Lecture 7

Floating-Point Format

Lecture

Slides (PDF, PPTX).

Outline

• Floating-point format.
• Standard IEEE 754.
• Floating-point instructions.
• Programs with floating-point operations.

Examples

• sqrt.s
• sqrtd.s
• e.s

Workshop

1. Find decimal values for the following binary values:

    0.0
    0.01
    0.010
    0.0011
    0.00110
    0.001101
    0.0011010
    0.00110011
   

2. Find binary values for the following fractions:

   1/2
   1/8
   3/4
   5/16
   11/32
   

3. Find binary values for the following decimal values:

   0.5
   0.25
   0.125
   1.125
   5.875
   3.1875
   

4. Write program fprint.s that inputs a single and double floating-point value and prints them in the binary format.

5. Write program fprint2.s that separately prints fields (sign, fraction, exponent) of single and double floating-point values. The code of the previous program can be partially reused.

6. Write program farithm.s that inputs three double values a, b, and c, calculates the result of expression a + b - c, and prints the result.

7. Write program even_back.s that does the following:

   Input an integer value N and then N float values. Output line by line only even ones, in reversed order. To decide whether a float number is even, it must be converted (rounded) to an integer value.

   Input:

   6
   12.3
   -11.0
   3.25
   88.01
   0.0
   1.25
   

   Output:

   0.0
   88.01
   12.3
   

8. Write program no_dups.s that does the following:

   Inputs an integer N value and then N double values. Outputs all the doubles, skipping duplicated ones.

   Input:

   8
   12.025
   34.5
   -12.0
   23.25
   12.025
   -12.0
   56.75
   9.125
   

   Output:

   12.025
   34.5
   -12.0
   23.25
   56.75
   9.125
   

Homework

1. Write program fraction_truncate.s that does the following:

   Input three cardinals — A, B and n. Output double float F that has exact n decimal places of A/B. You need to write a subroutine than accepts double f=A/B in fa0 and integer n in a0 and returns rounded double F in fa0.

   Hint: \(10^n*A/B < 2^{31}\)

   Input:

   123
   456
   7
   

   Output:

   0.2697368
   

   Spoiler: \(10^n*A/B < 2^{31}\) means that you can just take an integer part of it, then divide the result back to \(10^n\)

2. Write program cubic_root.s that does the following:

   Input double (positive or negative) values \(1 <= |A| <= 1000000\) and \(0.00001<= ɛ <=0.01\). Calculate a cubical root of A with closeness \(<=ɛ\) (you do not need to round the result).

   HINT: You always can calculate a cubic power of something!

   Input:

   1000
   0.0001
   

   Output:

   9.99995
   

   Spoiler: suppose solution is between M and N (M < N). Select \(K=(M+N)/2\) and if \(|K^3|>|A|\) then solution is between M and K, else it is between K and N.

3. Bonus task (2 bonus points). Write program leibpi.s that does the following:

   Calculate π value using Leibniz formula for π accurate to N decimal places. Input N, output the result. Use function defined in FractionTruncate to truncate out other digits. Keep in mind that the exact formula is calculating π/4, you probably should start with 4 instead 1 to gain exact accuracy. Warning: the algorithm is slow, do not panic, but keep code as simple as possible.

   Input:

   4
   

   Output:

   3.1416
   

   Hint: to gain performance, keep anything in registers.

References

• Standard IEEE 754 (Wikipedia).
• Standard IEEE 754-2008.
• Floating point. Section 3.5 in [CODR].
• Floating point. Section 2.4 in [CSPP].
• RISC-V Assembly Programmer’s Manual.
• RISC-V Formal Specifications in nML.