Update some of the docs.
This commit is contained in:
Simon Forman
@@ -105,10 +105,17 @@ Subtract $k$ from the index and take the absolute value:
|
||||
|
||||
```python
|
||||
for n in range(2 * k):
|
||||
print abs(n - k),
|
||||
print(abs(n - k),)
|
||||
```
|
||||
|
||||
4 3 2 1 0 1 2 3
|
||||
4
|
||||
3
|
||||
2
|
||||
1
|
||||
0
|
||||
1
|
||||
2
|
||||
3
|
||||
|
||||
|
||||
Not quite. Subtract $k - 1$ from the index and take the absolute value:
|
||||
@@ -116,22 +123,20 @@ Not quite. Subtract $k - 1$ from the index and take the absolute value:
|
||||
|
||||
```python
|
||||
for n in range(2 * k):
|
||||
print abs(n - (k - 1)),
|
||||
print(abs(n - (k - 1)), end=' ')
|
||||
```
|
||||
|
||||
3 2 1 0 1 2 3 4
|
||||
|
||||
3 2 1 0 1 2 3 4
|
||||
|
||||
Great, now add $k$...
|
||||
|
||||
|
||||
```python
|
||||
for n in range(2 * k):
|
||||
print abs(n - (k - 1)) + k,
|
||||
print(abs(n - (k - 1)) + k, end=' ')
|
||||
```
|
||||
|
||||
7 6 5 4 5 6 7 8
|
||||
|
||||
7 6 5 4 5 6 7 8
|
||||
|
||||
So to write a function that can give us the value of a row at a given index:
|
||||
|
||||
@@ -146,11 +151,10 @@ def row_value(k, i):
|
||||
```python
|
||||
k = 5
|
||||
for i in range(2 * k):
|
||||
print row_value(k, i),
|
||||
print(row_value(k, i), end=' ')
|
||||
```
|
||||
|
||||
9 8 7 6 5 6 7 8 9 10
|
||||
|
||||
9 8 7 6 5 6 7 8 9 10
|
||||
|
||||
(I'm leaving out details of how I figured this all out and just giving the relevent bits. It took a little while to zero in of the aspects of the pattern that were important for the task.)
|
||||
|
||||
@@ -182,7 +186,7 @@ def rank_and_offset(n):
|
||||
|
||||
```python
|
||||
for n in range(2, 51):
|
||||
print n, rank_and_offset(n)
|
||||
print(n, rank_and_offset(n))
|
||||
```
|
||||
|
||||
2 (1, 0)
|
||||
@@ -240,7 +244,7 @@ for n in range(2, 51):
|
||||
```python
|
||||
for n in range(2, 51):
|
||||
k, i = rank_and_offset(n)
|
||||
print n, row_value(k, i)
|
||||
print(n, row_value(k, i))
|
||||
```
|
||||
|
||||
2 1
|
||||
@@ -394,7 +398,7 @@ E
|
||||
|
||||
|
||||
|
||||
$$4 k \left(k + 1\right) + 2$$
|
||||
$\displaystyle 4 k \left(k + 1\right) + 2$
|
||||
|
||||
|
||||
|
||||
@@ -413,7 +417,7 @@ It gives correct answers:
|
||||
|
||||
```python
|
||||
for n in (9, 10, 25, 26, 49, 50):
|
||||
print n, rank_of(n)
|
||||
print(n, rank_of(n))
|
||||
```
|
||||
|
||||
9 1
|
||||
@@ -431,14 +435,14 @@ And it runs much faster (at least for large numbers):
|
||||
%time rank_of(23000000000000) # Compare runtime with rank_and_offset()!
