1. Arithmetic for Computers
Chapter 3
Arithmetic for Computers

3. Morgan Kaufmann Publishers
14 September, 2018
Floating Point
§3.5 Floating Point
Representation for non-integral numbers
Including very small and very large numbers
Like scientific notation
–2.34 × 1056
× 10–4
× 109
Normalized number: A number in floating-point notation that has no leading 0s
In binary
±1.xxxxxxx2 × 2yyyy
Types float and double in C
Floating point: Computer arithmetic that represents numbers in which the binary point is not fixed
normalized
not normalized
Chapter 3 — Arithmetic for Computers

4. LEGv8 floating-point number
Floating point numbers are of the form (-1)s x F x 2E
S: sign bit (0  non-negative, 1  negative), E: exponent, F: fraction
Numbers representable in computer
Smallest fractions 2.0ten × 10−38
Largest numbers 2.0ten × 1038
overflow (floating-point): a positive exponent becomes too large to fit in the exponent field. Raises exception
underflow (floating-point): a negative exponent becomes too large to fit in the exponent field. Raises interrupt 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 S
exponent
fraction
1 bit bits bits

5. Floating Point Standard
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Floating Point Standard
14 September, 2018
Defined by IEEE Std
Developed in response to divergence of representations
Portability issues for scientific code
Now almost universally adopted
Two representations
Single precision (32-bit)
Double precision (64-bit)
IEEE 754 makes the leading 1 bit of normalized binary numbers implicit.
Single precision: 24 bits (implied 1 and a 23-bit fraction), and double precision: 53 bits (1 +52).
Significand: represents the 24- or 53-bit number (1 plus the fraction)
Fraction: 23- or 52-bit number.
Since 0 has no leading 1, it is given the reserved exponent value 0 so that the hardware won’t attach a leading 1 to it.
Chapter 3 — Arithmetic for Computers

6. IEEE 754 Floating Point Standard
00 … 00two represents 0;
Representation of the rest of the numbers: (-1)s x (1+Fraction) x 2E
Special symbols to represent unusual events.
For example, instead of interrupting on a divide by 0, software can set the result to a bit pattern representing +∞ or −∞;
the largest exponent is reserved for these special symbols.
NaN (Not a Number): symbol for the result of invalid operations, such as 0/0 or subtracting infinity from infinity.

7. LEGv8 floating-point number
Floating point numbers are of the form (-1)s x F x 2E
S: sign bit (0  non-negative, 1  negative), E: exponent, F: fraction
Numbers representable in computer
Smallest fractions 2.0ten × 10−38
Largest numbers 2.0ten × 1038
overflow (floating-point): a positive exponent becomes too large to fit in the exponent field. Raises exception
underflow (floating-point): a negative exponent becomes too large to fit in the exponent field. Raises interrupt 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 S
exponent
fraction
1 bit bits bits

8. Floating Point Standard
Morgan Kaufmann Publishers
Floating Point Standard
14 September, 2018
Defined by IEEE Std
Developed in response to divergence of representations
Portability issues for scientific code
Now almost universally adopted
Two representations
Single precision (32-bit)
Double precision (64-bit)
IEEE 754 makes the leading 1 bit of normalized binary numbers implicit.
Single precision: 24 bits (implied 1 and a 23-bit fraction), and double precision: 53 bits (1 +52).
Significand: represents the 24- or 53-bit number (1 plus the fraction)
Fraction: 23- or 52-bit number.
Since 0 has no leading 1, it is given the reserved exponent value 0 so that the hardware won’t attach a leading 1 to it.
Chapter 3 — Arithmetic for Computers

9. IEEE 754 Floating Point Standard
00 … 00two represents 0;
Representation of the rest of the numbers: (-1)s x (1+Fraction) x 2E
Special symbols to represent unusual events.
For example, instead of interrupting on a divide by 0, software can set the result to a bit pattern representing +∞ or −∞;
the largest exponent is reserved for these special symbols.
NaN (Not a Number): symbol for the result of invalid operations, such as 0/0 or subtracting infinity from infinity.

