Previous: Introduction to Numbers, Up: Numbers [Contents][Index]
bfloat replaces integers, rationals, floating point numbers, and some symbolic constants
in expr with bigfloat (variable-precision floating point) numbers.
The constants %e, %gamma, %phi, and %pi
are replaced by a numerical approximation.
However, %e in %e^x is not replaced by a numeric value
unless bfloat(x) is a number.
bfloat also causes numerical evaluation of some built-in functions,
namely trigonometric functions, exponential functions, abs, and log.
The number of significant digits in the resulting bigfloats is specified by the
global variable fpprec.
Bigfloats already present in expr are replaced with values which have
precision specified by the current value of fpprec.
When float2bf is false, a warning message is printed when
a floating point number is replaced by a bigfloat number with less precision.
Examples:
bfloat replaces integers, rationals, floating point numbers, and some symbolic constants
in expr with bigfloat numbers.
(%i1) bfloat([123, 17/29, 1.75]);
(%o1) [1.23b2, 5.862068965517241b-1, 1.75b0]
(%i2) bfloat([%e, %gamma, %phi, %pi]);
(%o2) [2.718281828459045b0, 5.772156649015329b-1,
1.618033988749895b0, 3.141592653589793b0]
(%i3) bfloat((f(123) + g(h(17/29)))/(x + %gamma));
1.0b0 (g(h(5.862068965517241b-1)) + f(1.23b2))
(%o3) ----------------------------------------------
x + 5.772156649015329b-1
bfloat also causes numerical evaluation of some built-in functions.
(%i1) bfloat(sin(17/29)); (%o1) 5.532051841609784b-1 (%i2) bfloat(exp(%pi)); (%o2) 2.314069263277927b1 (%i3) bfloat(abs(-%gamma)); (%o3) 5.772156649015329b-1 (%i4) bfloat(log(%phi)); (%o4) 4.812118250596035b-1
Returns true if expr is a bigfloat number, otherwise false.
Default value: false
bftorat controls the conversion of bfloats to rational numbers. When
bftorat is false, ratepsilon will be used to control the
conversion (this results in relatively small rational numbers). When
bftorat is true, the rational number generated will accurately
represent the bfloat.
Note: bftorat has no effect on the transformation to rational numbers
with the function rationalize.
Example:
(%i1) ratepsilon:1e-4;
(%o1) 1.0e-4
(%i2) rat(bfloat(11111/111111)), bftorat:false;
`rat' replaced 9.99990999991B-2 by 1/10 = 1.0B-1
1
(%o2)/R/ --
10
(%i3) rat(bfloat(11111/111111)), bftorat:true;
`rat' replaced 9.99990999991B-2 by 11111/111111 = 9.99990999991B-2
11111
(%o3)/R/ ------
111111
Default value: true
bftrunc causes trailing zeroes in non-zero bigfloat numbers not to be
displayed. Thus, if bftrunc is false, bfloat (1)
displays as 1.000000000000000B0. Otherwise, this is displayed as
1.0B0.
Returns the number of bits of precision in a bigfloat number. This
value depends, of course, on the value of fpprec.
(%i1) fpprec:16; (%o1) 16 (%i2) bigfloat_bits(); (%o2) 56 (%i3) fpprec:32; (%o3) 32 (%i4) bigfloat_bits(); (%o4) 109
Returns the smallest bigfloat value, eps, such that
1+eps is not equal to 1. The value depends on fpprec,
of course.
(%i1) fpprec:16; (%o1) 16 (%i2) bigfloat_eps(); (%o2) 1.387778780781446b-17 (%i3) fpprec:32; (%o3) 32 (%i4) bigfloat_eps(); (%o4) 1.5407439555097886824447823540679b-33
decode_float takes a float f and returns a list of three
values that characterizes f, which must be either a float
or bfloat. The first value has the same type as f, but
is a number in the range [1, 2). The second value is an
exponent. The third value is a float of the same type as f and
has the value of 1 if f is greater than or equal to 0;
otherwise, -1.
If the returned list is [mantissa, expo, sign], then
scale_float(mantissa, exp)*sign is identical to f.
