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83
section_1/float/asm_rounding.s Normal file
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.global main
.text
.align 2
dless .req d20
dmore .req d21
ndless .req d22
ndmore .req d23
Emit: str x30, [sp, -16]!
bl printf
ldr x30, [sp], 16
ret
main: str x30, [sp, -16]!
stp dless, dmore, [sp, -16]!
stp ndless, ndmore, [sp, -16]!
ldr x0, =vless
ldr dless, [x0]
ldr dmore, [x0, 8]
ldr ndless, [x0, 16]
ldr ndmore, [x0, 24]
//-fcvtps----------------------
fcvtps x1, dless
fcvtps x2, dmore
ldr x0, =fmt1
bl Emit
fcvtps x1, ndless
fcvtps x2, ndmore
ldr x0, =fmt1
bl Emit
//-fcvtms-----------------------
fcvtms x1, dless
fcvtms x2, dmore
ldr x0, =fmt2
bl Emit
fcvtms x1, ndless
fcvtms x2, ndmore
ldr x0, =fmt2
bl Emit
//-fcvtzs-----------------------
fcvtzs x1, dless
fcvtzs x2, dmore
ldr x0, =fmt4
bl Emit
fcvtzs x1, ndless
fcvtzs x2, ndmore
ldr x0, =fmt4
bl Emit
//-fcvtas-----------------------
fcvtas x1, dless
fcvtas x2, dmore
ldr x0, =fmt3
bl Emit
fcvtas x1, ndless
fcvtas x2, ndmore
ldr x0, =fmt3
bl Emit
//------------------------------
ldp ndless, ndmore, [sp], 16
ldp dless, dmore, [sp], 16
ldr x30, [sp], 16
mov w0, wzr
ret
.section .rodata
vless: .double 5.49
vmore: .double 5.51
nvless: .double -5.49
nvmore: .double -5.51
fmt1: .asciz "fcvtps less: %d more: %d\n"
fmt2: .asciz "fcvtms less: %d more: %d\n"
fmt3: .asciz "fcvtta less: %d more: %d\n"
fmt4: .asciz "fcvtzs less: %d more: %d\n"
.end

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#include <iostream>
#include <iomanip>
#include <cmath>
using namespace std;
template <typename T>
int RoundAwayFromZero(T x) {
return int((x < 0) ? floor(x) : ceil(x));
}
int main() {
int32_t iv;
float fv = 5.1;
iv = (int(fv) == fv) ? int(fv) : int(fv) + ((fv < 0) ? -1 : 1);
cout << setw(4) << fv << " away from zero (should be 6): ";
cout << iv << endl;
fv = -fv;
iv = (int(fv) == fv) ? int(fv) : int(fv) + ((fv < 0) ? -1 : 1);
cout << setw(4) << fv << " away from zero (should be -6): ";
cout << iv << endl;
cout << endl;
cout << "Using MyRound()\n";
fv = -fv;
iv = RoundAwayFromZero(fv);
cout << setw(4) << fv << " away from zero (should be 6): ";
cout << iv << endl;
fv = -fv;
iv = RoundAwayFromZero(fv);
cout << setw(4) << fv << " away from zero (should be -6): ";
cout << iv << endl;
return 0;
}

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# Section 1 / Conversion of Floating Point and Integers
This chapter has been surprisingly difficult to research and write. Huh?
All we're talking about is taking a floating point value and turning it
into an integer - what could be hard?
It's hard because the AARCH64 has so many instructions that seemingly
do the aforementioned job and each of them come in many variations. Even
the language used is confusing.
For this chapter, I will use:
* Rounding means picking some fractional value and if the float's
fraction is higher, you go one way and if lower, you go the other.
* Truncation means you don't look too closely at the fractional value.
Instead, you just eliminate the fractional part and slam the whole
number ... one way or the other.
## Truncation Towards Zero
In C and C++, truncation is what we get from:
```c++
integer_variable = int(floating_variable); // C++
integer_variable = (int) floating_variable; // C
```
Diving a little deeper, there is a choice to be made as to whether or
not `integer_variable` is signed or unsigned. And, whether or not
`integer_variable` is a 32 bit or 64 bit value.
The instruction is `fcvtz` - convert towards zero. Then, the choice
as to whether to produce a signed or unsigned result is defined by the
final letterL `u` or `s`.
| Mnemonic | Meaning |
| -------- | ------- |
| fcvtzu | Truncate (always towards 0) producing an unsigned int |
| fcvtzs | Truncate (always towards 0) producing a signed int |
As an example of how the ARM documentation is confusing - this
instruction which completely discards the fractional value is said by
the ARM documentation as doing rounding.
