Average Error: 8.1 → 0.7
Time: 43.2s
Precision: 64
Internal Precision: 128
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}}{F}\]

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 8.1

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

  2. Initial simplification7.7

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

  4. Applied associate-/r*0.7

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

  6. Applied *-un-lft-identity0.7

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

  7. Applied associate-/l*0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2018360 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))

Details

Time bar (total: 8.8s)Debug log

start1.8s

Algorithm
intervals

setup175.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 7.1b

localize33.0ms

Local error

Found 4 expressions with local error:

6.3b
(/ (tan (* PI l)) (* F F))
5.4b
(tan (* PI l))
0.3b
(* PI l)
0.3b
(* PI l)

rewrite18.0ms

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)

series259.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

157.0ms
(/ (tan (* PI l)) (* F F))
47.0ms
(tan (* PI l))
28.0ms
(* PI l)
26.0ms
(* PI l)

simplify1.2s

Counts
23 → 56
Calls

23 calls. Slowest were:

478.0ms
(+ (/ (* 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)))))
386.0ms
(+ (* 2/15 (* (pow PI 5) (pow l 5))) (+ (* 1/3 (* (pow PI 3) (pow l 3))) (* PI l)))
212.0ms
(/ (* (* (tan (* PI l)) (tan (* PI l))) (tan (* PI l))) (* (* (* F F) (* F F)) (* F F)))

prune556.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 0.9b

localize8.0ms

Local error

Found 4 expressions with local error:

5.4b
(tan (* PI l))
0.3b
(* PI l)
0.3b
(* PI l)
0.2b
(/ (tan (* PI l)) F)

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 38
Calls

4 calls. Slowest were:

4.0ms
(/ (tan (* PI l)) F)
2.0ms
(tan (* PI l))
0.0ms
(* PI l)

series232.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

113.0ms
(/ (tan (* PI l)) F)
57.0ms
(tan (* PI l))
32.0ms
(* PI l)
29.0ms
(* PI l)

simplify644.0ms

Counts
15 → 50
Calls

15 calls. Slowest were:

420.0ms
(+ (* 2/15 (* (pow PI 5) (pow l 5))) (+ (* 1/3 (* (pow PI 3) (pow l 3))) (* PI l)))
174.0ms
(+ (* 1/3 (/ (* (pow PI 3) (pow l 3)) F)) (+ (/ (* PI l) F) (* 2/15 (/ (* (pow PI 5) (pow l 5)) F))))
6.0ms
(/ (sin (* PI l)) (* F (cos (* PI l))))

prune492.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 0.9b

localize25.0ms

Local error

Found 4 expressions with local error:

5.4b
(tan (* PI l))
0.8b
(/ 1 (/ F (tan (* PI l))))
0.3b
(* PI l)
0.3b
(* PI l)

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 43
Calls

4 calls. Slowest were:

3.0ms
(tan (* PI l))
3.0ms
(/ 1 (/ F (tan (* PI l))))
0.0ms
(* PI l)

series264.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

117.0ms
(/ 1 (/ F (tan (* PI l))))
63.0ms
(tan (* PI l))
46.0ms
(* PI l)
38.0ms
(* PI l)

simplify626.0ms

Counts
19 → 55
Calls

19 calls. Slowest were:

450.0ms
(+ (* 2/15 (* (pow PI 5) (pow l 5))) (+ (* 1/3 (* (pow PI 3) (pow l 3))) (* PI l)))
115.0ms
(+ (* 1/3 (/ (* (pow PI 3) (pow l 3)) F)) (/ (* PI l) F))
6.0ms
(/ (sin (* PI l)) (* F (cos (* PI l))))

prune604.0ms

Pruning

16 alts after pruning (15 fresh and 1 done)

Merged error: 0.8b

localize12.0ms

Local error

Found 4 expressions with local error:

5.4b
(tan (* PI l))
0.3b
(* PI l)
0.3b
(* PI l)
0.2b
(/ (tan (* PI l)) F)

rewrite12.0ms

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)

series202.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

85.0ms
(/ (tan (* PI l)) F)
60.0ms
(tan (* PI l))
30.0ms
(* PI l)
27.0ms
(* PI l)

simplify646.0ms

Counts
15 → 50
Calls

15 calls. Slowest were:

451.0ms
(+ (* 2/15 (* (pow PI 5) (pow l 5))) (+ (* 1/3 (* (pow PI 3) (pow l 3))) (* PI l)))
148.0ms
(+ (* 1/3 (/ (* (pow PI 3) (pow l 3)) F)) (+ (/ (* PI l) F) (* 2/15 (/ (* (pow PI 5) (pow l 5)) F))))
6.0ms
(/ (sin (* PI l)) (* F (cos (* PI l))))

prune568.0ms

Pruning

16 alts after pruning (14 fresh and 2 done)

Merged error: 0.8b

regimes474.0ms

Accuracy

0% (0.2b remaining)

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

bsearch2.0ms