(%i1) decode_float(4e0); (%o1) [1.0, 2, 1.0] (%i2) decode_float(4b0); (%o2) [1.0b0, 2, 1.0b0] (%i3) decode_float(%pi); decode_float is only defined for floats and bfloats: %pi -- an error. To debug this try: debugmode(true); (%i4) decode_float(float(%pi)); (%o4) [1.570796326794897, 1, 1.0] (%i5) decode_float(1.1e-5); (%o5) [1.441792, - 17, 1.0] (%i6) %[1]*2^%[2]; (%o6) 1.1e-5
This is a relatively simple interface to Common Lisp
decode_float. However we return a signficand in the range
[1,2) instead of [0.5, 1). The former matches
IEEE-754. Of course, this is extended to support bfloats.
Returns true if expr is a literal even integer, otherwise
false.
evenp returns false if expr is a symbol, even if expr
is declared even.
Converts integers, rational numbers and bigfloats in expr to floating
point numbers. It is also an evflag, float causes
non-integral rational numbers and bigfloat numbers to be converted to floating
point.
Default value: true
When float2bf is false, a warning message is printed when
a floating point number is replaced by a bigfloat number with less precision.
Returns the number of bits of precision of a floating-point number.
Returns the smallest floating-point value, eps, such that
1+eps is not equal to 1.
Returns the number of bits of precision of a floating-point number,
which can be either a float or bigfloat. This is basically the number
of bits used to represent the mantissa of a floating-point number.
For floats, this is 53 (for IEEE double-floats), but can be less when
denormal numbers occur. For bigfloats, this is equal to
fpprec, when converted from digits to bits.
Returns true if expr is a floating point number, otherwise
false.
Default value: 16
fpprec is the number of significant digits for arithmetic on bigfloat
numbers. fpprec does not affect computations on ordinary floating point
numbers.
See also bfloat and fpprintprec.
Default value: 0
fpprintprec is the number of digits to print when printing an ordinary
float or bigfloat number.
For ordinary floating point numbers,
when fpprintprec has a value between 2 and 16 (inclusive),
the number of digits printed is equal to fpprintprec.
Otherwise, fpprintprec is 0, or greater than 16,
and the number is printed "readably":
that is, it is printed with sufficient digits to exactly reconstruct the number on input.
For bigfloat numbers,
when fpprintprec has a value between 2 and fpprec (inclusive),
the number of digits printed is equal to fpprintprec.
Otherwise, fpprintprec is 0, or greater than fpprec,
and the number of digits printed is equal to fpprec.
For both ordinary floats and bigfloats,
trailing zero digits are suppressed.
The actual number of digits printed is less than fpprintprec
if there are trailing zero digits.
fpprintprec cannot be 1.
Returns true if expr is a literal numeric integer, otherwise
false.
integerp returns false if expr is a symbol, even if expr
is declared integer.
Examples:
(%i1) integerp (0); (%o1) true (%i2) integerp (1); (%o2) true (%i3) integerp (-17); (%o3) true (%i4) integerp (0.0); (%o4) false (%i5) integerp (1.0); (%o5) false (%i6) integerp (%pi); (%o6) false (%i7) integerp (n); (%o7) false (%i8) declare (n, integer); (%o8) done (%i9) integerp (n); (%o9) false
integer_decode_float takes a float f and returns a list of three
values that characterizes f, which must be either a float
or bfloat. The first value is an integer. The second value is an
exponent. The third value is 1 if f is positive or zero;
otherwise, -1.
If the returned list is [mantissa, expo, sign], then
scale_float(fl(mantissa), expo)*sign is identical to f.
Here, fl is either float or bfloat depending on
whether f is a float or a bfloat.
(%i1) integer_decode_float(4.0); (%o1) [4503599627370496, - 50, 1] (%i2) integer_decode_float(4b0); (%o2) [36028797018963968, - 53, 1] (%i3) scale_float(float(%o1[1]), %o1[2]); (%o3) 4.0 (%i4) scale_float(bfloat(%o2[1]), %o2[2]); (%o4) 4.0b0 (%i5) integer_decode_float(4); decode_float is only defined for floats and bfloats: 4 -- an error. To debug this try: debugmode(true); (%i6) integer_decode_float(1e-7); (%o6) [7555786372591432, - 76, 1] (%i7) integer_decode_float(1b-7); (%o7) [60446290980731459, - 79, 1] (%i8) scale_float(float(%o6[1]), %o6[2]); (%o8) 1.0e-7
For lisps that support denormal numbers, we have the following results.
