Average Error: 0.2 → 0.2
Time: 18.3s
Precision: 64
Internal Precision: 128
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1 - \cos B \cdot x}{\sin B}\]

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. 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. Using strategy rm

  7. Applied associate-*l/0.2

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

  8. Applied sub-div0.2

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

  9. Final simplification0.2

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

Reproduce

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

Details

Time bar (total: 17.1s)Debug log

sample235.0ms

Algorithm
intervals

simplify306.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

306.0ms
(+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B)))

prune11.0ms

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)))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
3 → 76
Calls

3 calls. Slowest were:

7.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
1.0ms
(/ 1 (sin B))
1.0ms
(/ x (tan B))

series248.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

142.0ms
(/ x (tan B))
89.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
17.0ms
(/ 1 (sin B))

simplify1.5s

Counts
66 → 85
Calls

66 calls. Slowest were:

66.0ms
(fma (sqrt (/ 1 (sin B))) (sqrt (/ 1 (sin B))) (- (* (/ 1 (tan B)) x)))
56.0ms
(fma (* (cbrt (/ 1 (sin B))) (cbrt (/ 1 (sin B)))) (cbrt (/ 1 (sin B))) (- (* (cos B) (/ x (sin B)))))
53.0ms
(fma (* (cbrt (/ 1 (sin B))) (cbrt (/ 1 (sin B)))) (cbrt (/ 1 (sin B))) (- (* (/ x (tan B)) 1)))

prune553.0ms

Pruning

11 alts after pruning (11 fresh and 0 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)))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 78
Calls

4 calls. Slowest were:

8.0ms
(- (/ 1 (sin B)) (* (/ x (sin B)) (cos B)))
4.0ms
(* (/ x (sin B)) (cos B))
1.0ms
(/ 1 (sin B))

series279.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

108.0ms
(* (/ x (sin B)) (cos B))
91.0ms
(- (/ 1 (sin B)) (* (/ x (sin B)) (cos B)))
67.0ms
(/ x (sin B))
14.0ms
(/ 1 (sin B))

simplify868.0ms

Counts
43 → 90
Calls

43 calls. Slowest were:

234.0ms
(- (* 1 (sin B)) (* (sin B) (* x (cos B))))
107.0ms
(* (* (* (/ x (sin B)) (/ x (sin B))) (/ x (sin B))) (* (* (cos B) (cos B)) (cos B)))
42.0ms
(+ (* 1/6 B) (+ (/ 1 B) (* 7/360 (pow B 3))))

prune675.0ms

Pruning

11 alts after pruning (10 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 → 45
Calls

2 calls. Slowest were:

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

series108.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

75.0ms
(/ (- 1 (* x (cos B))) (sin B))
33.0ms
(* x (cos B))

simplify895.0ms

Counts
37 → 51
Calls

37 calls. Slowest were:

240.0ms
(- (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)))
36.0ms
(- (* 1 1) (* (* x (cos B)) (* x (cos B))))

prune524.0ms

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 4 expressions with local error:

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

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 78
Calls

4 calls. Slowest were:

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

series300.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

142.0ms
(* x (/ 1 (tan B)))
107.0ms
(- (/ 1 (sin B)) (* x (/ 1 (tan B))))
36.0ms
(/ 1 (tan B))
15.0ms
(/ 1 (sin B))

simplify1.0s

Counts
43 → 90
Calls

43 calls. Slowest were:

198.0ms
(- (/ 1 B) (+ (* 1/3 B) (* 1/45 (pow B 3))))
89.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (tan B)) x)))
75.0ms
(fma (* (cbrt (/ 1 (sin B))) (cbrt (/ 1 (sin B)))) (cbrt (/ 1 (sin B))) (- (* (/ 1 (tan B)) x)))

prune774.0ms

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 0.0b

regimes121.0ms

Accuracy

0% (0.2b remaining)

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

bsearch1.0ms

end0.0ms

sample8.6s

Algorithm
intervals