Average Error: 0.2 → 0.2
Time: 34.6s
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: 18.8s)Debug log

start316.0ms

Algorithm
intervals

setup109.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize31.0ms

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)))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
3 → 76
Calls

3 calls. Slowest were:

17.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
2.0ms
(/ 1 (sin B))
1.0ms
(/ x (tan B))

series217.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

103.0ms
(- (/ 1 (sin B)) (/ x (tan B)))
100.0ms
(/ x (tan B))
13.0ms
(/ 1 (sin B))

simplify1.6s

Counts
66 → 85
Calls

66 calls. Slowest were:

81.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (tan B)) x)))
72.0ms
(+ (* 1/6 B) (+ (/ 1 B) (* 7/360 (pow B 3))))
66.0ms
(fma (* (cbrt (/ 1 (sin B))) (cbrt (/ 1 (sin B)))) (cbrt (/ 1 (sin B))) (- (* (cos B) (/ x (sin B)))))

prune901.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize22.0ms

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)))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 105
Calls

4 calls. Slowest were:

7.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
4.0ms
(/ (* x (cos B)) (sin B))
1.0ms
(* x (cos B))

series253.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

107.0ms
(/ (* x (cos B)) (sin B))
91.0ms
(- (/ 1 (sin B)) (/ (* x (cos B)) (sin B)))
31.0ms
(* x (cos B))
24.0ms
(/ 1 (sin B))

simplify8.5s

Counts
101 → 117
Calls

101 calls. Slowest were:

455.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (sin B)) (* x (cos B)))))
450.0ms
(/ (* (* (* x (cos B)) (* x (cos B))) (* x (cos B))) (* (* (sin B) (sin B)) (sin B)))
448.0ms
(fma 1 (/ 1 (sin B)) (- (* (/ 1 (sin B)) (* x (cos B)))))

prune1.3s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.0b

localize10.0ms

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:

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

series144.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

93.0ms
(/ (- 1 (* x (cos B))) (sin B))
50.0ms
(* x (cos B))

simplify1.3s

Counts
37 → 51
Calls

37 calls. Slowest were:

343.0ms
(- (pow 1 3) (pow (* x (cos B)) 3))
274.0ms
(/ (* (* (- 1 (* x (cos B))) (- 1 (* x (cos B)))) (- 1 (* x (cos B)))) (* (* (sin B) (sin B)) (sin B)))
72.0ms
(- (* 1 1) (* (* x (cos B)) (* x (cos B))))

prune598.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.0b

localize26.0ms

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))))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

6.0ms
(* (cbrt (* x (cos B))) (cbrt (* x (cos B))))
1.0ms
(cbrt (* x (cos B)))
1.0ms
(cbrt (* x (cos B)))

series659.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

183.0ms
(* (cbrt (* x (cos B))) (cbrt (* x (cos B))))
178.0ms
(cbrt (* x (cos B)))
153.0ms
(cbrt (* x (cos B)))
145.0ms
(cbrt (* x (cos B)))

simplify1.9s

Counts
31 → 71
Calls

31 calls. Slowest were:

238.0ms
(* (exp (* 1/3 (- (log (pow (cos B) 2)) (* 2 (log (/ -1 x)))))) (pow (cbrt -1) 2))
235.0ms
(exp (* 1/3 (- (log (pow (cos B) 2)) (* 2 (log (/ 1 x))))))
217.0ms
(* (cbrt -1) (exp (* 1/3 (- (log (cos B)) (log (/ -1 x))))))

prune856.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

regimes165.0ms

Accuracy

0% (0.1b remaining)

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

bsearch3.0ms