Average Error: 15.1 → 0.4
Time: 29.2s
Precision: 64
Internal Precision: 128
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{1}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot \left(r \cdot \sin b\right)\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

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Derivation

  1. Initial program 15.1

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

  2. Initial simplification15.1

    \[\leadsto \frac{r \cdot \sin b}{\cos \left(b + a\right)}\]
  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 div-inv0.4

    \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos b \cdot \cos a - \sin b \cdot \sin a}}\]

  7. Final simplification0.4

    \[\leadsto \frac{1}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot \left(r \cdot \sin b\right)\]

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))

Details

Time bar (total: 17.1s)Debug log

start272.0ms

Algorithm
intervals

setup146.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 15.7b

localize85.0ms

Local error

Found 3 expressions with local error:

2.3b
(cos (+ b a))
0.2b
(/ (* r (sin b)) (cos (+ b a)))
0.1b
(* r (sin b))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
3 → 44
Calls

3 calls. Slowest were:

11.0ms
(/ (* r (sin b)) (cos (+ b a)))
7.0ms
(cos (+ b a))
3.0ms
(* r (sin b))

series458.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

356.0ms
(/ (* r (sin b)) (cos (+ b a)))
67.0ms
(* r (sin b))
33.0ms
(cos (+ b a))

simplify1.0s

Counts
24 → 53
Calls

24 calls. Slowest were:

440.0ms
(/ (* (* (* r (sin b)) (* r (sin b))) (* r (sin b))) (* (* (cos (+ b a)) (cos (+ b a))) (cos (+ b a))))
256.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
169.0ms
(- 1 (+ (* 1/2 (pow b 2)) (* a b)))

prune705.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.2b

localize25.0ms

Local error

Found 4 expressions with local error:

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

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
4 → 83
Calls

4 calls. Slowest were:

10.0ms
(/ (* r (sin b)) (- (* (cos b) (cos a)) (* (sin b) (sin a))))
7.0ms
(- (* (cos b) (cos a)) (* (sin b) (sin a)))
7.0ms
(* (cos b) (cos a))

series480.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

257.0ms
(/ (* r (sin b)) (- (* (cos b) (cos a)) (* (sin b) (sin a))))
90.0ms
(- (* (cos b) (cos a)) (* (sin b) (sin a)))
74.0ms
(* (sin b) (sin a))
58.0ms
(* (cos b) (cos a))

simplify2.3s

Counts
51 → 95
Calls

51 calls. Slowest were:

441.0ms
(- (* (+ (cos (+ b a)) (cos (- b a))) 2) (* 2 (- (cos (- b a)) (cos (+ b a)))))
382.0ms
(- (+ (cos (+ b a)) (cos (- b a))) (- (cos (- b a)) (cos (+ b a))))
303.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))

prune1.4s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 4 expressions with local error:

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

rewrite63.0ms

Algorithm
rewrite-expression-head
Counts
4 → 99
Calls

4 calls. Slowest were:

23.0ms
(* r (/ (sin b) (- (* (cos b) (cos a)) (* (sin b) (sin a)))))
18.0ms
(/ (sin b) (- (* (cos b) (cos a)) (* (sin b) (sin a))))
14.0ms
(- (* (cos b) (cos a)) (* (sin b) (sin a)))

series512.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

223.0ms
(* r (/ (sin b) (- (* (cos b) (cos a)) (* (sin b) (sin a)))))
158.0ms
(/ (sin b) (- (* (cos b) (cos a)) (* (sin b) (sin a))))
77.0ms
(- (* (cos b) (cos a)) (* (sin b) (sin a)))
53.0ms
(* (sin b) (sin a))

simplify2.8s

Counts
73 → 111
Calls

73 calls. Slowest were:

412.0ms
(- (+ (cos (+ b a)) (cos (- b a))) (- (cos (- b a)) (cos (+ b a))))
330.0ms
(- (* (+ (cos (+ b a)) (cos (- b a))) 2) (* 2 (- (cos (- b a)) (cos (+ b a)))))
205.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))

prune1.7s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize24.0ms

Local error

Found 4 expressions with local error:

0.2b
(/ (sin b) (- (* (cos b) (cos a)) (expm1 (log1p (* (sin b) (sin a))))))
0.2b
(* (sin b) (sin a))
0.2b
(* r (/ (sin b) (- (* (cos b) (cos a)) (expm1 (log1p (* (sin b) (sin a)))))))
0.1b
(- (* (cos b) (cos a)) (expm1 (log1p (* (sin b) (sin a)))))

rewrite56.0ms

Algorithm
rewrite-expression-head
Counts
4 → 96
Calls

4 calls. Slowest were:

18.0ms
(/ (sin b) (- (* (cos b) (cos a)) (expm1 (log1p (* (sin b) (sin a))))))
17.0ms
(* r (/ (sin b) (- (* (cos b) (cos a)) (expm1 (log1p (* (sin b) (sin a)))))))
11.0ms
(- (* (cos b) (cos a)) (expm1 (log1p (* (sin b) (sin a)))))

series753.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

316.0ms
(* r (/ (sin b) (- (* (cos b) (cos a)) (expm1 (log1p (* (sin b) (sin a)))))))
204.0ms
(/ (sin b) (- (* (cos b) (cos a)) (expm1 (log1p (* (sin b) (sin a))))))
158.0ms
(- (* (cos b) (cos a)) (expm1 (log1p (* (sin b) (sin a)))))
76.0ms
(* (sin b) (sin a))

simplify2.2s

Counts
72 → 108
Calls

72 calls. Slowest were:

251.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
178.0ms
(+ (* (* (cos b) (cos a)) (* (cos b) (cos a))) (+ (* (expm1 (log1p (* (sin b) (sin a)))) (expm1 (log1p (* (sin b) (sin a))))) (* (* (cos b) (cos a)) (expm1 (log1p (* (sin b) (sin a)))))))
167.0ms
(+ (* (* (cos b) (cos a)) (* (cos b) (cos a))) (+ (* (expm1 (log1p (* (sin b) (sin a)))) (expm1 (log1p (* (sin b) (sin a))))) (* (* (cos b) (cos a)) (expm1 (log1p (* (sin b) (sin a)))))))

prune1.8s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

regimes347.0ms

Accuracy

0% (0.3b remaining)

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

bsearch4.0ms