3 * This file contains the interface to the underlying bignum package.
4 * Its most important design principle is to completely hide the inner
5 * working of that other package from the user of GiNaC. It must either
6 * provide implementation of arithmetic operators and numerical evaluation
7 * of special functions or implement the interface to the bignum package. */
10 * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
12 * This program is free software; you can redistribute it and/or modify
13 * it under the terms of the GNU General Public License as published by
14 * the Free Software Foundation; either version 2 of the License, or
15 * (at your option) any later version.
17 * This program is distributed in the hope that it will be useful,
18 * but WITHOUT ANY WARRANTY; without even the implied warranty of
19 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
20 * GNU General Public License for more details.
22 * You should have received a copy of the GNU General Public License
23 * along with this program; if not, write to the Free Software
24 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
33 #if defined(HAVE_SSTREAM)
35 #elif defined(HAVE_STRSTREAM)
38 #error Need either sstream or strstream
47 // CLN should not pollute the global namespace, hence we include it here
48 // instead of in some header file where it would propagate to other parts.
49 // Also, we only need a subset of CLN, so we don't include the complete cln.h:
51 #include <cln/cl_output.h>
52 #include <cln/cl_integer_io.h>
53 #include <cln/cl_integer_ring.h>
54 #include <cln/cl_rational_io.h>
55 #include <cln/cl_rational_ring.h>
56 #include <cln/cl_lfloat_class.h>
57 #include <cln/cl_lfloat_io.h>
58 #include <cln/cl_real_io.h>
59 #include <cln/cl_real_ring.h>
60 #include <cln/cl_complex_io.h>
61 #include <cln/cl_complex_ring.h>
62 #include <cln/cl_numtheory.h>
63 #else // def HAVE_CLN_CLN_H
64 #include <cl_output.h>
65 #include <cl_integer_io.h>
66 #include <cl_integer_ring.h>
67 #include <cl_rational_io.h>
68 #include <cl_rational_ring.h>
69 #include <cl_lfloat_class.h>
70 #include <cl_lfloat_io.h>
71 #include <cl_real_io.h>
72 #include <cl_real_ring.h>
73 #include <cl_complex_io.h>
74 #include <cl_complex_ring.h>
75 #include <cl_numtheory.h>
76 #endif // def HAVE_CLN_CLN_H
78 #ifndef NO_NAMESPACE_GINAC
80 #endif // ndef NO_NAMESPACE_GINAC
82 // linker has no problems finding text symbols for numerator or denominator
85 GINAC_IMPLEMENT_REGISTERED_CLASS(numeric, basic)
88 // default constructor, destructor, copy constructor assignment
89 // operator and helpers
94 /** default ctor. Numerically it initializes to an integer zero. */
95 numeric::numeric() : basic(TINFO_numeric)
97 debugmsg("numeric default constructor", LOGLEVEL_CONSTRUCT);
101 setflag(status_flags::evaluated |
102 status_flags::expanded |
103 status_flags::hash_calculated);
108 debugmsg("numeric destructor" ,LOGLEVEL_DESTRUCT);
112 numeric::numeric(const numeric & other)
114 debugmsg("numeric copy constructor", LOGLEVEL_CONSTRUCT);
118 const numeric & numeric::operator=(const numeric & other)
120 debugmsg("numeric operator=", LOGLEVEL_ASSIGNMENT);
121 if (this != &other) {
130 void numeric::copy(const numeric & other)
133 value = new cl_N(*other.value);
136 void numeric::destroy(bool call_parent)
139 if (call_parent) basic::destroy(call_parent);
143 // other constructors
148 numeric::numeric(int i) : basic(TINFO_numeric)
150 debugmsg("numeric constructor from int",LOGLEVEL_CONSTRUCT);
151 // Not the whole int-range is available if we don't cast to long
152 // first. This is due to the behaviour of the cl_I-ctor, which
153 // emphasizes efficiency:
154 value = new cl_I((long) i);
156 setflag(status_flags::evaluated|
157 status_flags::hash_calculated);
161 numeric::numeric(unsigned int i) : basic(TINFO_numeric)
163 debugmsg("numeric constructor from uint",LOGLEVEL_CONSTRUCT);
164 // Not the whole uint-range is available if we don't cast to ulong
165 // first. This is due to the behaviour of the cl_I-ctor, which
166 // emphasizes efficiency:
167 value = new cl_I((unsigned long)i);
169 setflag(status_flags::evaluated|
170 status_flags::hash_calculated);
174 numeric::numeric(long i) : basic(TINFO_numeric)
176 debugmsg("numeric constructor from long",LOGLEVEL_CONSTRUCT);
179 setflag(status_flags::evaluated|
180 status_flags::hash_calculated);
184 numeric::numeric(unsigned long i) : basic(TINFO_numeric)
186 debugmsg("numeric constructor from ulong",LOGLEVEL_CONSTRUCT);
189 setflag(status_flags::evaluated|
190 status_flags::hash_calculated);
193 /** Ctor for rational numerics a/b.
