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f32

This format conforms to the IEEE-754 binary32/single format. Notable differences from the other single format in this library:

"single" (as it appears in this library) binary32
exponent bias is +128 exponent bias is +127
All special values (0, inf, NaN) are stored with an exponent of -128 0 has an exponent of -127, inf and NaN have an exponent of +128
Sign bit is stored in the top bit of the significand so the exponent is not split across bytes. Sign bit is stored in the top byte, so the exponent is split across two bytes.

Format

binary32 floats are stored in little endian. The "most significant bit" is sign, the next 8 bits are exponent, and the next 23 bits encode the 24-bit significand (note that the top bit of the significand is always 1, so we don't need to store it explicitly):

'm' is significand
'e' is exponent
's' is sign
'x' is any value (0 or 1, doesn't matter)
'-' is any value (0 or 1, doesn't matter)

        seeeeeee emmmmmmm mmmmmmmm mmmmmmmm
+0      00000000 0xxxxxxx xxxxxxxx xxxxxxxx
-0      10000000 0xxxxxxx xxxxxxxx xxxxxxxx
 1      00111111 10000000 00000000 00000000
 2      01000000 00000000 00000000 00000000
-1      10111111 10000000 00000000 00000000
pi      01000000 01001001 00001111 11011011
+inf    01111111 10000000 00000000 00000000
-inf    11111111 10000000 00000000 00000000
NaN     -1111111 1xxxxxxx xxxxxxxx xxxxxxxx ;as long as at least 1 'x' is non-zero

Routines

(Note, I haven't tested very small and very large numbers.)

tested routine Description
yes f32abs |x| - absolute value of x
yes f32acos arccosine(x)
yes f32acosh hyperbolic arccosine(x)
yes f32add x+y
yes f32amean (x+y)/2 - arithmetic mean
yes f32asin arcsine(x)
yes f32asinh hyperbolic arcsine(x)
yes f32atan arctangent(x)
yes f32atanh hyperbolic arctangent(x)
yes f32bgi 1/BG(x,y) - BG(x,y) is the Borchardt-Gauss Mean
f32cmp compares x to y - returns Z and C flag
yes f32cos cosine(x)
yes f32cosh hyperbolic cosine(x)
yes f32div x/y
yes f32exp e^x
yes f32geomean sqrt(x*y) - geometric mean
yes f32log log(x) - natural logarithm
yes f32log10 log10(x) - log base 10
yes f32log2 log2(x) - log base 2
yes f32logy log_y(x) - log base y
yes f32mod1 x % 1
yes f32mul x*y
yes f32mul2 x*2
yes f32neg -x
yes f32pow x^y
yes f32pow10 10^x
yes f32pow2 2^x
yes f32rand rand - uniform random variable selected from [0,1)
yes f32randnorm randnorm - normal random variable centered about 0 with standard deviation of 1.
yes f32rsub -x+y
yes f32sin sine(x)
yes f32sinh hyperbolic sine(x)
yes f32sqrt sqrt(x) - square root
yes f32sub x-y
yes f32tan tangent(x)
f32tanh hyperbolic tangent(x) - seems to be broken at tanh(0)

Useful Subroutines

These are in the /f32/routines folder:

  • f32_muli8
    • multiply an f32 float by a signed 8-bit integer
    • this is faster than multiplying two floats
  • f32_mulu8_divpow2
    • multiply an f32 float by an unsigned 8-bit integer, and divide by a power of 2
    • this is faster than multiplying two floats.
  • f32mulpow2
    • multiply an f32 float by a power of 2
    • Significantly faster than multiplication. This just needs to do an 8-bit add along with edge-case testing (and special numbers).