Average Error: 8.6 → 0.7
Time: 31.3s
Precision: 64
Internal Precision: 128
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F}\]

Error

Bits error versus F

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 8.6

    \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

  2. Simplified8.2

    \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity8.2

    \[\leadsto \pi \cdot \ell - \frac{\color{blue}{1 \cdot \tan \left(\pi \cdot \ell\right)}}{F \cdot F}\]

  5. Applied times-frac0.7

    \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1}{F} \cdot \frac{\tan \left(\pi \cdot \ell\right)}{F}}\]

  6. Final simplification0.7

    \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))

Details

Time bar (total: 30.0s)Debug log

sample729.0ms

Algorithm
intervals

simplify242.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

241.0ms
(- (* PI l) (* (/ 1 (* F F)) (tan (* PI l))))

prune25.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 8.3b

localize33.0ms

Local error

Found 4 expressions with local error:

8.0b
(/ (tan (* PI l)) (* F F))
4.7b
(tan (* PI l))
0.2b
(* PI l)
0.2b
(* PI l)

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 44
Calls

4 calls. Slowest were:

9.0ms
(/ (tan (* PI l)) (* F F))
2.0ms
(tan (* PI l))
0.0ms
(* PI l)

series263.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

128.0ms
(/ (tan (* PI l)) (* F F))
49.0ms
(* PI l)
49.0ms
(tan (* PI l))
36.0ms
(* PI l)

simplify1.1s

Counts
23 → 56
Calls

23 calls. Slowest were:

435.0ms
(+ (/ (* PI l) (pow F 2)) (+ (* 2/15 (/ (* (pow PI 5) (pow l 5)) (pow F 2))) (* 1/3 (/ (* (pow PI 3) (pow l 3)) (pow F 2)))))
241.0ms
(+ (* 2/15 (* (pow PI 5) (pow l 5))) (+ (* 1/3 (* (pow PI 3) (pow l 3))) (* PI l)))
234.0ms
(/ (* (* (tan (* PI l)) (tan (* PI l))) (tan (* PI l))) (* (* (* F F) (* F F)) (* F F)))

prune581.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.4b

localize21.0ms

Local error

Found 4 expressions with local error:

4.7b
(tan (* PI l))
0.2b
(/ (tan (* PI l)) F)
0.2b
(* PI l)
0.2b
(* PI l)

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 38
Calls

4 calls. Slowest were:

8.0ms
(/ (tan (* PI l)) F)
3.0ms
(tan (* PI l))
0.0ms
(* PI l)

series235.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

114.0ms
(/ (tan (* PI l)) F)
45.0ms
(tan (* PI l))
42.0ms
(* PI l)
33.0ms
(* PI l)

simplify456.0ms

Counts
15 → 50
Calls

15 calls. Slowest were:

219.0ms
(+ (* 2/15 (* (pow PI 5) (pow l 5))) (+ (* 1/3 (* (pow PI 3) (pow l 3))) (* PI l)))
198.0ms
(+ (* 1/3 (/ (* (pow PI 3) (pow l 3)) F)) (+ (/ (* PI l) F) (* 2/15 (/ (* (pow PI 5) (pow l 5)) F))))
5.0ms
(/ (sin (* PI l)) (cos (* PI l)))

prune451.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 0.4b

localize12.0ms

Local error

Found 4 expressions with local error:

4.7b
(tan (* PI l))
0.3b
(* (tan (* PI l)) (/ 1 F))
0.2b
(* PI l)
0.2b
(* PI l)

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 46
Calls

4 calls. Slowest were:

7.0ms
(* (tan (* PI l)) (/ 1 F))
2.0ms
(tan (* PI l))
0.0ms
(* PI l)

series222.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

76.0ms
(* (tan (* PI l)) (/ 1 F))
58.0ms
(tan (* PI l))
45.0ms
(* PI l)
43.0ms
(* PI l)

simplify1.1s

Counts
24 → 58
Calls

24 calls. Slowest were:

459.0ms
(* (* (* (tan (* PI l)) (tan (* PI l))) (tan (* PI l))) (* (* (/ 1 F) (/ 1 F)) (/ 1 F)))
264.0ms
(+ (* 2/15 (* (pow PI 5) (pow l 5))) (+ (* 1/3 (* (pow PI 3) (pow l 3))) (* PI l)))
225.0ms
(+ (* 1/3 (/ (* (pow PI 3) (pow l 3)) F)) (+ (/ (* PI l) F) (* 2/15 (/ (* (pow PI 5) (pow l 5)) F))))

prune613.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.4b

localize8.0ms

Local error

Found 4 expressions with local error:

4.7b
(tan (* PI l))
0.2b
(* (/ 1 F) (/ (tan (* PI l)) F))
0.2b
(/ (tan (* PI l)) F)
0.2b
(* PI l)

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

5.0ms
(* (/ 1 F) (/ (tan (* PI l)) F))
4.0ms
(/ (tan (* PI l)) F)
2.0ms
(tan (* PI l))

series310.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

125.0ms
(* (/ 1 F) (/ (tan (* PI l)) F))
104.0ms
(/ (tan (* PI l)) F)
42.0ms
(tan (* PI l))
39.0ms
(* PI l)

simplify1.6s

Counts
27 → 64
Calls

27 calls. Slowest were:

464.0ms
(* (* (* (/ 1 F) (/ 1 F)) (/ 1 F)) (* (* (/ (tan (* PI l)) F) (/ (tan (* PI l)) F)) (/ (tan (* PI l)) F)))
420.0ms
(+ (/ (* PI l) (pow F 2)) (* 1/3 (/ (* (pow PI 3) (pow l 3)) (pow F 2))))
220.0ms
(+ (* 2/15 (* (pow PI 5) (pow l 5))) (+ (* 1/3 (* (pow PI 3) (pow l 3))) (* PI l)))

prune566.0ms

Pruning

14 alts after pruning (13 fresh and 1 done)

Merged error: 0.4b

regimes578.0ms

Accuracy

0% (0.2b remaining)

Error of 0.7b against oracle of 0.5b and baseline of 0.7b

bsearch3.0ms

end0.0ms

sample20.9s

Algorithm
intervals