The the choice of source register defined whether you are converting
a double or single precision floating point value.
| Source Register | Converts a |
| --------------- | ---------- |
| dX | `double` to an integer |
| sX | `float` to an integer |
| Destination Register | Converts a |
| --------------- | ---------- |
| xX | 64 bit integer |
| wX | 32 bit or less integer |
Examples where `d` is a `double` and `f` is a `float`:
| C++ | Instruction |
| --- | ----------- |
| `int32_t(d)` | `fcvtzs w0, d0` |
| `uint32_t(d)` | `fcvtzu w0, d0` |
| `int64_t(d)` | `fcvtzs x0, d0` |
| `uint64_t(d)` | `fcvtzu x0, d0` |
[Here](./asm_rounding.s) is a program which demonstrates various
ways of converting doubles to integers.
Let's look at:
```text
//-fcvtzs----------------------- // 45
fcvtzs x1, dless // 46
fcvtzs x2, dmore // 47
ldr x0, =fmt4 // 48
bl Emit // 49
// 50
fcvtzs x1, ndless // 51
fcvtzs x2, ndmore // 52
ldr x0, =fmt4 // 53
bl Emit // 54
```
Reminder:
* `dless` is 5.49
* `dmore` is 5.51
* `ndless` is -5.49
* `ndmore` is -5.51
Here is the relevant output:
```text
fcvtzs less: 5 more: 5
fcvtzs less: -5 more: -5
```
Notice all the values were truncated to the whole number that is
*closer to zero*.
## Truncation Away From Zero
Truncation away from zero is not as easy. In fact, it cannot be
performed with a single instruction.
In C and C++:
```c
iv = (int(fv) == fv) ? int(fv) : int(fv) + ((fv < 0) ? -1 : 1);
```
If the `fv` is already equal to a whole number, the
integer value will be that whole number. Other wise the `iv` is
the whole number further *away from zero*.
In C++, a more sophisticated version would require `<cmath>` and
could look like:
```c++
template <typename T>
int MyTruncate(T x) {
return int((x < 0) ? floor(x) : ceil(x));
}
```
* `floor()` always truncates downward (towards more negative).
* `ceil()` always truncates upwards (towards more positive).
[Here](./rounding.cpp) is a program which demonstrates this:
In assembly language, a function is used which implements
what is in essence, one instantiation of the templated function
given above.
```asm
RoundAwayFromZero:
fcmp d0, 0
ble 1f
// Value is positive, truncate towards positive infinity (ceil)
frintp d0, d0
b 2f
1: // Value is negative, truncate towards negative infinity (floor)
frintm d0, d0
2: fcvtzs x0, d0
ret
```
`frintp` and `frintm` will honor the source register already being
a whole number (no fractional part). Thus a value of 5 will not be
converted to 6 and -5 will not be converted to -6. But, a value of
5.000000001 **will** go to 6, etc.
[Here](./frintp.s) is a program that demonstrates this:
```text
.text // 1
.global main // 2
.align 2 // 3
// 4
main: str x30, [sp, -16]! // 5
// 6
ldr x0, =d // 7
ldr d0, [x0] // 8
frintp d0, d0 // 9
ldr x0, =fmt1 // 10
bl printf // 11
// 12
ldr x0, =h // 13
ldr d0, [x0] // 14
frintp d0, d0 // 15
ldr x0, =fmt2 // 16
bl printf // 17
// 18
ldr x30, [sp], 16 // 19
mov w0, wzr // 20
ret // 21
// 22
.data // 23
fmt1: .asciz "with fraction: %f\n" // 24
fmt2: .asciz "without fraction: %f\n" // 25
d: .double 5.00000001 // 26
h: .double 5.0 // 27
.end // 28
```
The output is:
```text
with fraction: 6.000000
without fraction: 5.000000
```
## Rounding Conversion
An instruction which does what we normally think of as rounding is
`frinta`. This is the conversion "to nearest with ties going away."
So, 5.5 goes to 6 as one would expect from "rounding."
## Converting an Integer to a Floating Point Value
In C / C++:
```c
double_var = double(integer_var); // C++
double_var = (double)integer_var; // C
```
Is handled by two instructions:
* `scvtf` converts a signed integer to a floating point value
* `ucvtf` converts an unsigned integer to a floating point value
The name of the destination register controls which kind of floating
point value is made. For example, specifying `dX` makes a double etc.