Average Error: 0.2 → 0.2
Time: 1.1m
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. Initial simplification0.2

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

  3. Taylor expanded around -inf 0.2

    \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}}\]
  4. Using strategy rm

  5. Applied sub-div0.2

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

  6. Final simplification0.2

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

Reproduce

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

Details

Time bar (total: 50.7s)Debug log

start662.0ms

Algorithm
intervals

setup126.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize19.0ms

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)))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
3 → 76
Calls

3 calls. Slowest were:

6.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
2.0ms
(/ x (tan B))
1.0ms
(/ 1 (sin B))

series313.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

151.0ms
(/ x (tan B))
137.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
24.0ms
(/ 1 (sin B))

simplify1.7s

Counts
66 → 85
Calls

66 calls. Slowest were:

80.0ms
(fma (sqrt (/ 1 (sin B))) (sqrt (/ 1 (sin B))) (- (* (/ 1 (tan B)) x)))
78.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (tan B)) x)))
60.0ms
(fma 1 (/ 1 (sin B)) (- (* (cos B) (/ x (sin B)))))

prune1.1s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

localize14.0ms

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)))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 105
Calls

4 calls. Slowest were:

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

series300.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

117.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
117.0ms
(/ (* x (cos B)) (sin B))
41.0ms
(* x (cos B))
24.0ms
(/ 1 (sin B))

simplify9.1s

Counts
101 → 117
Calls

101 calls. Slowest were:

582.0ms
(/ (* (* (* x (cos B)) (* x (cos B))) (* x (cos B))) (* (* (sin B) (sin B)) (sin B)))
499.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (sin B)) (* x (cos B)))))
417.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (sin B)) (* x (cos B)))))

prune1.4s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize5.0ms

Local error

Found 2 expressions with local error:

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
2 → 45
Calls

2 calls. Slowest were:

3.0ms
(/ (- 1 (* x (cos B))) (sin B))
2.0ms
(* x (cos B))

series132.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

81.0ms
(/ (- 1 (* x (cos B))) (sin B))
50.0ms
(* x (cos B))

simplify1.1s

Counts
37 → 51
Calls

37 calls. Slowest were:

402.0ms
(- (pow 1 3) (pow (* x (cos B)) 3))
163.0ms
(/ (* (* (- 1 (* x (cos B))) (- 1 (* x (cos B)))) (- 1 (* x (cos B)))) (* (* (sin B) (sin B)) (sin B)))
48.0ms
(/ (- 1 (* x (cos B))) (cbrt (sin B)))

prune641.0ms

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 4 expressions with local error:

0.3b
(/ (cos B) (/ (sin B) x))
0.2b
(/ (sin B) x)
0.1b
(/ 1 (sin B))
0.0b
(- (/ 1 (sin B)) (/ (cos B) (/ (sin B) x)))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 158
Calls

4 calls. Slowest were:

13.0ms
(- (/ 1 (sin B)) (/ (cos B) (/ (sin B) x)))
4.0ms
(/ (cos B) (/ (sin B) x))
2.0ms
(/ (sin B) x)

series275.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

113.0ms
(/ (cos B) (/ (sin B) x))
111.0ms
(- (/ 1 (sin B)) (/ (cos B) (/ (sin B) x)))
38.0ms
(/ (sin B) x)
13.0ms
(/ 1 (sin B))

simplify31.6s

Counts
197 → 170
Calls

197 calls. Slowest were:

840.0ms
(fma (- (/ (cbrt (cos B)) (/ (sin B) x))) (/ (* (cbrt (cos B)) (cbrt (cos B))) 1) (* (/ (cbrt (cos B)) (/ (sin B) x)) (/ (* (cbrt (cos B)) (cbrt (cos B))) 1)))
670.0ms
(fma (* (cbrt (/ 1 (sin B))) (cbrt (/ 1 (sin B)))) (cbrt (/ 1 (sin B))) (- (* (/ (cos B) (/ 1 x)) (/ 1 (sin B)))))
602.0ms
(fma (sqrt (/ 1 (sin B))) (sqrt (/ 1 (sin B))) (- (* (/ (sqrt (cos B)) (/ 1 x)) (/ (sqrt (cos B)) (sin B)))))

prune2.0s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.0b

regimes108.0ms

Accuracy

0% (0.2b remaining)

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

bsearch1.0ms