Method and apparatus for generating mathematical functions

Apparatus for generating an output signal which approximates a designated differentiable function of an input argument signal, comprising:

• means coupled to receive said input argument signal for partitioning between a most significant dominant portion and a next most significant adjunct portion;
• means comprising random access memory coupled to receive said dominant portion for mapping into a primary function signal in accordance with a primary map comprising said designated function of a primary combination comprising said dominant portion and a negative of a scaled predetermined bias, wherein said predetermined bias is selected to restrict an adjunct function signal and said scaling is related to a maximum magnitude of said adjunct portion;
• means coupled to receive said dominant portion for mapping into a secondary function signal in accordance with a secondary map comprising a logarithmic function of a magnitude of a signal comprising a first derivative of said designated function of said primary combination;
• means coupled to receive said adjunct portion for mapping into said adjunct function signal in accordance with an adjunct map comprising said logarithmic function of a magnitude of an adjunct combination comprising said predetermined bias and said adjunct portion;
• means coupled to receive said adjunct function signal and said secondary function signal for adding to produce a secondary sum;
• means coupled to receive said secondary sum for mapping into a primary correction signal in accordance with a primary correction map comprising an exponential function, corresponding to said logarithmic function, of said secondary sum;
• and
• means coupled to receive said primary function signal and said primary correction signal for adding to produce said output signal.

Apparatus for generating an output signal g which approximates a designated differentiable function .function. of an input argument signal x in accordance with an approximation g(x).apprxeq..function.(x) wherein

• g(x)=.function.(u*-.xi.)+axp(alg .function.(u*-.xi.)+alg(.xi.+v*))
• wherein, regarding notation:
• axp(y).ident.a.sup.y and alg(y).ident.log.sub.a (y) wherein a>0, and wherein x.apprxeq.u*+v*, .vertline.v*.vertline.<<.vertline.u*.vertline..sub.max, said designated function .function. has generally variable slope .function.' which is restricted to ensure that .function.'(u*-.xi.) is positively displaced from zero to restrict a corresponding value alg .function.'(u*-.xi.) and predetermined bias .xi. is restricted to ensure that .xi.+v* is positively displaced from zero to restrict a corresponding value alg(.xi.+v*), comprising:
• means coupled to receive said input argument signal x for partitioning between a most significant dominant portion u* and a next most significant adjunct portion v*;
• means comprising random access memory coupled to receive said dominant portion u* for mapping into a primary function signal .function.* in accordance with a primary map .function.*=.function.(u*-.xi.);
• means coupled to receive said dominant portion u* for mapping into a secondary function signal c in accordance with a secondary map c=alg .function.'(u*-.xi.);
• means coupled to receive said adjunct portion v* for mapping into an adjunct function signal d in accordance with an adjunct map d=alg(.xi.+v*);
• means coupled to receive said secondary function signal c and said adjunct function signal d to produce a first combination signal w=c+d;
• means coupled to receive said first combination signal w for mapping into a correction signal y in accordance with a correction map y=axp(w);
• and
• means coupled to receive said primary function signal .function.* and said correction signal y for combining to produce said output signal g as a second combination g=.function.*+y.

Apparatus for generating an output signal g which approximates a designated differentiable function .function. of an input argument signal x in accordance with an approximation g(x).apprxeq..function.(x) wherein ##EQU33## wherein, regarding notation:

