Average Error: 8.4 → 0.6
Time: 28.1s
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 8.4

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

  2. Simplified8.0

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

  4. Applied *-un-lft-identity8.0

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

  5. Applied times-frac0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019004 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))

Details

Time bar (total: 27.0s)Debug log

sample1.2s

Algorithm
intervals

simplify199.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

198.0ms
(- (* PI l) (* (/ 1 (* F F)) (tan (* PI l))))

prune24.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 5.7b

localize32.0ms

Local error

Found 4 expressions with local error:

5.7b
(/ (tan (* PI l)) (* F F))
4.6b
(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)

series247.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

139.0ms
(/ (tan (* PI l)) (* F F))
42.0ms
(* PI l)
38.0ms
(tan (* PI l))
26.0ms
(* PI l)

simplify997.0ms

Counts
23 → 56
Calls

23 calls. Slowest were:

361.0ms
(+ (/ (* 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)))))
278.0ms
(+ (* 2/15 (* (pow PI 5) (pow l 5))) (+ (* 1/3 (* (pow PI 3) (pow l 3))) (* PI l)))
233.0ms
(/ (* (* (tan (* PI l)) (tan (* PI l))) (tan (* PI l))) (* (* (* F F) (* F F)) (* F F)))

prune591.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.2b

localize10.0ms

Local error

Found 4 expressions with local error:

4.6b
(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)

series211.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

77.0ms
(/ (tan (* PI l)) F)
51.0ms
(tan (* PI l))
47.0ms
(* PI l)
37.0ms
(* PI l)

simplify377.0ms

Counts
15 → 50
Calls

15 calls. Slowest were:

178.0ms
(+ (* 2/15 (* (pow PI 5) (pow l 5))) (+ (* 1/3 (* (pow PI 3) (pow l 3))) (* PI l)))
166.0ms
(+ (* 1/3 (/ (* (pow PI 3) (pow l 3)) F)) (+ (/ (* PI l) F) (* 2/15 (/ (* (pow PI 5) (pow l 5)) F))))
7.0ms
(/ (sin (* PI l)) (* F (cos (* PI l))))

prune525.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 0.2b

localize16.0ms

Local error

Found 4 expressions with local error:

4.6b
(tan (* PI l))
1.5b
(/ 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)

series281.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

142.0ms
(/ 1 (/ F (tan (* PI l))))
66.0ms
(tan (* PI l))
45.0ms
(* PI l)
28.0ms
(* PI l)

simplify457.0ms

Counts
19 → 55
Calls

19 calls. Slowest were:

225.0ms
(+ (* 2/15 (* (pow PI 5) (pow l 5))) (+ (* 1/3 (* (pow PI 3) (pow l 3))) (* PI l)))
170.0ms
(+ (* 1/3 (/ (* (pow PI 3) (pow l 3)) F)) (/ (* PI l) F))
7.0ms
(/ (sin (* PI l)) (* F (cos (* PI l))))

prune520.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.2b

localize18.0ms

Local error

Found 4 expressions with local error:

4.6b
(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)

series205.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

94.0ms
(/ (tan (* PI l)) F)
56.0ms
(tan (* PI l))
29.0ms
(* PI l)
26.0ms
(* PI l)

simplify430.0ms

Counts
15 → 50
Calls

15 calls. Slowest were:

221.0ms
(+ (* 2/15 (* (pow PI 5) (pow l 5))) (+ (* 1/3 (* (pow PI 3) (pow l 3))) (* PI l)))
168.0ms
(+ (* 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))))

prune479.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.2b

regimes321.0ms

Accuracy

0% (0.2b remaining)

Error of 0.6b against oracle of 0.4b and baseline of 0.6b

bsearch1.0ms

end0.0ms

sample19.9s

Algorithm
intervals