(%i1) integer_decode_float(least_positive_float); (%o1) [1, - 1074, 1] (%i2) integer_decode_float(100*least_positive_float); (%o2) [100, - 1074, 1] (%i3) integer_decode_float(least_positive_normalized_float); (%o3) [4503599627370496, - 1074, 1]
The number of bits in the integer part decreases as the denormal number decreases. Bfloat numbers do not have denormals because the exponent is not bounded.
This is a relatively simple interface to Common Lisp integer_decode_float. However, the integer part can vary depending on the Lisp implementation; we return the same value, independent of the Lisp implementation. Of course, this is extended to support bfloats.
is_power_to_two returns true if n is a power of
two and false otherwise. n may be an integer, a
rational, a float, or a big float.
Some examples:
(%i1) is_power_of_two(0); (%o1) false (%i2) is_power_of_two(4); (%o2) true (%i3) is_power_of_two(355/113); (%o3) false (%i4) is_power_of_two(1/32); (%o4) true (%i5) is_power_of_two(1048576); (%o5) true (%i6) is_power_of_two(1048575); (%o6) false (%i7) is_power_of_two(0.0); (%o7) false (%i8) is_power_of_two(1048576.0); (%o8) true (%i9) is_power_of_two(1048575.0); (%o9) false (%i10) is_power_of_two(1/256.0); (%o10) true (%i11) is_power_of_two(0b0); (%o11) false (%i12) is_power_of_two(1048576b0); (%o12) true (%i13) is_power_of_two(1048575b0); (%o13) false (%i14) is_power_of_two(1/256b0); (%o14) true
Default value: false
m1pbranch is the principal branch for -1 to a power.
Quantities such as (-1)^(1/3) (that is, an "odd" rational exponent) and
(-1)^(1/4) (that is, an "even" rational exponent) are handled as follows:
domain:real
(-1)^(1/3): -1
(-1)^(1/4): (-1)^(1/4)
domain:complex
m1pbranch:false m1pbranch:true
(-1)^(1/3) 1/2+%i*sqrt(3)/2
(-1)^(1/4) sqrt(2)/2+%i*sqrt(2)/2
Return true if and only if n >= 0 and n is an integer.
Returns true if expr is a literal integer, rational number,
floating point number, or bigfloat, otherwise false.
numberp returns false if expr is a symbol, even if expr
is a symbolic number such as %pi or %i, or declared to be
even, odd, integer, rational, irrational,
real, imaginary, or complex.
Examples:
(%i1) numberp (42);
(%o1) true
(%i2) numberp (-13/19);
(%o2) true
(%i3) numberp (3.14159);
(%o3) true
(%i4) numberp (-1729b-4);
(%o4) true
(%i5) map (numberp, [%e, %pi, %i, %phi, inf, minf]);
(%o5) [false, false, false, false, false, false]
(%i6) declare (a, even, b, odd, c, integer, d, rational,
e, irrational, f, real, g, imaginary, h, complex);
(%o6) done
(%i7) map (numberp, [a, b, c, d, e, f, g, h]);
(%o7) [false, false, false, false, false, false, false, false]
numer causes some mathematical functions (including exponentiation)
with numerical arguments to be evaluated in floating point. It causes
variables in expr which have been given numerals to be replaced by
their values. It also sets the float switch on.
See also %enumer.
Examples:
(%i1) [sqrt(2), sin(1), 1/(1+sqrt(3))];
1
(%o1) [sqrt(2), sin(1), -----------]
sqrt(3) + 1
(%i2) [sqrt(2), sin(1), 1/(1+sqrt(3))],numer; (%o2) [1.414213562373095, 0.8414709848078965, 0.3660254037844387]
Default value: false
The option variable numer_pbranch controls the numerical evaluation of
the power of a negative integer, rational, or floating point number. When
numer_pbranch is true and the exponent is a floating point number
or the option variable numer is true too, Maxima evaluates
the numerical result using the principal branch. Otherwise a simplified, but
not an evaluated result is returned.
