6.5.8 Floating Point ¶

For the rules used by the text interpreter for recognising floating-point numbers see Number Conversion.

Gforth has a separate floating point stack, but the documentation uses the unified notation.¹⁰

Floating point numbers have a number of unpleasant surprises for the unwary (e.g., floating point addition is not associative) and even a few for the wary. You should not use them unless you know what you are doing or you don’t care that the results you get may be totally bogus. If you want to learn about the problems of floating point numbers (and how to avoid them), you might start with David Goldberg, What Every Computer Scientist Should Know About Floating-Point Arithmetic, ACM Computing Surveys 23(1):5−48, March 1991.

Conversion between integers and floating-point:

s>f ( n – r ) floating-ext “s-to-f”

d>f ( d – r ) floating “d-to-f”

f>s ( r – n ) floating-ext “f-to-s”

f>d ( r – d ) floating “f-to-d”

Arithmetics:

f+ ( r1 r2 – r3 ) floating “f-plus”

f- ( r1 r2 – r3 ) floating “f-minus”

f* ( r1 r2 – r3 ) floating “f-star”

f/ ( r1 r2 – r3 ) floating “f-slash”

fnegate ( r1 – r2 ) floating “f-negate”

fabs ( r1 – r2 ) floating-ext “f-abs”

fcopysign ( r1 r2 – r3  ) gforth-1.0 “fcopysign”

r3 takes its absolute value from r1 and its sign from r2

fmax ( r1 r2 – r3 ) floating “f-max”

fmin ( r1 r2 – r3 ) floating “f-min”

floor ( r1 – r2 ) floating “floor”

Round towards the next smaller integral value, i.e., round toward negative infinity.

fround ( r1 – r2 ) floating “f-round”

Round to the nearest integral value.

ftrunc ( r1 – r2  ) floating-ext “f-trunc”

round towards 0

f** ( r1 r2 – r3 ) floating-ext “f-star-star”

r3 = r1^r2

fsqrt ( r1 – r2 ) floating-ext “f-square-root”

fexp ( r1 – r2 ) floating-ext “f-e-x-p”

r2 = e^r1

fexpm1 ( r1 – r2 ) floating-ext “f-e-x-p-m-one”

r2=e^r1−1

fln ( r1 – r2 ) floating-ext “f-l-n”

Natural logarithm: r1 = e^r2

flnp1 ( r1 – r2 ) floating-ext “f-l-n-p-one”

Inverse of fexpm1: r1+1 = e^r2

flog ( r1 – r2 ) floating-ext “f-log”

The decimal logarithm: r1 = 10^r2

falog ( r1 – r2 ) floating-ext “f-a-log”

r2=10^r1

f2* ( r1 – r2  ) gforth-0.2 “f2*”

Multiply r1 by 2.0e0

f2/ ( r1 – r2  ) gforth-0.2 “f2/”

Multiply r1 by 0.5e0

1/f ( r1 – r2  ) gforth-0.2 “1/f”

Divide 1.0e0 by r1.

Vector arithmetics:

v* ( f-addr1 nstride1 f-addr2 nstride2 ucount – r ) gforth-0.5 “v-star”

dot-product: r=v1*v2. The first element of v1 is at f_addr1, the next at f_addr1+nstride1 and so on (similar for v2). Both vectors have ucount elements.

faxpy ( ra f-x nstridex f-y nstridey ucount – ) gforth-0.5 “faxpy”

vy=ra*vx+vy, where vy is the vector starting at f_y with stride nstridey bytes, and vx is the vector starting at f_x with stride nstridex, and both vectors contain ucount elements.

Angles in floating point operations are given in radians (a full circle has 2 pi radians).

fsin ( r1 – r2 ) floating-ext “f-sine”

fcos ( r1 – r2 ) floating-ext “f-cos”

fsincos ( r1 – r2 r3 ) floating-ext “f-sine-cos”

r2=sin(r1), r3=cos(r1)

ftan ( r1 – r2 ) floating-ext “f-tan”

fasin ( r1 – r2 ) floating-ext “f-a-sine”

facos ( r1 – r2 ) floating-ext “f-a-cos”

fatan ( r1 – r2 ) floating-ext “f-a-tan”

fatan2 ( r1 r2 – r3 ) floating-ext “f-a-tan-two”

r1/r2=tan(r3). Forth-2012 does not require, but probably intends this to be the inverse of fsincos. In Gforth it is.

fsinh ( r1 – r2 ) floating-ext “f-cinch”

fcosh ( r1 – r2 ) floating-ext “f-cosh”

ftanh ( r1 – r2 ) floating-ext “f-tan-h”

fasinh ( r1 – r2 ) floating-ext “f-a-cinch”

facosh ( r1 – r2 ) floating-ext “f-a-cosh”

fatanh ( r1 – r2 ) floating-ext “f-a-tan-h”

pi ( – r  ) gforth-0.2 “pi”

Fconstant – r is the value pi; the ratio of a circle’s area to its diameter.

Special values in IEEE754 can be derived by for example dividing by zero. The most common ones are defined as floating point constants for easy usage.

infinity ( – r  ) gforth-1.0 “infinity”

floating point infinity

inf ( – r  ) gforth-1.0 “inf”

Synonym of infinity to allow copying and pasting from the output of ..., See Examining data.

-infinity ( – r  ) gforth-1.0 “-infinity”

floating point -infinity

-inf ( – r  ) gforth-1.0 “-inf”

Synonym of -infinity to allow copying and pasting from the output of ..., See Examining data.

NaN ( – r  ) gforth-1.0 “NaN”

floating point Not a Number