Algorithm
rewrite-expression-head
Counts
4 → 93
Calls

4 calls. Slowest were:

20.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b)))))))
11.0ms
(/ (sin b) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b))))))
5.0ms
(* (sin a) (sin b))

series504.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

273.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b)))))))
108.0ms
(/ (sin b) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b))))))
78.0ms
(log (exp (* (sin a) (sin b))))
45.0ms
(* (sin a) (sin b))

simplify2.8s

Counts
70 → 105
Calls

70 calls. Slowest were:

388.0ms
(+ (* (* (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)))))))
281.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
257.0ms
(+ (* (* (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)))))))

prune1.6s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (* r (sin b)) (/ 1 (- (* (cos b) (cos a)) (* (sin a) (sin b)))))
0.2b
(* (sin a) (sin b))
0.2b
(/ 1 (- (* (cos b) (cos a)) (* (sin a) (sin b))))
0.1b
(- (* (cos b) (cos a)) (* (sin a) (sin b)))

rewrite59.0ms

Algorithm
rewrite-expression-head
Counts
4 → 85
Calls

4 calls. Slowest were:

18.0ms
(/ 1 (- (* (cos b) (cos a)) (* (sin a) (sin b))))
17.0ms
(* (* r (sin b)) (/ 1 (- (* (cos b) (cos a)) (* (sin a) (sin b)))))
14.0ms
(- (* (cos b) (cos a)) (* (sin a) (sin b)))

series506.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

242.0ms
(* (* r (sin b)) (/ 1 (- (* (cos b) (cos a)) (* (sin a) (sin b)))))
130.0ms
(/ 1 (- (* (cos b) (cos a)) (* (sin a) (sin b))))
89.0ms
(- (* (cos b) (cos a)) (* (sin a) (sin b)))
46.0ms
(* (sin a) (sin b))

simplify2.6s

Counts
48 → 97
Calls

48 calls. Slowest were:

399.0ms
(- (* (+ (cos (+ b a)) (cos (- b a))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
324.0ms
(* (* (* (* r (sin b)) (* r (sin b))) (* r (sin b))) (* (* (/ 1 (- (* (cos b) (cos a)) (* (sin a) (sin b)))) (/ 1 (- (* (cos b) (cos a)) (* (sin a) (sin b))))) (/ 1 (- (* (cos b) (cos a)) (* (sin a) (sin b))))))
260.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))

prune1.2s

Pruning

12 alts after pruning (10 fresh and 2 done)

Merged error: 0.0b

regimes376.0ms

Accuracy

0% (0.3b remaining)

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

bsearch3.0ms

end0.0ms

sample7.6s

Algorithm
intervals