Examples:
(%i1) (-2)^0.75;
0.75
(%o1) (- 2)
(%i2) (-2)^0.75,numer_pbranch:true; (%o2) 1.189207115002721 %i - 1.189207115002721
(%i3) (-2)^(3/4);
3/4 3/4
(%o3) (- 1) 2
(%i4) (-2)^(3/4),numer;
0.75
(%o4) 1.681792830507429 (- 1)
(%i5) (-2)^(3/4),numer,numer_pbranch:true; (%o5) 1.189207115002721 %i - 1.189207115002721
Declares the variables x_1, …, x_n to have
numeric values equal to expr_1, …, expr_n.
The numeric value is evaluated and substituted for the variable
in any expressions in which the variable occurs if the numer flag is
true. See also ev.
The expressions expr_1, …, expr_n can be any expressions,
not necessarily numeric.
Returns true if expr is a literal odd integer, otherwise
false.
oddp returns false if expr is a symbol, even if expr
is declared odd.
Default value: 2.0e-15
ratepsilon is the tolerance used in the conversion
of floating point numbers to rational numbers, when the option variable
bftorat has the value false. See bftorat for an example.
Convert all double floats and big floats in the Maxima expression expr to
their exact rational equivalents. If you are not familiar with the binary
representation of floating point numbers, you might be surprised that
rationalize (0.1) does not equal 1/10. This behavior isn’t special to
Maxima – the number 1/10 has a repeating, not a terminating, binary
representation.
(%i1) rationalize (0.5);
1
(%o1) -
2
(%i2) rationalize (0.1);
3602879701896397
(%o2) -----------------
36028797018963968
(%i3) fpprec : 5$
(%i4) rationalize (0.1b0);
209715
(%o4) -------
2097152
(%i5) fpprec : 20$
(%i6) rationalize (0.1b0);
236118324143482260685
(%o6) ----------------------
2361183241434822606848
(%i7) rationalize (sin (0.1*x + 5.6));
3602879701896397 x 3152519739159347
(%o7) sin(------------------ + ----------------)
36028797018963968 562949953421312
Returns true if expr is a literal integer or ratio of literal
integers, otherwise false.
scale_float scales the float f by the value
2^n. This is done carefully so that no round-off every
occurs. If f is a float, then it is possible to underflow to 0
or overflow, depending on the value of f and n. Bigfloats
cannot underflow or overflow.
(%i1) scale_float(2d0, 2); (%o1) 8.0 (%i2) scale_float(2d0, -2); (%o2) 0.5 (%i3) scale_float(-2d0, -10); (%o3) - 0.001953125 (%i4) scale_float(1d0, -2000); (%o4) 0.0 (%i5) scale_float(2b0, 2); (%o5) 8.0b0 (%i6) scale_float(1b0, -2000); (%o6) 8.709809816217217b-603 (%i7) scale_float(1, 5); scale_float: first arg must be a float or bfloat: 1 -- an error. To debug this try: debugmode(true); (%i8) scale_float(1.0, n); scale_float: second arg must be an integer: n -- an error. To debug this try: debugmode(true);
This is a relatively simple interface to Common Lisp scale_float. Of course, this is extended to support bfloats.
unit_in_last_place returns a value that is the gap between
n and the nearest other number. See, for example,
Kahan, FOOTNOTE 1. unit_in_last_place supports rational numbers,
floating-point numbers and bigfloat numbers. For integer, the result
is always 1, and for rational numbers the result is always 0.
The examples below assume IEEE-754 arithmetic that supports denormal numbers. Some lisps like Clisp do not have denormal numbers.
(%i1) unit_in_last_place(0); (%o1) 1 (%i2) unit_in_last_place(-123); (%o2) 1 (%i3) unit_in_last_place(2/3); (%o3) 0 (%i4) unit_in_last_place(355/113); (%o4) 0 (%i5) unit_in_last_place(0b0); (%o5) 0.0b0 (%i6) unit_in_last_place(0.0); (%o6) 4.940656458412465e-324 (%i7) unit_in_last_place(1.0); (%o7) 1.110223024625157e-16 (%i8) unit_in_last_place(1b0); (%o8) 1.387778780781446b-17 (%i9) unit_in_last_place(100.0); (%o9) 1.4210854715202e-14 (%i10) unit_in_last_place(100b0); (%o10) 1.77635683940025b-15 (%i11) fpprec:32; (%o11) 32 (%i12) unit_in_last_place(1b0); (%o12) 1.5407439555097886824447823540679b-33 (%i13) unit_in_last_place(100b0); (%o13) 1.972152263052529513529321413207b-31
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