10. IEEE 754 Floating Point Standard
Floating-point representation to process integer comparisons
Consequence: sign is in the most significant bit
Placing the exponent before the significand
simplifies the sorting of floating-point numbers using integer comparison instructions
numbers with bigger exponents look larger than numbers with smaller exponents, as long as both exponents have the same sign
Negative exponents pose a challenge to simplified sorting
Single precision representation of 1.0two x 2-1: exponent looks like a big number
The value 1.0two × 2+1 would look like the smaller binary number 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

11. IEEE 754 Floating-Point Format
Morgan Kaufmann Publishers
IEEE 754 Floating-Point Format
14 September, 2018
single: 8 bits double: 11 bits
single: 23 bits double: 52 bits S
Exponent
Fraction
Most negative exponent represented as and
most positive exponent represented as
Biased notation:
Exponent: unsigned representation = actual exponent + Bias
Ensures exponent is unsigned
Single precision: Bias = 127; Double precision: Bias = 1203
exponent −1 represented by −1 +127ten = 126ten = two, and
Exponent +1 represented by = 128ten = two
Chapter 3 — Arithmetic for Computers

12. Single-Precision Range
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14 September, 2018
Single-Precision Range
Exponents 0000…00 and 1111…11 reserved
Smallest value
Exponent:  actual exponent = 1 – 127 = –126
Fraction: 000…00  significand = 1.0
±1.0 × 2–126 ≈ ±1.2 × 10–38
Largest value
exponent:  actual exponent = 254 – 127 = +127
Fraction: 111…11  significand ≈ 2.0
±2.0 × ≈ ±3.4 × 10+38
Chapter 3 — Arithmetic for Computers

13. Double-Precision Range
Morgan Kaufmann Publishers
14 September, 2018
Double-Precision Range
Exponents 0000…00 and 1111…11 reserved
Smallest value
Exponent:  actual exponent = 1 – 1023 = –1022
Fraction: 000…00  significand = 1.0
±1.0 × 2–1022 ≈ ±2.2 × 10–308
Largest value
Exponent:  actual exponent = 2046 – 1023 = +1023
Fraction: 111…11  significand ≈ 2.0
±2.0 × ≈ ±1.8 ×
Chapter 3 — Arithmetic for Computers

14. Floating-Point Precision
Morgan Kaufmann Publishers
14 September, 2018
Floating-Point Precision
Relative precision
all fraction bits are significant
Single: approx 2–23
Equivalent to 23 × log102 ≈ 23 × 0.3 ≈ 6 decimal digits of precision
Double: approx 2–52
Equivalent to 52 × log102 ≈ 52 × 0.3 ≈ 16 decimal digits of precision
Chapter 3 — Arithmetic for Computers

15. Floating-Point Example
Morgan Kaufmann Publishers
14 September, 2018
Floating-Point Example
Represent –0.75
–0.75 = (–1)1 × 1.12 × 2–1
S = 1
Fraction = 1000…002
Exponent = –1 + Bias
Single: – = 126 =
Double: – = 1022 =
Single: …00
Double: …00
Chapter 3 — Arithmetic for Computers

16. Floating-Point Example
Morgan Kaufmann Publishers
14 September, 2018
Floating-Point Example
What number is represented by the single-precision float
…00
S = 1
Fraction = 01000…002 = 1 ×2−2 = ¼ = 0.25
exponent = = 129
x = (–1)1 × ( ) × 2(129 – 127)
= (–1) × 1.25 × 22
= –5.0
Chapter 3 — Arithmetic for Computers

17. Morgan Kaufmann Publishers
Denormal Numbers
14 September, 2018
Exponent =  hidden bit is 0
Smaller than normal numbers
allow for gradual underflow, with diminishing precision
Denormal with fraction =
Two representations of 0.0!
Chapter 3 — Arithmetic for Computers

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14 September, 2018
Infinities and NaNs
Exponent = , Fraction =
±Infinity
Can be used in subsequent calculations, avoiding need for overflow check
Exponent = , Fraction ≠
Not-a-Number (NaN)
Indicates illegal or undefined result
e.g., 0.0 / 0.0
Can be used in subsequent calculations
Chapter 3 — Arithmetic for Computers

19. Floating-Point Addition
Morgan Kaufmann Publishers
14 September, 2018
Floating-Point Addition
Consider a 4-digit decimal example
9.999 × × 10–1
1. Align decimal points
Shift number with smaller exponent
9.999 × × 101
2. Add significands
9.999 × × 101 = × 101
3. Normalize result & check for over/underflow
× 102
4. Round and renormalize if necessary
1.002 × 102
Chapter 3 — Arithmetic for Computers

