Exponential series Functions

Functions
float64_t	csky_exp_f64 (float64_t x)
	Fast approximation to the exponential function based on e for float64_t -point_t data. More...
float64_t	csky_pow2_f64 (float64_t x)
	Fast approximation to the 2 based exponential function for float64_t -point data. More...
float64_t	csky_pow_f64 (float64_t x, float64_t y)
	Fast approximation to the exponential function for float64_t-point data. More...

Description

Exponentional series functions includes pow, pow2 and exp, which are represented any value based, 2 based and e based Exponentional function repectively. The common methods to compute these functions are ploynomial approximation and table loop-up. And, it is a double floating version for the functions, say, the input and output are both double.

Function Documentation

float64_t csky_exp_f64

(

float64_t

x

)

Parameters

[in]

x

input value in double.

Returns

e^x.

The implementation is based on table lookup using 712 coarse values together with 1024 fine values which are both in double format.

The steps used are:

1. First, compute the result with table lookup when smallint < x < bigint.
2. Then, process the special case namely x > badint or x < smallint.
3. Finally, compute the result with table loopup when bigint < x < badint.

float64_t csky_pow2_f64

(

float64_t

x

)

Parameters

[in]

x

input value in double.

Returns

2^x.

The implementation is based on table lookup using 512 delta values combining with 512 accurate values.

The steps used are:

1. processing of special cases.
2. argument reduction. Choose integers ex, -256 <= t < 256, and some real -1/1024 <= x1 <= 1024 so that

    x = ex + t/512 + x1

3. adjust for accurate table entry. Find e so that

    x = ex + t/512 + e + x2

   where -1e6 < e < 1e6, and (float64_t)(2^(t/512+e)) is accurate to one part in 2^-64.
4. approximate 2^x2 - 1, using a fourth-degree polynomial, with maximum error in [-2^-10-2^-30,2^-10+2^-30] less than 10^-19.

float64_t csky_pow_f64	(	float64_t	x,
		float64_t	y
	)

Parameters

[in]	x	input value in double, the base of pow.
[in]	x	input value in double, the exponention of pow.

Returns

x^y.

The implementation is based on the transform

 x^y = e^(y*log(x))

The steps used are:

1. First, calculation of the nearest value of log(x) with series expansion or table look up depending on the range of x.
2. Then, compute of the nearest value of e^(y*log(x)) with the function e^(x+xx), in which, defferent methods are used based on the range of x.
3. Finally, some special cases are processed.