Average Error: 0.2 → 0.2
Time: 25.3s
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. 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 2019004 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))

Details

Time bar (total: 24.1s)Debug log

sample195.0ms

Algorithm
intervals

simplify109.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

109.0ms
(+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B)))

prune18.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize35.0ms

Local error

Found 3 expressions with local error:

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

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
3 → 76
Calls

3 calls. Slowest were:

11.0ms
(- (/ 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:

126.0ms
(/ x (tan B))
101.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
21.0ms
(/ 1 (sin B))

simplify1.6s

Counts
66 → 85
Calls

66 calls. Slowest were:

79.0ms
(fma (* (cbrt (/ 1 (sin B))) (cbrt (/ 1 (sin B)))) (cbrt (/ 1 (sin B))) (- (* (/ 1 (tan B)) x)))
72.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (tan B)) x)))
72.0ms
(fma (sqrt (/ 1 (sin B))) (sqrt (/ 1 (sin B))) (- (* (/ 1 (tan B)) x)))

prune967.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 3 expressions with local error:

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

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
3 → 63
Calls

3 calls. Slowest were:

3.0ms
(/ (- x) (sin B))
1.0ms
(/ 1 (sin B))
0.0ms
(fma (cos B) (/ (- x) (sin B)) (/ 1 (sin B)))

series174.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

92.0ms
(fma (cos B) (/ (- x) (sin B)) (/ 1 (sin B)))
66.0ms
(/ (- x) (sin B))
16.0ms
(/ 1 (sin B))

simplify389.0ms

Counts
46 → 72
Calls

46 calls. Slowest were:

144.0ms
(/ (* (* (- x) (- x)) (- x)) (* (* (sin B) (sin B)) (sin B)))
39.0ms
(+ (* 1/6 B) (+ (/ 1 B) (* 7/360 (pow B 3))))
28.0ms
(- (+ (/ x B) (* 1/6 (* x B))))

prune709.0ms

Pruning

10 alts after pruning (9 fresh and 1 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)))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 105
Calls

4 calls. Slowest were:

11.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
7.0ms
(/ (* x (cos B)) (sin B))
2.0ms
(* x (cos B))

series289.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

130.0ms
(/ (* x (cos B)) (sin B))
98.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
41.0ms
(* x (cos B))
19.0ms
(/ 1 (sin B))

simplify7.9s

Counts
101 → 117
Calls

101 calls. Slowest were:

438.0ms
(/ (* (* (* x (cos B)) (* x (cos B))) (* x (cos B))) (* (* (sin B) (sin B)) (sin B)))
407.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (sin B)) (* x (cos B)))))
395.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (sin B)) (* x (cos B)))))

prune1.2s

Pruning

8 alts after pruning (7 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))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
2 → 45
Calls

2 calls. Slowest were:

6.0ms
(/ (- 1 (* x (cos B))) (sin B))
2.0ms
(* x (cos B))

series110.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

72.0ms
(/ (- 1 (* x (cos B))) (sin B))
38.0ms
(* x (cos B))

simplify1.0s

Counts
37 → 51
Calls

37 calls. Slowest were:

307.0ms
(- (pow 1 3) (pow (* x (cos B)) 3))
168.0ms
(/ (* (* (- 1 (* x (cos B))) (- 1 (* x (cos B)))) (- 1 (* x (cos B)))) (* (* (sin B) (sin B)) (sin B)))
47.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))

prune437.0ms

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0.0b

regimes118.0ms

Accuracy

0% (0.1b remaining)

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

bsearch2.0ms

end0.0ms

sample8.6s

Algorithm
intervals