• axp(y).ident.a.sup.y and alg(y).ident.log.sub.a (y) wherein a>0, and wherein x.apprxeq.u*+v*, .vertline.v*.vertline.<<.vertline.u*.vertline..sub.max, wherein said designated function .function. is characterized by slope .function.' which is restricted to ensure that .function.'(u*-.xi.)-K is positively displaced from zero to restrict a corresponding value alg(.function.'(u*-.xi.)-K), K is selected for simultaneous scaling of v* and biasing of .function.'(u*-.xi.), and predetermined bias .xi. is restricted to ensure that .xi.+v* is positively displaced from zero to restrict a corresponding value alg(.xi.+v*), comprising:
• means coupled to receive said input argument signal x for partitioning between a most significant dominant portion u* and a next most significant adjunct portion v*;
• means comprising random access memory coupled to receive said dominant portion u* for mapping into a primary function signal .function.* in accordance with a primary map .function.*=.function.(u*-K.xi.);
• means coupled to receive said dominant portion u* for mapping into a secondary function signal c in accordance with a secondary map c=alg(.function.'(u*-.xi.)-K);
• means coupled to receive said adjunct portion v* for mapping into an adjunct function signal d in accordance with an adjunct map d=alg(.xi.+v*);
• means coupled to receive said secondary function signal c and said adjunct function signal d to produce a first combination signal w=c+d;
• means coupled to receive said first combination signal w for mapping into a principal correction signal y in accordance with a correction map y=axp(w);
• means coupled to receive said adjunct portion v* for scaling to produce an adjunct correction Kv*;
• and
• means coupled to receive said primary function signal .function.*, said correction signal y and said adjunct correction Kv* for combining to produce said output signal g as a second combination g=.function.*+y+Kv*.

Apparatus for generating an output signal g which approximates a designated differentiable function .function. of an input argument signal x in accordance with an approximation g(x).apprxeq..function.(x) wherein ##EQU34## wherein, regarding notation:

• axp(y).ident.a.sup.y and alg(y).ident.log.sub.a (y) wherein a>0, and wherein x.apprxeq.u*+v*, .vertline.v*.vertline.<<.vertline.u*.vertline..sub.max, wherein said designated function .function. is characterized by slope .function.' which is restricted to ensure that .vertline..function.'(u*-.xi.)-K.vertline. is positively displaced from zero to restrict a corresponding value alg.vertline..function.'(u*-.xi.)-K.vertline., wherein K is selected for simultaneous scaling of v* and biasing of .function.'(u*-.xi.), predetermined bias .xi. is restricted to ensure that .vertline..xi.+v*.vertline. is positively displaced from zero to restrict a corresponding value alg.vertline..xi.+v*.vertline., and s=sign{(.xi.+v*)(.function.'(u*-.xi.)-K)}, comprising:
• means coupled to receive said input argument signal x for partitioning between a most significant dominant portion u* and a next most significant adjunct portion v*;
• means comprising random access memory coupled to receive said dominant portion u* for mapping into a primary function signal .function.* in accordance with a primary map .function.*=.function.(u*-.xi.)+K.xi.;
• means coupled to receive said dominant portion u* for mapping into a secondary function signal c in accordance with a secondary map c=alg.vertline..function.'(u*-.xi.)-K.vertline.;
• means coupled to receive said adjunct portion v* for mapping into an adjunct function signal d in accordance with an adjunct map d=alg.vertline..xi.+v*.vertline.;
• means coupled to receive said secondary function signal c and said adjunct function signal d for combining to produce a first combination signal w=c+d;
• means coupled to receive said first combination signal w for mapping into an unsigned principal correction signal y in accordance with a correction map y=axp(w);
• means comprising combinatorial logic coupled to receive said dominant portion u* and said adjunct portion v* for producing a sign signal s in accordance with a relation s=sign{(.xi.+v*)(.function.'(u*-.xi.)-K)};
• means coupled to receive said sign signal s and said unsigned principal correction signal y for attaching a sign to produce a signed principal correction signal sy;
• means coupled to receive said adjunct portion v* for scaling to produce an adjunct correction signal Kv*;
• and
• means coupled to receive said primary function signal .function.*, said signed principal correction signal sy and said adjunct correction Kv* for combining to produce said output signal g as a second combination g=.function.*+sy+Kv*.