|
||||
```
|
||||
|
||||
CPU times: user 68 ms, sys: 8 ms, total: 76 ms
|
||||
Wall time: 73.8 ms
|
||||
CPU times: user 27.8 ms, sys: 5 µs, total: 27.8 ms
|
||||
Wall time: 27.3 ms
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
$$2397916$$
|
||||
$\displaystyle 2397916$
|
||||
|
||||
|
||||
|
||||
@@ -447,14 +451,14 @@ $$2397916$$
|
||||
%time rank_and_offset(23000000000000)
|
||||
```
|
||||
|
||||
CPU times: user 308 ms, sys: 0 ns, total: 308 ms
|
||||
Wall time: 306 ms
|
||||
CPU times: user 216 ms, sys: 89 µs, total: 216 ms
|
||||
Wall time: 215 ms
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
$$\left ( 2397916, \quad 223606\right )$$
|
||||
$\displaystyle \left( 2397916, \ 223606\right)$
|
||||
|
||||
|
||||
|
||||
@@ -485,7 +489,7 @@ g
|
||||
|
||||
|
||||
|
||||
$$- \frac{1}{2} \sqrt{y - 1} - \frac{1}{2}$$
|
||||
$\displaystyle - \frac{\sqrt{y - 1}}{2} - \frac{1}{2}$
|
||||
|
||||
|
||||
|
||||
@@ -497,7 +501,7 @@ f
|
||||
|
||||
|
||||
|
||||
$$\frac{1}{2} \sqrt{y - 1} - \frac{1}{2}$$
|
||||
$\displaystyle \frac{\sqrt{y - 1}}{2} - \frac{1}{2}$
|
||||
|
||||
|
||||
|
||||
@@ -511,7 +515,7 @@ floor(f) + 1
|
||||
|
||||
|
||||
|
||||
$$\lfloor{\frac{1}{2} \sqrt{y - 1} - \frac{1}{2}}\rfloor + 1$$
|
||||
$\displaystyle \left\lfloor{\frac{\sqrt{y - 1}}{2} - \frac{1}{2}}\right\rfloor + 1$
|
||||
|
||||
|
||||
|
||||
@@ -523,7 +527,7 @@ F = lambdify(y, floor(f) + 1)
|
||||
|
||||
```python
|
||||
for n in (9, 10, 25, 26, 49, 50):
|
||||
print n, int(F(n))
|
||||
print(n, int(F(n)))
|
||||
```
|
||||
|
||||
9 1
|
||||
@@ -541,14 +545,14 @@ It's pretty fast.
|
||||
%time int(F(23000000000000)) # The clear winner.
|
||||
```
|
||||
|
||||
CPU times: user 0 ns, sys: 0 ns, total: 0 ns
|
||||
Wall time: 11.9 µs
|
||||
CPU times: user 60 µs, sys: 4 µs, total: 64 µs
|
||||
Wall time: 67 µs
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
$$2397916$$
|
||||
$\displaystyle 2397916$
|
||||
|
||||
|
||||
|
||||
@@ -559,7 +563,7 @@ Knowing the equation we could write our own function manually, but the speed is
|
||||
from math import floor as mfloor, sqrt
|
||||
|
||||
def mrank_of(n):
|
||||
return int(mfloor(sqrt(23000000000000 - 1) / 2 - 0.5) + 1)
|
||||
return int(mfloor(sqrt(n - 1) / 2 - 0.5) + 1)
|
||||
```
|
||||
|
||||
|
||||
@@ -567,14 +571,14 @@ def mrank_of(n):
|
||||
%time mrank_of(23000000000000)
|
||||
```
|
||||
|
||||
CPU times: user 0 ns, sys: 0 ns, total: 0 ns
|
||||
Wall time: 12.9 µs
|
||||
CPU times: user 7 µs, sys: 1 µs, total: 8 µs
|
||||
Wall time: 10 µs
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
$$2397916$$
|
||||
$\displaystyle 2397916$
|
||||
|
||||
|
||||
|
||||
@@ -598,7 +602,7 @@ offset_of(23000000000000, 2397916)
|
||||
|
||||
|
||||
|
||||
$$223606$$
|
||||
$\displaystyle 223606$
|
||||
|
||||
|
||||
|
||||
@@ -632,7 +636,7 @@ aoc20173(23)
|
||||
|
||||
|
||||
|
||||
$$2$$
|
||||
$\displaystyle 2$
|
||||
|
||||
|
||||
|
||||
@@ -644,7 +648,7 @@ aoc20173(23000)
|
||||
|
||||
|
||||
|
||||
$$105$$
|
||||
$\displaystyle 105$
|
||||
|
||||
|
||||
|
||||
@@ -656,7 +660,7 @@ aoc20173(23000000000000)
|
||||
|
||||
|
||||
|
||||
$$4572225$$
|
||||
$\displaystyle 4572225$
|
||||
|
||||
|
||||
|
||||
@@ -665,14 +669,14 @@ $$4572225$$
|
||||
%time aoc20173(23000000000000000000000000) # Fast for large values.
|
||||
```
|
||||
|
||||
CPU times: user 0 ns, sys: 0 ns, total: 0 ns
|
||||
Wall time: 20 µs
|
||||
CPU times: user 22 µs, sys: 2 µs, total: 24 µs
|
||||
Wall time: 26.7 µs
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
$$2690062495969$$
|
||||
$\displaystyle 2690062495969$
|
||||
|
||||
|
||||
|
||||
@@ -698,7 +702,7 @@ The translation is straightforward.