195 * @exception overflow_error (division by zero) */
196 numeric::numeric(long numer, long denom) : basic(TINFO_numeric)
198 debugmsg("numeric constructor from long/long",LOGLEVEL_CONSTRUCT);
200 throw (std::overflow_error("division by zero"));
201 value = new cl_I(numer);
202 *value = *value / cl_I(denom);
204 setflag(status_flags::evaluated|
205 status_flags::hash_calculated);
209 numeric::numeric(double d) : basic(TINFO_numeric)
211 debugmsg("numeric constructor from double",LOGLEVEL_CONSTRUCT);
212 // We really want to explicitly use the type cl_LF instead of the
213 // more general cl_F, since that would give us a cl_DF only which
214 // will not be promoted to cl_LF if overflow occurs:
216 *value = cl_float(d, cl_default_float_format);
218 setflag(status_flags::evaluated|
219 status_flags::hash_calculated);
223 numeric::numeric(const char *s) : basic(TINFO_numeric)
224 { // MISSING: treatment of complex and ints and rationals.
225 debugmsg("numeric constructor from string",LOGLEVEL_CONSTRUCT);
227 value = new cl_LF(s);
231 setflag(status_flags::evaluated|
232 status_flags::hash_calculated);
235 /** Ctor from CLN types. This is for the initiated user or internal use
237 numeric::numeric(const cl_N & z) : basic(TINFO_numeric)
239 debugmsg("numeric constructor from cl_N", LOGLEVEL_CONSTRUCT);
242 setflag(status_flags::evaluated|
243 status_flags::hash_calculated);
250 /** Construct object from archive_node. */
251 numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
253 debugmsg("numeric constructor from archive_node", LOGLEVEL_CONSTRUCT);
256 // Read number as string
258 if (n.find_string("number", str)) {
259 istringstream s(str);
260 cl_idecoded_float re, im;
264 case 'N': // Ordinary number
265 case 'R': // Integer-decoded real number
266 s >> re.sign >> re.mantissa >> re.exponent;
267 *value = re.sign * re.mantissa * ::expt(cl_float(2.0, cl_default_float_format), re.exponent);
269 case 'C': // Integer-decoded complex number
270 s >> re.sign >> re.mantissa >> re.exponent;
271 s >> im.sign >> im.mantissa >> im.exponent;
272 *value = ::complex(re.sign * re.mantissa * ::expt(cl_float(2.0, cl_default_float_format), re.exponent),
273 im.sign * im.mantissa * ::expt(cl_float(2.0, cl_default_float_format), im.exponent));
275 default: // Ordinary number
282 // Read number as string
284 if (n.find_string("number", str)) {
285 istrstream f(str.c_str(), str.size() + 1);
286 cl_idecoded_float re, im;
290 case 'R': // Integer-decoded real number
291 f >> re.sign >> re.mantissa >> re.exponent;
292 *value = re.sign * re.mantissa * ::expt(cl_float(2.0, cl_default_float_format), re.exponent);
294 case 'C': // Integer-decoded complex number
295 f >> re.sign >> re.mantissa >> re.exponent;
296 f >> im.sign >> im.mantissa >> im.exponent;
297 *value = ::complex(re.sign * re.mantissa * ::expt(cl_float(2.0, cl_default_float_format), re.exponent),
298 im.sign * im.mantissa * ::expt(cl_float(2.0, cl_default_float_format), im.exponent));
300 default: // Ordinary number
308 setflag(status_flags::evaluated|
309 status_flags::hash_calculated);
312 /** Unarchive the object. */
313 ex numeric::unarchive(const archive_node &n, const lst &sym_lst)
315 return (new numeric(n, sym_lst))->setflag(status_flags::dynallocated);
318 /** Archive the object. */
319 void numeric::archive(archive_node &n) const
321 inherited::archive(n);
323 // Write number as string
325 if (this->is_crational())
328 // Non-rational numbers are written in an integer-decoded format
329 // to preserve the precision
330 if (this->is_real()) {
331 cl_idecoded_float re = integer_decode_float(The(cl_F)(*value));
333 s << re.sign << " " << re.mantissa << " " << re.exponent;
