Average Error: 8.8 → 0.7
Time: 30.0s
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 8.8

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

  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 2019002 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))

Details

Time bar (total: 28.7s)Debug log

sample670.0ms

Algorithm
intervals

simplify202.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

201.0ms
(- (* PI l) (* (/ 1 (* F F)) (tan (* PI l))))

prune12.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 8.0b

localize16.0ms

Local error

Found 4 expressions with local error:

7.9b
(/ (tan (* PI l)) (* F F))
3.4b
(tan (* PI l))
0.2b
(* PI l)
0.2b
(* PI l)

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 44
Calls

4 calls. Slowest were:

11.0ms
(/ (tan (* PI l)) (* F F))
3.0ms
(tan (* PI l))
0.0ms
(* PI l)

series248.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

134.0ms
(/ (tan (* PI l)) (* F F))
53.0ms
(tan (* PI l))
34.0ms
(* PI l)
27.0ms
(* PI l)

simplify988.0ms

Counts
23 → 56
Calls

23 calls. Slowest were:

418.0ms
(+ (/ (* 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)))))
252.0ms
(+ (* 2/15 (* (pow PI 5) (pow l 5))) (+ (* 1/3 (* (pow PI 3) (pow l 3))) (* PI l)))
203.0ms
(/ (* (* (tan (* PI l)) (tan (* PI l))) (tan (* PI l))) (* (* (* F F) (* F F)) (* F F)))

prune583.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.2b

localize19.0ms

Local error

Found 4 expressions with local error:

3.4b
(tan (* PI l))
0.2b
(* PI l)
0.2b
(* PI l)
0.2b
(/ (tan (* PI l)) F)

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 38
Calls

4 calls. Slowest were:

7.0ms
(/ (tan (* PI l)) F)
3.0ms
(tan (* PI l))
0.0ms
(* PI l)

series199.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

89.0ms
(/ (tan (* PI l)) F)
45.0ms
(* PI l)
40.0ms
(tan (* PI l))
26.0ms
(* PI l)

simplify514.0ms

Counts
15 → 50
Calls

15 calls. Slowest were:

298.0ms
(+ (* 2/15 (* (pow PI 5) (pow l 5))) (+ (* 1/3 (* (pow PI 3) (pow l 3))) (* PI l)))
181.0ms
(+ (* 1/3 (/ (* (pow PI 3) (pow l 3)) F)) (+ (/ (* PI l) F) (* 2/15 (/ (* (pow PI 5) (pow l 5)) F))))
4.0ms
(* (cbrt (tan (* PI l))) (cbrt (tan (* PI l))))

prune486.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 0.2b

localize8.0ms

Local error

Found 4 expressions with local error:

3.4b
(tan (* PI l))
0.2b
(* PI l)
0.2b
(* PI l)
0.2b
(/ (tan (* PI l)) F)

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 38
Calls

4 calls. Slowest were:

3.0ms
(/ (tan (* PI l)) F)
2.0ms
(tan (* PI l))
0.0ms
(* PI l)

series198.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

84.0ms
(/ (tan (* PI l)) F)
44.0ms
(tan (* PI l))
44.0ms
(* PI l)
26.0ms
(* PI l)

simplify507.0ms

Counts
15 → 50
Calls

15 calls. Slowest were:

246.0ms
(+ (* 2/15 (* (pow PI 5) (pow l 5))) (+ (* 1/3 (* (pow PI 3) (pow l 3))) (* PI l)))
224.0ms
(+ (* 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)))

prune544.0ms

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 0.2b

localize15.0ms

Local error

Found 4 expressions with local error:

3.4b
(tan (* PI l))
0.3b
(* (tan (* PI l)) (/ 1 F))
0.2b
(* PI l)
0.2b
(* PI l)

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 46
Calls

4 calls. Slowest were:

6.0ms
(* (tan (* PI l)) (/ 1 F))
2.0ms
(tan (* PI l))
0.0ms
(* PI l)

series225.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

95.0ms
(* (tan (* PI l)) (/ 1 F))
52.0ms
(tan (* PI l))
47.0ms
(* PI l)
30.0ms
(* PI l)

simplify1.1s

Counts
24 → 58
Calls

24 calls. Slowest were:

462.0ms
(* (* (* (tan (* PI l)) (tan (* PI l))) (tan (* PI l))) (* (* (/ 1 F) (/ 1 F)) (/ 1 F)))
324.0ms
(+ (* 2/15 (* (pow PI 5) (pow l 5))) (+ (* 1/3 (* (pow PI 3) (pow l 3))) (* PI l)))
217.0ms
(+ (* 1/3 (/ (* (pow PI 3) (pow l 3)) F)) (+ (/ (* PI l) F) (* 2/15 (/ (* (pow PI 5) (pow l 5)) F))))

prune597.0ms

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 0.2b

regimes405.0ms

Accuracy

0% (0.2b remaining)

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

bsearch1.0ms

end0.0ms

sample21.1s

Algorithm
intervals