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Assembly Language & x86 Low-Level Systems Programming · 课时

浮点精度、舍入与异常

理解 IEEE 754 表示法和舍入模式，以及 FPU 和 SSE 单元如何报告溢出、下溢和无效操作等异常。

浮点精度、舍入与异常 是 CoddyKit 上的免费 Assembly Language & x86 Low-Level Systems Programming 课时。 这是第 4 节课，共 4 节。 你可以在下方免费阅读本课时的完整内容 — 然后在浏览器中使用内置代码编辑器和全天候 AI 导师进行实践。 这是 Assembly Language & x86 Low-Level Systems Programming 学习路径的一部分，你的进度在网页和 CoddyKit 应用中同步。 Assembly Language & x86 Low-Level Systems Programming 课程共包含 4 节课。

本课时的部分内容尚未翻译，以英文显示。

IEEE 754 Format

x86 floating-point follows the IEEE 754 standard. A number is stored as sign, exponent, and mantissa:

  • Single (32-bit): 1 + 8 + 23
  • Double (64-bit): 1 + 11 + 52
  • Extended (80-bit): used internally by the x87 FPU

Why Precision Matters

Most decimal fractions cannot be represented exactly in binary. For example 0.1 + 0.2 does not equal exactly 0.3. Accumulated rounding error is a core hazard in numerical code.

The Four Rounding Modes

IEEE 754 defines four rounding modes:

  • Round to nearest, ties to even (default)
  • Round toward negative infinity (floor)
  • Round toward positive infinity (ceil)
  • Round toward zero (truncate)

Controlling Rounding on x87

The x87 control word holds the rounding-control (RC) bits. You read it with fstcw, modify, then load with fldcw.

sub esp, 4
fstcw [esp]          ; store control word
or word [esp], 0x0C00 ; RC = 11 -> round toward zero
fldcw [esp]          ; load it back
add esp, 4

SSE MXCSR Register

SSE has its own control/status register, MXCSR. It holds rounding-control bits plus exception masks and flags. Manage it with ldmxcsr and stmxcsr.

sub rsp, 4
stmxcsr [rsp]        ; save MXCSR
ldmxcsr [rsp]        ; restore MXCSR
add rsp, 4

Floating-Point Exceptions

The standard defines six exceptions:

  • Invalid (e.g. 0/0, sqrt of negative)
  • Denormal operand
  • Divide by zero
  • Overflow
  • Underflow
  • Inexact (rounding occurred)

Masked vs Unmasked

Each exception can be masked. A masked exception produces a default result (like infinity or NaN) and sets a flag. An unmasked exception raises a CPU trap so your handler can intervene.

Special Values: NaN and Infinity

IEEE 754 reserves bit patterns for special values:

  • +Inf / -Inf from overflow or divide-by-zero
  • NaN (Not a Number) from invalid operations

Any comparison with NaN is false — even NaN == NaN.

Detecting NaN

Because NaN is unordered, you detect it with an unordered compare. In SSE, ucomiss sets the parity flag when an operand is NaN.

ucomiss xmm0, xmm0   ; compare value with itself
jp is_nan            ; PF set => NaN detected

A Runnable C Demonstration

This self-contained C program shows that floating-point addition is not exact and detects a NaN.

#include <stdio.h>
#include <math.h>
int main(void) {
    double a = 0.1 + 0.2;
    printf("0.1+0.2 = %.17f\n", a);
    printf("equals 0.3? %d\n", a == 0.3);
    double nan_val = 0.0 / 0.0;
    printf("isnan? %d\n", isnan(nan_val));
    return 0;
}

Practical Advice

To write robust floating-point code:

  • Never compare floats with ==; use an epsilon tolerance
  • Be aware of accumulation order in sums
  • Keep the default round-to-nearest unless you have a reason
  • Check status flags after risky operations

Quick Check

Test your floating-point knowledge.

Recap

You explored floating-point behavior:

  • IEEE 754 defines single, double, and 80-bit extended formats
  • Four rounding modes are configured via the x87 control word or MXCSR
  • Six exceptions can be masked (default result + flag) or unmasked (trap)
  • NaN and infinity are special values; NaN compares false to everything

常见问题解答

「浮点精度、舍入与异常」课时是免费的吗？

是的 — 「浮点精度、舍入与异常」的完整文本可在网页上免费阅读。要进行交互式练习（内置代码编辑器和全天候 AI 导师）并解锁 Assembly Language & x86 Low-Level Systems Programming 课程的其余内容，请升级到 CoddyKit PRO。 Assembly Language & x86 Low-Level Systems Programming 课程共包含 4 节课。

「浮点精度、舍入与异常」这节课中我会学到什么？

理解 IEEE 754 表示法和舍入模式，以及 FPU 和 SSE 单元如何报告溢出、下溢和无效操作等异常。 你通过在浏览器中直接运行的动手代码来练习 Assembly Language & x86 Low-Level Systems Programming，全天候 AI 导师会在你学习这节课的过程中回答你的问题。

学习 Assembly Language & x86 Low-Level Systems Programming 需要有经验吗？

无需任何先前经验。CoddyKit 上的 Assembly Language & x86 Low-Level Systems Programming 课程适合初学者到高级学习者，你可以从这里开始或从头开始，按照自己的节奏学习。 这是第 4 节课，共 4 节。

「浮点精度、舍入与异常」课时需要多长时间？

大多数 CoddyKit 课程大约需要 5–10 分钟。每节课都很精短且互动，所以你能稳步进步，并在网页和应用中从离开的地方继续。

我能在这节 Assembly Language & x86 Low-Level Systems Programming 课中编写并运行代码吗？

能。每节 Assembly Language & x86 Low-Level Systems Programming 课都包含内置代码编辑器，你可以在浏览器中直接编写并运行真实代码，并获得即时 AI 反馈 — 无需本地设置。

此课程中的所有课时

  1. x87 FPU 编程基础
  2. SSE/AVX 指令集简介
  3. 使用 SIMD 将代码向量化
  4. 浮点精度、舍入与异常
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