Single precision Vs Double precision floating point numbers

From the IEEE standard for floating point arithmetic

Single Precision

The IEEE single precision floating point standard representation requires a 32 bit word, which may be represented as numbered from 0 to 31, left to right.

• The first bit is the sign bit, S,
• the next eight bits are the exponent bits, 'E', and
• the final 23 bits are the fraction 'F':

  S EEEEEEEE FFFFFFFFFFFFFFFFFFFFFFF
  0 1      8 9                    31
  

The value V represented by the word may be determined as follows:

• If E=255 and F is nonzero, then V=NaN ("Not a number")
• If E=255 and F is zero and S is 1, then V=-Infinity
• If E=255 and F is zero and S is 0, then V=Infinity
• If 0<E<255 then V=(-1)**S * 2 ** (E-127) * (1.F) where "1.F" is intended to represent the binary number created by prefixing F with an implicit leading 1 and a binary point.
• If E=0 and F is nonzero, then V=(-1)**S * 2 ** (-126) * (0.F). These are "unnormalized" values.
• If E=0 and F is zero and S is 1, then V=-0
• If E=0 and F is zero and S is 0, then V=0

In particular,

0 00000000 00000000000000000000000 = 0
1 00000000 00000000000000000000000 = -0

0 11111111 00000000000000000000000 = Infinity
1 11111111 00000000000000000000000 = -Infinity

0 11111111 00000100000000000000000 = NaN
1 11111111 00100010001001010101010 = NaN

0 10000000 00000000000000000000000 = +1 * 2**(128-127) * 1.0 = 2
0 10000001 10100000000000000000000 = +1 * 2**(129-127) * 1.101 = 6.5
1 10000001 10100000000000000000000 = -1 * 2**(129-127) * 1.101 = -6.5

0 00000001 00000000000000000000000 = +1 * 2**(1-127) * 1.0 = 2**(-126)
0 00000000 10000000000000000000000 = +1 * 2**(-126) * 0.1 = 2**(-127) 
0 00000000 00000000000000000000001 = +1 * 2**(-126) * 
                                     0.00000000000000000000001 = 
                                     2**(-149)  (Smallest positive value)

Double Precision

The IEEE double precision floating point standard representation requires a 64 bit word, which may be represented as numbered from 0 to 63, left to right.

• The first bit is the sign bit, S,
• the next eleven bits are the exponent bits, 'E', and
• the final 52 bits are the fraction 'F':

  S EEEEEEEEEEE FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
  0 1        11 12                                                63
  

The value V represented by the word may be determined as follows:

• If E=2047 and F is nonzero, then V=NaN ("Not a number")
• If E=2047 and F is zero and S is 1, then V=-Infinity
• If E=2047 and F is zero and S is 0, then V=Infinity
• If 0<E<2047 then V=(-1)**S * 2 ** (E-1023) * (1.F) where "1.F" is intended to represent the binary number created by prefixing F with an implicit leading 1 and a binary point.
• If E=0 and F is nonzero, then V=(-1)**S * 2 ** (-1022) * (0.F) These are "unnormalized" values.
• If E=0 and F is zero and S is 1, then V=-0
• If E=0 and F is zero and S is 0, then V=0

Reference:
ANSI/IEEE Standard 754-1985,
Standard for Binary Floating Point Arithmetic.

Optimization using __restrict keyword

well explained in this link..

http://cellperformance.beyond3d.com/articles/2006/05/demystifying-the-restrict-keyword.html

SUMMARY

• Strict aliasing means that two objects of different types cannot refer to the same location in memory. Enable this option in GCC with the -fstrict-aliasing flag. Be sure that all code can safely run with this rule enabled. Enable strict aliasing related warnings with -Wstrict-aliasing, but do not expect to be warned in all cases.
• Compare the assembly output of the function with restricted pointers and file scope arrays to ensure that all of the possible aliasing information has been used.
• Only use restricted leaf pointers. Use of parent pointers may break the restrict contract.
• Publish as many assumptions as possible about aliasing information in the function declaration.
• Memory windows may be overlapping and still be without aliases. Do not limit the data design to non-overlapping windows.
• Begin using the restrict keyword immediately. Retrofit old code as soon as possible.
• Keep loads and stores separated from calculations. This results in better scheduling in GCC, and makes the relationship between the output assembly and the original source clearer.

Advantage of zero length array in C

https://www.quora.com/What-is-the-advantage-of-using-zero-length-arrays-in-C