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IEEE-754 float & My GFloat

• IEEE-754 float

  sign	exponents（8 bits）					fractions（23 bits）
  31	30	29	...	24	23	22	21	...	1	0

  • $\mathbf{X}_{IEEE754} = (-1)^\mathbf{sign} \times (1.\mathbf{fraction}) \times 2 ^{\mathbf{exponent} - 127}$

• GFloat

  sign	fractions（23 bits）					exponents（8 bits）
  31	30	29	...	9	8	7	6	...	1	0

  • $\mathbf{X}_{GFloat} = (-1)^\mathbf{sign} \times (\mathbf{fraction}) \times 2 ^{\mathbf{exponent} - 127}$

Transcendental function

• Sin、Cos : Mathematica MiniMaxApproximation or use look-up table
  • https://reference.wolfram.com/language/FunctionApproximations/tutorial/FunctionApproximations.html
• ASin、ACos、Atan : Taylor expansion
  • https://www.wolframalpha.com/input?i=taylor+series+of++asin%28x%29+
• InvSqrt、Sqrt : Mathematica MiniMaxApproximation + Newton's method
• Exp、Pow : convert to power of 2 to calculate
• Log、Log10 : convert to logarithm of 2 to calculate
• pow2、log2 ：scanforwardbits to get exponent of the number. and use MiniMaxApproximation to caculate [0.5~1.5] value

BenchMark

• random input