|
||||
|
||||
|
||||
```python
|
||||
define('rank_of == -- sqrt 2 / 0.5 - floor ++')
|
||||
define('rank_of -- sqrt 2 / 0.5 - floor ++')
|
||||
```
|
||||
|
||||
### `offset_of`
|
||||
@@ -732,7 +736,7 @@ Ergo:
|
||||
|
||||
|
||||
```python
|
||||
define('offset_of == dup 2 * [dup -- 4 * * 2 + -] dip %')
|
||||
define('offset_of dup 2 * [dup -- 4 * * 2 + -] dip %')
|
||||
```
|
||||
|
||||
### `row_value`
|
||||
@@ -752,7 +756,7 @@ define('offset_of == dup 2 * [dup -- 4 * * 2 + -] dip %')
|
||||
|
||||
|
||||
```python
|
||||
define('row_value == over -- - abs +')
|
||||
define('row_value over -- - abs +')
|
||||
```
|
||||
|
||||
### `aoc2017.3`
|
||||
@@ -770,7 +774,7 @@ define('row_value == over -- - abs +')
|
||||
|
||||
|
||||
```python
|
||||
define('aoc2017.3 == dup rank_of [offset_of] dupdip swap row_value')
|
||||
define('aoc2017.3 dup rank_of [offset_of] dupdip swap row_value')
|
||||
```
|
||||
|
||||
|
||||
@@ -794,45 +798,45 @@ J('23000 aoc2017.3')
|
||||
V('23000000000000 aoc2017.3')
|
||||
```
|
||||
|
||||
. 23000000000000 aoc2017.3
|
||||
23000000000000 . aoc2017.3
|
||||
23000000000000 . dup rank_of [offset_of] dupdip swap row_value
|
||||
23000000000000 23000000000000 . rank_of [offset_of] dupdip swap row_value
|
||||
23000000000000 23000000000000 . -- sqrt 2 / 0.5 - floor ++ [offset_of] dupdip swap row_value
|
||||
23000000000000 22999999999999 . sqrt 2 / 0.5 - floor ++ [offset_of] dupdip swap row_value
|
||||
23000000000000 4795831.523312615 . 2 / 0.5 - floor ++ [offset_of] dupdip swap row_value
|
||||
23000000000000 4795831.523312615 2 . / 0.5 - floor ++ [offset_of] dupdip swap row_value
|
||||
23000000000000 2397915.7616563076 . 0.5 - floor ++ [offset_of] dupdip swap row_value
|
||||
23000000000000 2397915.7616563076 0.5 . - floor ++ [offset_of] dupdip swap row_value
|
||||
23000000000000 2397915.2616563076 . floor ++ [offset_of] dupdip swap row_value
|
||||
23000000000000 2397915 . ++ [offset_of] dupdip swap row_value
|
||||
23000000000000 2397916 . [offset_of] dupdip swap row_value
|
||||
23000000000000 2397916 [offset_of] . dupdip swap row_value
|
||||
23000000000000 2397916 . offset_of 2397916 swap row_value
|
||||
23000000000000 2397916 . dup 2 * [dup -- 4 * * 2 + -] dip % 2397916 swap row_value
|
||||
23000000000000 2397916 2397916 . 2 * [dup -- 4 * * 2 + -] dip % 2397916 swap row_value
|
||||
23000000000000 2397916 2397916 2 . * [dup -- 4 * * 2 + -] dip % 2397916 swap row_value
|
||||
23000000000000 2397916 4795832 . [dup -- 4 * * 2 + -] dip % 2397916 swap row_value
|
||||
23000000000000 2397916 4795832 [dup -- 4 * * 2 + -] . dip % 2397916 swap row_value
|
||||
23000000000000 2397916 . dup -- 4 * * 2 + - 4795832 % 2397916 swap row_value
|
||||
23000000000000 2397916 2397916 . -- 4 * * 2 + - 4795832 % 2397916 swap row_value
|
||||
23000000000000 2397916 2397915 . 4 * * 2 + - 4795832 % 2397916 swap row_value
|
||||
23000000000000 2397916 2397915 4 . * * 2 + - 4795832 % 2397916 swap row_value
|
||||
23000000000000 2397916 9591660 . * 2 + - 4795832 % 2397916 swap row_value
|
||||
23000000000000 22999994980560 . 2 + - 4795832 % 2397916 swap row_value
|
||||
23000000000000 22999994980560 2 . + - 4795832 % 2397916 swap row_value
|
||||
23000000000000 22999994980562 . - 4795832 % 2397916 swap row_value
|
||||
5019438 . 4795832 % 2397916 swap row_value
|
||||
5019438 4795832 . % 2397916 swap row_value
|
||||
223606 . 2397916 swap row_value
|
||||
223606 2397916 . swap row_value
|
||||
2397916 223606 . row_value
|
||||
2397916 223606 . over -- - abs +
|
||||
2397916 223606 2397916 . -- - abs +
|
||||
2397916 223606 2397915 . - abs +
|
||||
2397916 -2174309 . abs +
|
||||
2397916 2174309 . +
|
||||
4572225 .
