Average Error: 0.2 → 0.2
Time: 21.9s
Precision: 64
Internal Precision: 128
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\]

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. Final simplification0.2

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

Reproduce

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

Details

Time bar (total: 6.6s)Debug log

start341.0ms

Algorithm
intervals

setup98.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize30.0ms

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)))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
3 → 46
Calls

3 calls. Slowest were:

10.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
1.0ms
(/ 1 (sin B))
1.0ms
(/ x (tan B))

series197.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

95.0ms
(/ x (tan B))
85.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
16.0ms
(/ 1 (sin B))

simplify279.0ms

Counts
24 → 55
Calls

24 calls. Slowest were:

44.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
42.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
33.0ms
(+ (* 1/6 B) (+ (/ 1 B) (* 7/360 (pow B 3))))

prune459.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

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)))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
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))

series209.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

86.0ms
(/ (* x (cos B)) (sin B))
80.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
30.0ms
(* x (cos B))
13.0ms
(/ 1 (sin B))

simplify784.0ms

Counts
43 → 77
Calls

43 calls. Slowest were:

358.0ms
(/ (* (* (* x (cos B)) (* x (cos B))) (* x (cos B))) (* (* (sin B) (sin B)) (sin B)))
112.0ms
(- (* 1 (sin B)) (* (sin B) (* x (cos B))))
36.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))

prune713.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.0b

localize17.0ms

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)))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 62
Calls

4 calls. Slowest were:

13.0ms
(- (/ 1 (sin B)) (* (/ x (sin B)) (cos B)))
7.0ms
(* (/ x (sin B)) (cos B))
2.0ms
(/ 1 (sin B))

series333.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

128.0ms
(- (/ 1 (sin B)) (* (/ x (sin B)) (cos B)))
124.0ms
(* (/ x (sin B)) (cos B))
61.0ms
(/ x (sin B))
20.0ms
(/ 1 (sin B))

simplify551.0ms

Counts
33 → 74
Calls

33 calls. Slowest were:

134.0ms
(- (* 1 (sin B)) (* (sin B) (* x (cos B))))
100.0ms
(* (* (* (/ x (sin B)) (/ x (sin B))) (/ x (sin B))) (* (* (cos B) (cos B)) (cos B)))
48.0ms
(/ (exp (/ 1 (sin B))) (exp (* (/ x (sin B)) (cos B))))

prune566.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

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

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

5.0ms
(* (/ 1 (/ (sin B) x)) (cos B))
2.0ms
(/ (sin B) x)
1.0ms
(/ 1 (/ (sin B) x))

series147.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

66.0ms
(* (/ 1 (/ (sin B) x)) (cos B))
42.0ms
(/ 1 (/ (sin B) x))
26.0ms
(/ (sin B) x)
13.0ms
(/ 1 (sin B))

simplify771.0ms

Counts
34 → 80
Calls

34 calls. Slowest were:

246.0ms
(- (+ (/ B x) (* 1/120 (/ (pow B 5) x))) (* 1/6 (/ (pow B 3) x)))
243.0ms
(* (* (* (/ 1 (/ (sin B) x)) (/ 1 (/ (sin B) x))) (/ 1 (/ (sin B) x))) (* (* (cos B) (cos B)) (cos B)))
60.0ms
(+ (log (/ 1 (/ (sin B) x))) (log (cos B)))

prune886.0ms

Pruning

9 alts after pruning (7 fresh and 2 done)

Merged error: 0.0b

regimes145.0ms

Accuracy

0% (0.2b remaining)

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

bsearch1.0ms