335 cl_idecoded_float re = integer_decode_float(The(cl_F)(::realpart(*value)));
336 cl_idecoded_float im = integer_decode_float(The(cl_F)(::imagpart(*value)));
338 s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
339 s << im.sign << " " << im.mantissa << " " << im.exponent;
342 n.add_string("number", s.str());
344 // Write number as string
346 ostrstream f(buf, 1024);
347 if (this->is_crational())
350 // Non-rational numbers are written in an integer-decoded format
351 // to preserve the precision
352 if (this->is_real()) {
353 cl_idecoded_float re = integer_decode_float(The(cl_F)(*value));
355 f << re.sign << " " << re.mantissa << " " << re.exponent << ends;
357 cl_idecoded_float re = integer_decode_float(The(cl_F)(::realpart(*value)));
358 cl_idecoded_float im = integer_decode_float(The(cl_F)(::imagpart(*value)));
360 f << re.sign << " " << re.mantissa << " " << re.exponent << " ";
361 f << im.sign << " " << im.mantissa << " " << im.exponent << ends;
365 n.add_string("number", str);
370 // functions overriding virtual functions from bases classes
375 basic * numeric::duplicate() const
377 debugmsg("numeric duplicate", LOGLEVEL_DUPLICATE);
378 return new numeric(*this);
382 /** Helper function to print a real number in a nicer way than is CLN's
383 * default. Instead of printing 42.0L0 this just prints 42.0 to ostream os
384 * and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as
385 * long as it only uses cl_LF and no other floating point types.
387 * @see numeric::print() */
388 void print_real_number(ostream & os, const cl_R & num)
390 cl_print_flags ourflags;
391 if (::instanceof(num, ::cl_RA_ring)) {
392 // case 1: integer or rational, nothing special to do:
393 ::print_real(os, ourflags, num);
396 // make CLN believe this number has default_float_format, so it prints
397 // 'E' as exponent marker instead of 'L':
398 ourflags.default_float_format = ::cl_float_format(The(cl_F)(num));
399 ::print_real(os, ourflags, num);
404 /** This method adds to the output so it blends more consistently together
405 * with the other routines and produces something compatible to ginsh input.
407 * @see print_real_number() */
408 void numeric::print(ostream & os, unsigned upper_precedence) const
410 debugmsg("numeric print", LOGLEVEL_PRINT);
411 if (this->is_real()) {
412 // case 1, real: x or -x
413 if ((precedence<=upper_precedence) && (!this->is_pos_integer())) {
415 print_real_number(os, The(cl_R)(*value));
418 print_real_number(os, The(cl_R)(*value));
421 // case 2, imaginary: y*I or -y*I
422 if (::realpart(*value) == 0) {
423 if ((precedence<=upper_precedence) && (::imagpart(*value) < 0)) {
424 if (::imagpart(*value) == -1) {
428 print_real_number(os, The(cl_R)(::imagpart(*value)));
432 if (::imagpart(*value) == 1) {
435 if (::imagpart (*value) == -1) {
438 print_real_number(os, The(cl_R)(::imagpart(*value)));
444 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
445 if (precedence <= upper_precedence)
447 print_real_number(os, The(cl_R)(::realpart(*value)));
448 if (::imagpart(*value) < 0) {
449 if (::imagpart(*value) == -1) {
452 print_real_number(os, The(cl_R)(::imagpart(*value)));
456 if (::imagpart(*value) == 1) {
460 print_real_number(os, The(cl_R)(::imagpart(*value)));
464 if (precedence <= upper_precedence)
471 void numeric::printraw(ostream & os) const
473 // The method printraw doesn't do much, it simply uses CLN's operator<<()
474 // for output, which is ugly but reliable. e.g: 2+2i
475 debugmsg("numeric printraw", LOGLEVEL_PRINT);
476 os << "numeric(" << *value << ")";
480 void numeric::printtree(ostream & os, unsigned indent) const
482 debugmsg("numeric printtree", LOGLEVEL_PRINT);
483 os << string(indent,' ') << *value