Apparatus for generating a radix-r.sub.2 output signal g which approximates a designated differentiable function .function. of a radix-r.sub.1 input argument signal x in accordance with an approximation g(x).apprxeq..function.(x) wherein

• g(x)=.function.(u*-h.xi.)+h axp(alg(.xi.+v*)+alg .function.'(u*-h.xi.))
• wherein, regarding notation:
• axp(y).ident.a.sup.y and alg(y).ident.log.sub.a (y) wherein a>0, and wherein x.apprxeq.u*+hv*, h=r.sub.1.sup.-n wherein n is a positive integer, .vertline.h.vertline.<<1, .vertline.hv*.vertline.<<.vertline.u*.vertline.. sub.max, wherein said designated function .function. is characterized by slope .function.' which is restricted to ensure that .function.'(u*-h.xi.) is positively displaced from zero to restrict a corresponding value alg(.function.'(u*-.xi.)), and predetermined bias .xi. is restricted to ensure that .xi.+v* is positively displaced from zero to restrict a corresponding value alg(.xi.+v*), comprising:
• means coupled to receive said input argument signal x for partitioning between a most significant dominant portion u* and an unscaled next most significant adjunct portion v* which is consistent with a relation x=u*+hv*;
• means comprising random access memory coupled to receive said dominant portion u* for mapping into a radix-r.sub.2 primary function signal .function.* in accordance with a primary map .function.*=.function.(u*-h.xi.);
• means coupled to receive said dominant portion u* for mapping into a secondary function signal c in accordance with a secondary map c=alg .function.'(u*-h.xi.);
• means coupled to receive said adjunct portion v* for mapping into an adjunct function signal d in accordance with an adjunct map d=alg(.xi.+v*);
• means coupled to receive said secondary function signal c and said adjunct function signal d for combining to produce a first combination signal w=c+d;
• means coupled to receive said first combination signal w for mapping into an unscaled correction signal y in accordance with an unscaled correction map y=axp(w);
• means coupled to receive said unscaled correction signal y for scaling to produce a scaled radix-r.sub.2 correction signal hy;
• and
• means coupled to receive said primary function signal .function.* and said scaled correction signal hy for combining to produce said output signal g as a radix-r.sub.2 second combination g=.function.*+hy.

Apparatus for generating a radix-r.sub.2 output signal g which approximates a designated differentiable function .function. of a radix-r.sub.1 input argument signal x in accordance with an approximation g(x).apprxeq..function.(x) wherein ##EQU35## wherein, regarding notation:

• axp(y).ident.a.sup.y and alg(y).ident.log.sub.a (y) wherein a>0, and wherein x.apprxeq.u*+hv*, h=r.sub.1.sup.-n wherein n is a positive integer, .vertline.h.vertline.<<1, .vertline.hv*.vertline.<<.vertline.u*.vertline.. sub.max, wherein said designated function .function. is characterized by slope .function.' which is restricted to ensure that .function.'(u*-h.xi.)-K is positively displaced from zero to restrict a corresponding value alg(.function.'(u*-h.xi.)-K), K is selected for simultaneous scaling of hv* and biasing of .function.'(u*-h.xi.), and predetermined bias .xi. is restricted to ensure that .xi.+v* is positively displaced from zero to restrict a corresponding value alg(.xi.+v*), comprising:
• means coupled to receive said input argument signal x for partitioning between a most significant dominant portion u* and a next most significant adjunct portion v* which is consistent with a relation x=u*+hv*;
• means comprising random access memory coupled to receive said dominant portion u* for mapping into a radix-r.sub.2 primary function signal .function.* in accordance with a primary map .function.*=.function.(u*-h.xi.)+hK.xi.;
• means coupled to receive said dominant portion u* for mapping into a secondary function signal c in accordance with c=alg(.function.'(u*-h.xi.)-K) as a secondary map;
• means coupled to receive said adjunct portion v* for mapping into an adjunct function signal d in accordance with an adjunct map d=alg(.xi.+v*);
• means coupled to receive said secondary function signal c and said adjunct function signal d for combining to produce a first combination signal w=c+d;
• means coupled to receive said first combination signal w for mapping into an unscaled principal correction signal y in accordance with an unscaled correction map y=axp(w);
• means coupled to receive said unscaled correction signal y for scaling to produce a scaled radix-r.sub.2 correction signal hy;
• means coupled to receive said adjunct portion v* for scaling to produce a radix-r.sub.2 adjunct correction signal hKv*;
• and
• means coupled to receive primary function signal .function.*, said scaled principal correction signal hy and said adjunct correction signal hKv* for combining to produce said output signal g as a second combination g=.function.*+hy+hKv*.

