Average Error: 15.3 → 0.4
Time: 23.2s
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

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

Details

Time bar (total: 21.3s)Debug log

sample290.0ms

Algorithm
intervals

simplify77.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

77.0ms
(* r (/ (sin b) (cos (+ a b))))

prune14.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 13.4b

localize36.0ms

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))))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
3 → 57
Calls

3 calls. Slowest were:

5.0ms
(cos (+ a b))
4.0ms
(/ (sin b) (cos (+ a b)))
2.0ms
(* r (/ (sin b) (cos (+ a b))))

series158.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

94.0ms
(* r (/ (sin b) (cos (+ a b))))
34.0ms
(/ (sin b) (cos (+ a b)))
30.0ms
(cos (+ a b))

simplify629.0ms

Counts
49 → 66
Calls

49 calls. Slowest were:

141.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
119.0ms
(+ b (+ (* 1/3 (pow b 3)) (* a (pow b 2))))
91.0ms
(/ (* (* (sin b) (sin b)) (sin b)) (* (* (cos (+ a b)) (cos (+ a b))) (cos (+ a b))))

prune839.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.2b

localize47.0ms

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)))

rewrite53.0ms

Algorithm
rewrite-expression-head
Counts
4 → 89
Calls

4 calls. Slowest were:

21.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
17.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
7.0ms
(* (sin a) (sin b))

series500.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

224.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
124.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
96.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
56.0ms
(* (sin a) (sin b))

simplify2.1s

Counts
70 → 101
Calls

70 calls. Slowest were:

330.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
284.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
198.0ms
(/ (* (* (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)))))

prune1.3s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.0b

localize28.0ms

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)))))))

rewrite53.0ms

Algorithm
rewrite-expression-head
Counts
4 → 85
Calls

4 calls. Slowest were:

21.0ms
(/ (sin b) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b))))))
20.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b)))))))
7.0ms
(* (sin a) (sin b))

series500.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

228.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b)))))))
159.0ms
(/ (sin b) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b))))))
57.0ms
(* (sin a) (sin b))
56.0ms
(log (exp (* (sin a) (sin b))))

simplify2.4s

Counts
70 → 97
Calls

70 calls. Slowest were:

250.0ms
(+ (* (* (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)))))))
203.0ms
(+ (* (* (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)))))))
201.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))

prune1.5s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 4 expressions with local error:

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

rewrite47.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

21.0ms
(/ (* (sin b) r) (- (* (cos a) (cos b)) (* (sin b) (sin a))))
13.0ms
(- (* (cos a) (cos b)) (* (sin b) (sin a)))
7.0ms
(* (sin b) (sin a))

series462.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

281.0ms
(/ (* (sin b) r) (- (* (cos a) (cos b)) (* (sin b) (sin a))))
107.0ms
(- (* (cos a) (cos b)) (* (sin b) (sin a)))
44.0ms
(* (sin b) (sin a))
29.0ms
(* (sin b) r)

simplify1.3s

Counts
42 → 78
Calls

42 calls. Slowest were:

272.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- b a)) (cos (+ b a)))))
184.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
156.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- b a)) (cos (+ b a))))

prune935.0ms

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.0b

regimes385.0ms

Accuracy

0% (0.3b remaining)

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

bsearch2.0ms

end0.0ms

sample7.7s

Algorithm
intervals