Average Error: 0.2 → 0.2
Time: 16.7s
Precision: 64
Internal Precision: 128
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\]

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. Simplified0.2

    \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}}\]
  3. Using strategy rm

  4. Applied tan-quot0.2

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

  5. Applied associate-/r/0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019005 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))

Details

Time bar (total: 15.6s)Debug log

sample236.0ms

Algorithm
intervals

simplify319.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

318.0ms
(+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B)))

prune11.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize22.0ms

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)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
3 → 46
Calls

3 calls. Slowest were:

5.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
1.0ms
(/ 1 (sin B))
1.0ms
(/ x (tan B))

series233.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

128.0ms
(/ x (tan B))
85.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
20.0ms
(/ 1 (sin B))

simplify180.0ms

Counts
24 → 55
Calls

24 calls. Slowest were:

25.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
24.0ms
(+ (* 1/6 B) (+ (/ 1 B) (* 7/360 (pow B 3))))
24.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))

prune405.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize21.0ms

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))

series298.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

124.0ms
(* (/ x (sin B)) (cos B))
83.0ms
(- (/ 1 (sin B)) (* (/ x (sin B)) (cos B)))
77.0ms
(/ x (sin B))
13.0ms
(/ 1 (sin B))

simplify628.0ms

Counts
33 → 74
Calls

33 calls. Slowest were:

180.0ms
(- (* 1 (sin B)) (* (sin B) (* x (cos B))))
126.0ms
(* (* (* (/ x (sin B)) (/ x (sin B))) (/ x (sin B))) (* (* (cos B) (cos B)) (cos B)))
48.0ms
(+ (* 1/6 B) (+ (/ 1 B) (* 7/360 (pow B 3))))

prune696.0ms

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 2 expressions with local error:

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

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 41
Calls

2 calls. Slowest were:

3.0ms
(/ (- 1 (* x (cos B))) (sin B))
1.0ms
(* x (cos B))

series113.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

79.0ms
(/ (- 1 (* x (cos B))) (sin B))
34.0ms
(* x (cos B))

simplify1.1s

Counts
37 → 47
Calls

37 calls. Slowest were:

308.0ms
(- (pow 1 3) (pow (* x (cos B)) 3))
198.0ms
(/ (* (* (- 1 (* x (cos B))) (- 1 (* x (cos B)))) (- 1 (* x (cos B)))) (* (* (sin B) (sin B)) (sin B)))
54.0ms
(/ (- 1 (* x (cos B))) (sin B))

prune562.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 4 expressions with local error:

0.3b
(cbrt (cos B))
0.3b
(cbrt (cos B))
0.3b
(cbrt (cos B))
0.3b
(* (cbrt (cos B)) (cbrt (cos B)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

5.0ms
(* (cbrt (cos B)) (cbrt (cos B)))
1.0ms
(cbrt (cos B))
1.0ms
(cbrt (cos B))

series740.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

255.0ms
(* (cbrt (cos B)) (cbrt (cos B)))
190.0ms
(cbrt (cos B))
160.0ms
(cbrt (cos B))
135.0ms
(cbrt (cos B))

simplify374.0ms

Counts
41 → 73
Calls

41 calls. Slowest were:

95.0ms
(- 1 (+ (* 1/72 (pow B 4)) (* 1/6 (pow B 2))))
89.0ms
(- 1 (+ (* 1/405 (pow B 6)) (* 1/3 (pow B 2))))
69.0ms
(- 1 (+ (* 1/72 (pow B 4)) (* 1/6 (pow B 2))))

prune796.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.0b

regimes140.0ms

Accuracy

0% (0.2b remaining)

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

bsearch2.0ms

end0.0ms

sample8.7s

Algorithm
intervals