Average Error: 0.2 → 0.2
Time: 24.8s
Precision: 64
Internal Precision: 128
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\]

Error

Bits error versus B

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]

  2. Initial simplification0.2

    \[\leadsto \frac{1}{\sin B} - \frac{x}{\tan B}\]

  3. Taylor expanded around inf 0.2

    \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x \cdot \cos B}{\sin B}}\]

  4. Final simplification0.2

    \[\leadsto \frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\]

Reproduce

herbie shell --seed 2018365 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))

Details

Time bar (total: 9.5s)Debug log

start708.0ms

Algorithm
intervals

setup941.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize42.0ms

Local error

Found 3 expressions with local error:

0.1b
(/ x (tan B))
0.1b
(/ 1 (sin B))
0.0b
(- (/ 1 (sin B)) (/ x (tan B)))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
3 → 46
Calls

3 calls. Slowest were:

11.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
3.0ms
(/ x (tan B))
2.0ms
(/ 1 (sin B))

series511.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

322.0ms
(/ x (tan B))
162.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
27.0ms
(/ 1 (sin B))

simplify262.0ms

Counts
24 → 55
Calls

24 calls. Slowest were:

52.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
29.0ms
(/ (exp (/ 1 (sin B))) (exp (/ x (tan B))))
26.0ms
(+ (* 1/6 B) (+ (/ 1 B) (* 7/360 (pow B 3))))

prune674.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 4 expressions with local error:

0.2b
(/ (* x (cos B)) (sin B))
0.1b
(* x (cos B))
0.1b
(/ 1 (sin B))
0.0b
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

10.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
7.0ms
(/ (* x (cos B)) (sin B))
2.0ms
(* x (cos B))

series306.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

126.0ms
(/ (* x (cos B)) (sin B))
122.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
38.0ms
(* x (cos B))
20.0ms
(/ 1 (sin B))

simplify1.1s

Counts
43 → 77
Calls

43 calls. Slowest were:

517.0ms
(/ (* (* (* x (cos B)) (* x (cos B))) (* x (cos B))) (* (* (sin B) (sin B)) (sin B)))
195.0ms
(- (* 1 (sin B)) (* (sin B) (* x (cos B))))
52.0ms
(+ (* 1/6 B) (+ (/ 1 B) (* 7/360 (pow B 3))))

prune873.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (/ x (sin B)) (cos B))
0.1b
(/ 1 (sin B))
0.1b
(/ x (sin B))
0.0b
(- (/ 1 (sin B)) (* (/ x (sin B)) (cos B)))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 62
Calls

4 calls. Slowest were:

6.0ms
(- (/ 1 (sin B)) (* (/ x (sin B)) (cos B)))
4.0ms
(* (/ x (sin B)) (cos B))
1.0ms
(/ 1 (sin B))

series289.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

117.0ms
(* (/ x (sin B)) (cos B))
87.0ms
(- (/ 1 (sin B)) (* (/ x (sin B)) (cos B)))
62.0ms
(/ x (sin B))
23.0ms
(/ 1 (sin B))

simplify671.0ms

Counts
33 → 74
Calls

33 calls. Slowest were:

158.0ms
(- (* 1 (sin B)) (* (sin B) (* x (cos B))))
139.0ms
(* (* (* (/ x (sin B)) (/ x (sin B))) (/ x (sin B))) (* (* (cos B) (cos B)) (cos B)))
51.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))

prune808.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 4 expressions with local error:

0.3b
(/ 1 (/ (sin B) x))
0.2b
(* (/ 1 (/ (sin B) x)) (cos B))
0.2b
(/ (sin B) x)
0.1b
(/ 1 (sin B))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

7.0ms
(* (/ 1 (/ (sin B) x)) (cos B))
2.0ms
(/ (sin B) x)
1.0ms
(/ 1 (/ (sin B) x))

series171.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

80.0ms
(* (/ 1 (/ (sin B) x)) (cos B))
52.0ms
(/ 1 (/ (sin B) x))
26.0ms
(/ (sin B) x)
14.0ms
(/ 1 (sin B))

simplify886.0ms

Counts
34 → 80
Calls

34 calls. Slowest were:

255.0ms
(* (* (* (/ 1 (/ (sin B) x)) (/ 1 (/ (sin B) x))) (/ 1 (/ (sin B) x))) (* (* (cos B) (cos B)) (cos B)))
246.0ms
(- (+ (/ B x) (* 1/120 (/ (pow B 5) x))) (* 1/6 (/ (pow B 3) x)))
88.0ms
(* (/ 1 (/ (sin B) x)) (cos B))

prune893.0ms

Pruning

9 alts after pruning (7 fresh and 2 done)

Merged error: 0.0b

regimes205.0ms

Accuracy

0% (0.2b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 0.2b

bsearch2.0ms