Average Error: 0.2 → 0.2
Time: 14.7s
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 2019002 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))

Details

Time bar (total: 13.8s)Debug log

sample182.0ms

Algorithm
intervals

simplify46.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

46.0ms
(+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B)))

prune269.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize16.0ms

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))

series226.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

124.0ms
(/ x (tan B))
88.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
14.0ms
(/ 1 (sin B))

simplify174.0ms

Counts
24 → 55
Calls

24 calls. Slowest were:

23.0ms
(+ (* 1/6 B) (+ (/ 1 B) (* 7/360 (pow B 3))))
23.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
23.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))

prune384.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

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

series231.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

89.0ms
(* (/ x (sin B)) (cos B))
79.0ms
(- (/ 1 (sin B)) (* (/ x (sin B)) (cos B)))
49.0ms
(/ x (sin B))
13.0ms
(/ 1 (sin B))

simplify477.0ms

Counts
33 → 74
Calls

33 calls. Slowest were:

141.0ms
(- (* 1 (sin B)) (* (sin B) (* x (cos B))))
102.0ms
(* (* (* (/ x (sin B)) (/ x (sin B))) (/ x (sin B))) (* (* (cos B) (cos B)) (cos B)))
33.0ms
(- (/ x B) (* 1/3 (* x B)))

prune568.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

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

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

5.0ms
(- (/ 1 (sin B)) (/ 1 (/ (tan B) x)))
2.0ms
(/ (tan B) x)
1.0ms
(/ 1 (/ (tan B) x))

series284.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

99.0ms
(- (/ 1 (sin B)) (/ 1 (/ (tan B) x)))
94.0ms
(/ 1 (/ (tan B) x))
68.0ms
(/ (tan B) x)
23.0ms
(/ 1 (sin B))

simplify424.0ms

Counts
33 → 78
Calls

33 calls. Slowest were:

61.0ms
(+ (/ B x) (+ (* 1/3 (/ (pow B 3) x)) (* 2/15 (/ (pow B 5) x))))
31.0ms
(/ (exp (/ 1 (sin B))) (exp (/ 1 (/ (tan B) x))))
30.0ms
(- (/ 1 (sin B)) (/ 1 (/ (tan B) x)))

prune623.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (sin B))
0.6b
(cbrt (sin B))
0.6b
(cbrt (sin B))
0.5b
(* (cbrt (sin B)) (cbrt (sin B)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

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

series607.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

178.0ms
(* (cbrt (sin B)) (cbrt (sin B)))
172.0ms
(cbrt (sin B))
129.0ms
(cbrt (sin B))
128.0ms
(cbrt (sin B))

simplify378.0ms

Counts
41 → 73
Calls

41 calls. Slowest were:

89.0ms
(- (pow B 1/3) (+ (* 1/18 (pow (pow B 7) 1/3)) (* 1/3240 (pow (pow B 13) 1/3))))
88.0ms
(- (pow B 1/3) (+ (* 1/18 (pow (pow B 7) 1/3)) (* 1/3240 (pow (pow B 13) 1/3))))
86.0ms
(- (pow B 1/3) (+ (* 1/18 (pow (pow B 7) 1/3)) (* 1/3240 (pow (pow B 13) 1/3))))

prune776.0ms

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 0.0b

regimes118.0ms

Accuracy

0% (0.2b remaining)

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

bsearch1.0ms

end0.0ms

sample7.9s

Algorithm
intervals