IEEE 754 Standard

Introduction

Before 1985, every computer manufacturer had their own floating-point format — IBM had one, DEC had another, Cray had a third. Programs would produce different results on different machines! The IEEE 754 standard, published in 1985 (revised 2008 and 2019), fixed this chaos by defining a universal floating-point format. Today, virtually every processor — from smartphones to supercomputers — implements IEEE 754.

The Standard Formats

Single Precision (32 bits) — float in C/Java

• Sign: 1 bit
• Exponent: 8 bits, bias = 127
• Mantissa: 23 bits (24 bits effective with hidden bit)
• Range: ±1.18×10⁻³⁸ to ±3.4×10³⁸
• Precision: ~7 decimal digits

Double Precision (64 bits) — double in C/Java

• Sign: 1 bit
• Exponent: 11 bits, bias = 1023
• Mantissa: 52 bits (53 bits effective with hidden bit)
• Range: ±2.23×10⁻³⁰⁸ to ±1.8×10³⁰⁸
• Precision: ~15-16 decimal digits

Encoding Rules

Normal Numbers (most common)

• Exponent: 1 to 254 (single) or 1 to 2046 (double) — not all 0s or all 1s
• Value = (-1)^S × 1.Mantissa × 2^(Exponent - Bias)
• The leading 1 is implicit (hidden bit)

Denormalized Numbers (subnormal)

• Exponent: all zeros (0)
• Value = (-1)^S × 0.Mantissa × 2^(1 - Bias)
• The leading bit is 0 (not 1) — no hidden bit
• Allows gradual underflow toward zero

Zero

• Exponent: all zeros
• Mantissa: all zeros
• +0 (sign=0) and -0 (sign=1) are both valid and compare equal

Infinity

• Exponent: all ones (255 or 2047)
• Mantissa: all zeros
• +∞ (sign=0) and -∞ (sign=1)
• Results from overflow or division by zero

NaN (Not a Number)

• Exponent: all ones
• Mantissa: non-zero
• Quiet NaN (qNaN): Propagates through operations without signaling
• Signaling NaN (sNaN): Triggers an exception when used

Worked Examples

Example 1: Represent +1.5 in Single Precision

1. 1.5₁₀ = 1.1₂ (1 + 0.5)
2. Already normalized: 1.1 × 2⁰
3. Sign = 0, Exponent = 0 + 127 = 127 = 01111111₂, Mantissa = 10000...0
4. Result: 0 01111111 10000000000000000000000
5. Hex: 0x3FC00000

Example 2: Represent -0.375 in Single Precision

1. 0.375₁₀ = 0.011₂ (0.25 + 0.125)
2. Normalize: 1.1 × 2⁻²
3. Sign = 1, Exponent = -2 + 127 = 125 = 01111101₂, Mantissa = 10000...0
4. Result: 1 01111101 10000000000000000000000
5. Hex: 0xBEC00000

Example 3: Decode 0x41200000

1. Binary: 0 10000010 01000000000000000000000
2. Sign = 0 (positive)
3. Exponent = 10000010₂ = 130, Actual = 130-127 = 3
4. Mantissa = 1.01 (with hidden bit)
5. Value = 1.01 × 2³ = 1010.0₂ = 10.0₁₀
6. Result: +10.0

Rounding Modes

IEEE 754 defines four rounding modes:

1. Round to Nearest, Even (default): Round to the nearest representable value. If exactly halfway, round to the value with an even last bit. This is the most common mode and minimizes average error.

1. Round Toward +∞ (ceiling): Always round up

1. Round Toward -∞ (floor): Always round down

1. Round Toward Zero (truncation): Always round toward zero

The "round to nearest, even" default is carefully chosen to prevent systematic bias in repeated calculations.

Precision Limits

What Can Be Represented Exactly?

• Integers up to 2²⁴ (16,777,216) in single precision
• Integers up to 2⁵³ (9,007,199,254,740,992) in double precision
• Fractions that are sums of negative powers of 2: 0.5, 0.25, 0.125, 0.375, etc.

What Cannot Be Represented Exactly?

• 0.1, 0.2, 0.3 (repeating binary fractions)
• Integers larger than 2²⁴ (single) or 2⁵³ (double)
• Irrational numbers (π, e, √2)

Machine Epsilon

The smallest value ε such that 1.0 + ε ≠ 1.0:

• Single precision: ε ≈ 1.19 × 10⁻⁷
• Double precision: ε ≈ 2.22 × 10⁻¹⁶

Exception Handling

IEEE 754 defines five exception conditions:

1. Overflow: Result too large → ±∞
2. Underflow: Result too small → denormalized or zero
3. Division by zero: → ±∞
4. Invalid operation: → NaN (e.g., 0/0, √-1, ∞-∞)
5. Inexact: Result was rounded (happens very frequently)

Key Takeaways

1. IEEE 754 standardized floating-point across all platforms — same format, same results
2. Single precision: 1+8+23 = 32 bits, ~7 decimal digits of precision
3. Double precision: 1+11+52 = 64 bits, ~15 decimal digits of precision
4. The biased exponent and hidden bit are key design features
5. Special encodings handle ±0, ±∞, NaN, and denormalized numbers
6. Not all decimal numbers can be represented exactly — this is inherent, not a bug
7. The standard defines rounding modes and exception handling for predictable behavior