Average Error: 0.2 → 0.2
Time: 35.9s
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

    \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x \cdot \cos B}{\sin B}}\]
  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 2018362 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))

Details

Time bar (total: 20.5s)Debug log

start1.1s

Algorithm
intervals

setup120.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize26.0ms

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)))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
3 → 76
Calls

3 calls. Slowest were:

7.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
2.0ms
(/ x (tan B))
1.0ms
(/ 1 (sin B))

series275.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

135.0ms
(/ x (tan B))
117.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
24.0ms
(/ 1 (sin B))

simplify1.8s

Counts
66 → 85
Calls

66 calls. Slowest were:

88.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (tan B)) x)))
77.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (tan B)) x)))
73.0ms
(fma (* (cbrt (/ 1 (sin B))) (cbrt (/ 1 (sin B)))) (cbrt (/ 1 (sin B))) (- (* (/ 1 (tan B)) x)))

prune899.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize23.0ms

Local error

Found 4 expressions with local error:

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

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 105
Calls

4 calls. Slowest were:

14.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
8.0ms
(/ (* x (cos B)) (sin B))
2.0ms
(* x (cos B))

series266.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

106.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
104.0ms
(/ (* x (cos B)) (sin B))
34.0ms
(* x (cos B))
21.0ms
(/ 1 (sin B))

simplify9.1s

Counts
101 → 117
Calls

101 calls. Slowest were:

702.0ms
(/ (* (* (* x (cos B)) (* x (cos B))) (* x (cos B))) (* (* (sin B) (sin B)) (sin B)))
519.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (sin B)) (* x (cos B)))))
459.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (sin B)) (* x (cos B)))))

prune1.4s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 2 expressions with local error:

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

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
2 → 45
Calls

2 calls. Slowest were:

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

series131.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

84.0ms
(/ (- 1 (* x (cos B))) (sin B))
47.0ms
(* x (cos B))

simplify1.2s

Counts
37 → 51
Calls

37 calls. Slowest were:

339.0ms
(- (pow 1 3) (pow (* x (cos B)) 3))
224.0ms
(/ (* (* (- 1 (* x (cos B))) (- 1 (* x (cos B)))) (- 1 (* x (cos B)))) (* (* (sin B) (sin B)) (sin B)))
58.0ms
(- (* 1 1) (* (* x (cos B)) (* x (cos B))))

prune515.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (* x (cos B)))
0.6b
(cbrt (* x (cos B)))
0.6b
(cbrt (* x (cos B)))
0.5b
(* (cbrt (* x (cos B))) (cbrt (* x (cos B))))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

11.0ms
(* (cbrt (* x (cos B))) (cbrt (* x (cos B))))
2.0ms
(cbrt (* x (cos B)))
2.0ms
(cbrt (* x (cos B)))

series703.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

187.0ms
(* (cbrt (* x (cos B))) (cbrt (* x (cos B))))
186.0ms
(cbrt (* x (cos B)))
172.0ms
(cbrt (* x (cos B)))
157.0ms
(cbrt (* x (cos B)))

simplify1.8s

Counts
31 → 71
Calls

31 calls. Slowest were:

230.0ms
(* (exp (* 1/3 (- (log (pow (cos B) 2)) (* 2 (log (/ -1 x)))))) (pow (cbrt -1) 2))
208.0ms
(exp (* 1/3 (- (log (pow (cos B) 2)) (* 2 (log (/ 1 x))))))
207.0ms
(* (cbrt -1) (exp (* 1/3 (- (log (cos B)) (log (/ -1 x))))))

prune968.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

regimes173.0ms

Accuracy

0% (0.1b remaining)

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

bsearch2.0ms