485 << "hash=" << hashvalue << " (0x" << hex << hashvalue << dec << ")"
486 << ", flags=" << flags << endl;
490 void numeric::printcsrc(ostream & os, unsigned type, unsigned upper_precedence) const
492 debugmsg("numeric print csrc", LOGLEVEL_PRINT);
493 ios::fmtflags oldflags = os.flags();
494 os.setf(ios::scientific);
495 if (this->is_rational() && !this->is_integer()) {
496 if (compare(_num0()) > 0) {
498 if (type == csrc_types::ctype_cl_N)
499 os << "cl_F(\"" << numer().evalf() << "\")";
501 os << numer().to_double();
504 if (type == csrc_types::ctype_cl_N)
505 os << "cl_F(\"" << -numer().evalf() << "\")";
507 os << -numer().to_double();
510 if (type == csrc_types::ctype_cl_N)
511 os << "cl_F(\"" << denom().evalf() << "\")";
513 os << denom().to_double();
516 if (type == csrc_types::ctype_cl_N)
517 os << "cl_F(\"" << evalf() << "\")";
525 bool numeric::info(unsigned inf) const
528 case info_flags::numeric:
529 case info_flags::polynomial:
530 case info_flags::rational_function:
532 case info_flags::real:
534 case info_flags::rational:
535 case info_flags::rational_polynomial:
536 return is_rational();
537 case info_flags::crational:
538 case info_flags::crational_polynomial:
539 return is_crational();
540 case info_flags::integer:
541 case info_flags::integer_polynomial:
543 case info_flags::cinteger:
544 case info_flags::cinteger_polynomial:
545 return is_cinteger();
546 case info_flags::positive:
547 return is_positive();
548 case info_flags::negative:
549 return is_negative();
550 case info_flags::nonnegative:
551 return !is_negative();
552 case info_flags::posint:
553 return is_pos_integer();
554 case info_flags::negint:
555 return is_integer() && is_negative();
556 case info_flags::nonnegint:
557 return is_nonneg_integer();
558 case info_flags::even:
560 case info_flags::odd:
562 case info_flags::prime:
568 /** Disassemble real part and imaginary part to scan for the occurrence of a
569 * single number. Also handles the imaginary unit. It ignores the sign on
570 * both this and the argument, which may lead to what might appear as funny
571 * results: (2+I).has(-2) -> true. But this is consistent, since we also
572 * would like to have (-2+I).has(2) -> true and we want to think about the
573 * sign as a multiplicative factor. */
574 bool numeric::has(const ex & other) const
576 if (!is_exactly_of_type(*other.bp, numeric))
578 const numeric & o = static_cast<numeric &>(const_cast<basic &>(*other.bp));
579 if (this->is_equal(o) || this->is_equal(-o))
581 if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
582 return (this->real().is_equal(o) || this->imag().is_equal(o) ||
583 this->real().is_equal(-o) || this->imag().is_equal(-o));
585 if (o.is_equal(I)) // e.g scan for I in 42*I
586 return !this->is_real();
587 if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
588 return (this->real().has(o*I) || this->imag().has(o*I) ||
589 this->real().has(-o*I) || this->imag().has(-o*I));
595 /** Evaluation of numbers doesn't do anything at all. */
596 ex numeric::eval(int level) const
598 // Warning: if this is ever gonna do something, the ex ctors from all kinds
599 // of numbers should be checking for status_flags::evaluated.
604 /** Cast numeric into a floating-point object. For example exact numeric(1) is
605 * returned as a 1.0000000000000000000000 and so on according to how Digits is
608 * @param level ignored, but needed for overriding basic::evalf.
609 * @return an ex-handle to a numeric. */
610 ex numeric::evalf(int level) const
612 // level can safely be discarded for numeric objects.
613 return numeric(::cl_float(1.0, ::cl_default_float_format) * (*value)); // -> CLN
618 /** Implementation of ex::diff() for a numeric. It always returns 0.