|
||||
• 23000000000000 aoc2017.3
|
||||
23000000000000 • aoc2017.3
|
||||
23000000000000 • dup rank_of [offset_of] dupdip swap row_value
|
||||
23000000000000 23000000000000 • rank_of [offset_of] dupdip swap row_value
|
||||
23000000000000 23000000000000 • -- sqrt 2 / 0.5 - floor ++ [offset_of] dupdip swap row_value
|
||||
23000000000000 22999999999999 • sqrt 2 / 0.5 - floor ++ [offset_of] dupdip swap row_value
|
||||
23000000000000 4795831.523312615 • 2 / 0.5 - floor ++ [offset_of] dupdip swap row_value
|
||||
23000000000000 4795831.523312615 2 • / 0.5 - floor ++ [offset_of] dupdip swap row_value
|
||||
23000000000000 2397915.7616563076 • 0.5 - floor ++ [offset_of] dupdip swap row_value
|
||||
23000000000000 2397915.7616563076 0.5 • - floor ++ [offset_of] dupdip swap row_value
|
||||
23000000000000 2397915.2616563076 • floor ++ [offset_of] dupdip swap row_value
|
||||
23000000000000 2397915 • ++ [offset_of] dupdip swap row_value
|
||||
23000000000000 2397916 • [offset_of] dupdip swap row_value
|
||||
23000000000000 2397916 [offset_of] • dupdip swap row_value
|
||||
23000000000000 2397916 • offset_of 2397916 swap row_value
|
||||
23000000000000 2397916 • dup 2 * [dup -- 4 * * 2 + -] dip % 2397916 swap row_value
|
||||
23000000000000 2397916 2397916 • 2 * [dup -- 4 * * 2 + -] dip % 2397916 swap row_value
|
||||
23000000000000 2397916 2397916 2 • * [dup -- 4 * * 2 + -] dip % 2397916 swap row_value
|
||||
23000000000000 2397916 4795832 • [dup -- 4 * * 2 + -] dip % 2397916 swap row_value
|
||||
23000000000000 2397916 4795832 [dup -- 4 * * 2 + -] • dip % 2397916 swap row_value
|
||||
23000000000000 2397916 • dup -- 4 * * 2 + - 4795832 % 2397916 swap row_value
|
||||
23000000000000 2397916 2397916 • -- 4 * * 2 + - 4795832 % 2397916 swap row_value
|
||||
23000000000000 2397916 2397915 • 4 * * 2 + - 4795832 % 2397916 swap row_value
|
||||
23000000000000 2397916 2397915 4 • * * 2 + - 4795832 % 2397916 swap row_value
|
||||
23000000000000 2397916 9591660 • * 2 + - 4795832 % 2397916 swap row_value
|
||||
23000000000000 22999994980560 • 2 + - 4795832 % 2397916 swap row_value
|
||||
23000000000000 22999994980560 2 • + - 4795832 % 2397916 swap row_value
|
||||
23000000000000 22999994980562 • - 4795832 % 2397916 swap row_value
|
||||
5019438 • 4795832 % 2397916 swap row_value
|
||||
5019438 4795832 • % 2397916 swap row_value
|
||||
223606 • 2397916 swap row_value
|
||||
223606 2397916 • swap row_value
|
||||
2397916 223606 • row_value
|
||||
2397916 223606 • over -- - abs +
|
||||
2397916 223606 2397916 • -- - abs +
|
||||
2397916 223606 2397915 • - abs +
|
||||
2397916 -2174309 • abs +
|
||||
2397916 2174309 • +
|
||||
4572225 •
|
||||
|
||||
|
||||
rank_of == -- sqrt 2 / 0.5 - floor ++
|
||||
|
||||
Reference in New Issue
Block a user