20. Algorithm for binary floating-point addition

21. Binary Floating-Point Addition
Morgan Kaufmann Publishers
14 September, 2018
Binary Floating-Point Addition
Now consider a 4-digit binary example
× 2–1 + – × 2–2 (0.5 + –0.4375)
1. Align binary points
Shift number with smaller exponent
× 2–1 + – × 2–1
2. Add significands
× 2–1 + – × 2–1 = × 2–1
3. Normalize result & check for over/underflow
× 2–4, Since 127 ≥−4 ≥−126, no overflow/underflow
4. Round and renormalize if necessary
× 2–4 (no change) =
Chapter 3 — Arithmetic for Computers

22. Morgan Kaufmann Publishers
14 September, 2018
FP Adder Hardware
Much more complex than integer adder
Doing it in one clock cycle would take too long
Much longer than integer operations
Slower clock would penalize all instructions
FP adder usually takes several cycles
Can be pipelined
Chapter 3 — Arithmetic for Computers

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FP Adder Hardware
14 September, 2018
Step 1
Step 2
Step 3
Step 4
Chapter 3 — Arithmetic for Computers

24. Floating-Point Multiplication
Morgan Kaufmann Publishers
14 September, 2018
Floating-Point Multiplication
Consider a 4-digit decimal example
1.110 × 1010 × × 10–5
1. Add exponents
For biased exponents, subtract bias from sum
New exponent = 10 + –5 = 5
2. Multiply significands
1.110 × =  × 105
3. Normalize result & check for over/underflow
× 106
4. Round and renormalize if necessary
1.021 × 106
5. Determine sign of result from signs of operands
× 106
Chapter 3 — Arithmetic for Computers

25. Floating-Point Multiplication

26. Binary Floating-Point Multiplication
Morgan Kaufmann Publishers
14 September, 2018
Binary Floating-Point Multiplication
Now consider a 4-digit binary example
× 2–1 × – × 2–2 (0.5 × –0.4375)
1. Add exponents
Unbiased: –1 + –2 = –3
Biased: (– ) + (– ) = – – 127 = –
2. Multiply significands
× =  × 2–3
3. Normalize result & check for over/underflow
× 2–3 (no change), since 127 ≥−3 ≥−126, no overflow/underflow
4. Round and renormalize if necessary
× 2–3 (no change)
5. Determine sign: +ve × –ve  –ve
– × 2–3 = two = - 7/25ten = - 7/32ten = – ten
Chapter 3 — Arithmetic for Computers

27. FP Arithmetic Hardware
Morgan Kaufmann Publishers
14 September, 2018
FP Arithmetic Hardware
FP multiplier is of similar complexity to FP adder
But uses a multiplier for significands instead of an adder
FP arithmetic hardware usually does
Addition, subtraction, multiplication, division, reciprocal, square-root
FP  integer conversion
Operations usually takes several cycles
Can be pipelined
Chapter 3 — Arithmetic for Computers

28. FP Instructions in LEGv8
Morgan Kaufmann Publishers
14 September, 2018
FP Instructions in LEGv8
Separate FP registers
32 single-precision: S0, …, S31
32 double-precision: DS0, …, D31
Sn stored in the lower 32 bits of Dn
FP instructions operate only on FP registers
Programs generally don’t do integer ops on FP data, or vice versa
More registers with minimal code-size impact
FP load and store instructions
LDURS, LDURD
STURS, STURD
Chapter 3 — Arithmetic for Computers

29. FP Instructions in LEGv8
Morgan Kaufmann Publishers
14 September, 2018
FP Instructions in LEGv8
Single-precision arithmetic
FADDS, FSUBS, FMULS, FDIVS
e.g., FADDS S2, S4, S6
Double-precision arithmetic
FADDD, FSUBD, FMULD, FDIVD
e.g., FADDD D2, D4, D6
Single- and double-precision comparison
FCMPS, FCMPD
Sets or clears FP condition-code bits
Branch on FP condition code true or false
B.cond
Chapter 3 — Arithmetic for Computers

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FP Example: °F to °C
C code:
float f2c (float fahr) { return ((5.0/9.0)*(fahr )); }
fahr in S12, result in S0, literals in global memory space
Compiled LEGv8 code:
f2c:
LDURS S16, [X27,const5] // S16 = 5.0 (5.0 in memory)
LDURS S18, [X27,const9] // S18 = 9.0 (9.0 in memory)
FDIVS S16, S16, S // S16 = 5.0 / 9.0
LDURS S18, [X27,const32] // S18 = 32.0
FSUBS S18, S12, S // S18 = fahr – 32.0
FMULS S0, S16, S // S0 = (5/9)*(fahr – 32.0)
BR LR // return
Chapter 3 — Arithmetic for Computers