Apparatus for generating a radix-r.sub.2 output signal g which approximates a designated differentiable function .function. of a radix-r.sub.1 input argument signal x in accordance with an approximation g(x).apprxeq..function.(x) wherein:

• ##EQU36## wherein, regarding notation:
• axp(y).ident.a.sup.y and alg(y).ident.log.sub.a (y) wherein a>0, and wherein x.apprxeq.u*+hv*, h=r.sub.1.sup.-n wherein n is a positive integer, .vertline.h.vertline.<<1, .vertline.hv*.vertline.<<.vertline.u*.vertline.. sub.max, wherein said designated function .function. is characterized by slope .function.' which is restricted to ensure that .vertline..function.'(u*-h.xi.)-K.vertline. is positively displaced from zero to restrict a corresponding value alg.vertline..function.'(u*-h.xi.)-K.vertline., s=sign{(.xi.+v*)(.function.'(u*-h.xi.)-K)}, K is selected for simultaneous scaling of v* and biasing of .function.'(u*-h.xi.), and predetermined bias .xi. is restricted to ensure that .vertline..xi.+v*.vertline. is positively displaced from zero to restrict a corresponding value alg.vertline..xi.+v*.vertline., comprising:
• means coupled to receive said input argument signal x for partitioning between a most significant dominant portion u* and a next most significant adjunct portion v* which is consistent with a relation x=u*+hv*;
• means comprising random access memory coupled to receive said dominant portion u* for mapping into a radix-r.sub.2 primary function signal .function.* in accordance with a primary map .function.*=.function.(u*-h.xi.)+hK.xi.;
• means coupled to receive said dominant portion u* for mapping into a secondary function signal c in accordance with a secondary map c=alg.vertline..function.'(u*-h.xi.)-K.vertline.;
• means coupled to receive said adjunct portion v* for mapping into an adjunct function signal d in accordance with an adjunct map d=alg.vertline..xi.+v*.vertline.;
• means coupled to receive said secondary function signal c and said adjunct function signal d for combining to produce a first combination signal w=c+d;
• means coupled to receive said first combination signal w for mapping into an unscaled unsigned principal correction signal y in accordance with an unscaled correction map y.ident.axp(w);
• means comprising combinatorial logic coupled to receive said dominant portion u* and said adjunct portion v* for mapping into a sign signal s in accordance with a sign map s=sign{.xi.+v*).function.'(u*-h.xi.)};
• means coupled to receive said sign signal s and said unscaled unsigned principal correction signal y for scaling to produce a scaled signed radix-r.sub.2 principal correction signal hsy;
• means coupled to receive said adjunct portion v* for scaling to produce a scaled radix-r.sub.2 adjunct correction hKv*;
• and
• means coupled to receive said primary function signal .function.*, said scaled signed principal correction signal hsy and said scaled adjunct correction hKv* for combining to produce said output signal g as a second combination g=.function.*+hsy+hKv*.

Apparatus for generating an output signal which approximates a designated differentiable function of an input argument signal, comprising:

• means coupled to receive said input argument signal for partitioning between a most significant dominant portion and a next most significant adjunct portion;
• means comprising random access memory coupled to receive said dominant portion for mapping into said output primary function signal in accordance with a primary map comprising said designated function of said dominant portion;
• means coupled to receive said dominant portion for extracting a most significant primary dominant portion;
• means coupled to receive said primary dominant portion and said adjunct portion for arranging into a hybrid signal as a hybrid arrangement;
• means coupled to receive said hybrid signal for mapping into a hybrid correction signal in accordance with a hybrid correction map comprising selected hybrid correction function signals of a hybrid correction function for said designated function, said hybrid correction function signal comprising a product of said adjunct portion and a first derivative of said designated function of said primary dominant portion, each said selected hybrid correction function signal corresponding to each possible ones of said hybrid signals as an input argument, to produce a hybrid correction signal as an interpolation correction signal;
• and
• means coupled to receive said interpolation correction signal and said primary function signal for combining to produce said output signal.