5.5.6 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.¹

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 are 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.

d>f       d – r        float       “d-to-f”

f>d       r – d        float       “f-to-d”

f+       r1 r2 – r3        float       “f-plus”

f-       r1 r2 – r3        float       “f-minus”

f*       r1 r2 – r3        float       “f-star”

f/       r1 r2 – r3        float       “f-slash”

fnegate       r1 – r2        float       “f-negate”

fabs       r1 – r2        float-ext       “f-abs”

fmax       r1 r2 – r3        float       “f-max”

fmin       r1 r2 – r3        float       “f-min”

floor       r1 – r2        float       “floor”

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

fround       r1 – r2        float       “f-round”

Round to the nearest integral value.

f**       r1 r2 – r3        float-ext       “f-star-star”

r3 is r1 raised to the r2th power.

fsqrt       r1 – r2        float-ext       “f-square-root”

fexp       r1 – r2        float-ext       “f-e-x-p”

fexpm1       r1 – r2        float-ext       “f-e-x-p-m-one”

r2=e**r1−1

fln       r1 – r2        float-ext       “f-l-n”

flnp1       r1 – r2        float-ext       “f-l-n-p-one”

r2=ln(r1+1)

flog       r1 – r2        float-ext       “f-log”

The decimal logarithm.

falog       r1 – r2        float-ext       “f-a-log”

r2=10**r1

f2*       r1 – r2         gforth       “f2*”

Multiply r1 by 2.0e0

f2/       r1 – r2         gforth       “f2/”

Multiply r1 by 0.5e0

1/f       r1 – r2         gforth       “1/f”

Divide 1.0e0 by r1.

precision       – u         float-ext       “precision”

u is the number of significant digits currently used by F. FE. and FS.

set-precision       u –         float-ext       “set-precision”

Set the number of significant digits currently used by F. FE. and FS. to u.

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

fsin       r1 – r2        float-ext       “f-sine”

fcos       r1 – r2        float-ext       “f-cos”

fsincos       r1 – r2 r3        float-ext       “f-sine-cos”

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

ftan       r1 – r2        float-ext       “f-tan”

fasin       r1 – r2        float-ext       “f-a-sine”

facos       r1 – r2        float-ext       “f-a-cos”

fatan       r1 – r2        float-ext       “f-a-tan”

fatan2       r1 r2 – r3        float-ext       “f-a-tan-two”

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

fsinh       r1 – r2        float-ext       “f-cinch”

fcosh       r1 – r2        float-ext       “f-cosh”

ftanh       r1 – r2        float-ext       “f-tan-h”

fasinh       r1 – r2        float-ext       “f-a-cinch”

facosh       r1 – r2        float-ext       “f-a-cosh”

fatanh       r1 – r2        float-ext       “f-a-tan-h”

pi       – r         gforth       “pi”

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

One particular problem with floating-point arithmetic is that comparison for equality often fails when you would expect it to succeed. For this reason approximate equality is often preferred (but you still have to know what you are doing). Also note that IEEE NaNs may compare differently from what you might expect. The comparison words are:

f~rel       r1 r2 r3 – flag         gforth       “f~rel”

Approximate equality with relative error: |r1-r2|<r3*|r1+r2|.

f~abs       r1 r2 r3 – flag         gforth       “f~abs”

Approximate equality with absolute error: |r1-r2|<r3.

f~       r1 r2 r3 – flag         float-ext       “f-proximate”

ANS Forth medley for comparing r1 and r2 for equality: r3>0: f~abs; r3=0: bitwise comparison; r3<0: fnegate f~rel.

f=       r1 r2 – f        gforth       “f-equals”

f<>       r1 r2 – f        gforth       “f-not-equals”

f<       r1 r2 – f        float       “f-less-than”

f<=       r1 r2 – f        gforth       “f-less-or-equal”

f>       r1 r2 – f        gforth       “f-greater-than”

f>=       r1 r2 – f        gforth       “f-greater-or-equal”

f0<       r – f        float       “f-zero-less-than”

f0<=       r – f        gforth       “f-zero-less-or-equal”

f0<>       r – f        gforth       “f-zero-not-equals”

f0=       r – f        float       “f-zero-equals”

f0>       r – f        gforth       “f-zero-greater-than”

f0>=       r – f        gforth       “f-zero-greater-or-equal”