621 ex numeric::derivative(const symbol & s) const
627 int numeric::compare_same_type(const basic & other) const
629 GINAC_ASSERT(is_exactly_of_type(other, numeric));
630 const numeric & o = static_cast<numeric &>(const_cast<basic &>(other));
632 if (*value == *o.value) {
640 bool numeric::is_equal_same_type(const basic & other) const
642 GINAC_ASSERT(is_exactly_of_type(other,numeric));
643 const numeric *o = static_cast<const numeric *>(&other);
645 return this->is_equal(*o);
648 unsigned numeric::calchash(void) const
650 return (hashvalue=cl_equal_hashcode(*value) | 0x80000000U);
652 cout << *value << "->" << hashvalue << endl;
653 hashvalue=HASHVALUE_NUMERIC+1000U;
654 return HASHVALUE_NUMERIC+1000U;
659 unsigned numeric::calchash(void) const
661 double d=to_double();
667 return 0x88000000U+s*unsigned(d/0x07FF0000);
673 // new virtual functions which can be overridden by derived classes
679 // non-virtual functions in this class
684 /** Numerical addition method. Adds argument to *this and returns result as
685 * a new numeric object. */
686 numeric numeric::add(const numeric & other) const
688 return numeric((*value)+(*other.value));
691 /** Numerical subtraction method. Subtracts argument from *this and returns
692 * result as a new numeric object. */
693 numeric numeric::sub(const numeric & other) const
695 return numeric((*value)-(*other.value));
698 /** Numerical multiplication method. Multiplies *this and argument and returns
699 * result as a new numeric object. */
700 numeric numeric::mul(const numeric & other) const
702 static const numeric * _num1p=&_num1();
705 } else if (&other==_num1p) {
708 return numeric((*value)*(*other.value));
711 /** Numerical division method. Divides *this by argument and returns result as
712 * a new numeric object.
714 * @exception overflow_error (division by zero) */
715 numeric numeric::div(const numeric & other) const
717 if (::zerop(*other.value))
718 throw (std::overflow_error("division by zero"));
719 return numeric((*value)/(*other.value));
722 numeric numeric::power(const numeric & other) const
724 static const numeric * _num1p = &_num1();
727 if (::zerop(*value)) {
728 if (::zerop(*other.value))
729 throw (std::domain_error("numeric::eval(): pow(0,0) is undefined"));
730 else if (::zerop(::realpart(*other.value)))
731 throw (std::domain_error("numeric::eval(): pow(0,I) is undefined"));
732 else if (::minusp(::realpart(*other.value)))
733 throw (std::overflow_error("numeric::eval(): division by zero"));
737 return numeric(::expt(*value,*other.value));
740 /** Inverse of a number. */
741 numeric numeric::inverse(void) const
743 return numeric(::recip(*value)); // -> CLN
746 const numeric & numeric::add_dyn(const numeric & other) const
748 return static_cast<const numeric &>((new numeric((*value)+(*other.value)))->
749 setflag(status_flags::dynallocated));
752 const numeric & numeric::sub_dyn(const numeric & other) const
754 return static_cast<const numeric &>((new numeric((*value)-(*other.value)))->
755 setflag(status_flags::dynallocated));
758 const numeric & numeric::mul_dyn(const numeric & other) const
760 static const numeric * _num1p=&_num1();
763 } else if (&other==_num1p) {
766 return static_cast<const numeric &>((new numeric((*value)*(*other.value)))->
767 setflag(status_flags::dynallocated));
770 const numeric & numeric::div_dyn(const numeric & other) const
772 if (::zerop(*other.value))
773 throw (std::overflow_error("division by zero"));
774 return static_cast<const numeric &>((new numeric((*value)/(*other.value)))->
775 setflag(status_flags::dynallocated));
778 const numeric & numeric::power_dyn(const numeric & other) const
780 static const numeric * _num1p=&_num1();
783 if (::zerop(*value)) {
784 if (::zerop(*other.value))
785 throw (std::domain_error("numeric::eval(): pow(0,0) is undefined"));
786 else if (::zerop(::realpart(*other.value)))
787 throw (std::domain_error("numeric::eval(): pow(0,I) is undefined"));
788 else if (::minusp(::realpart(*other.value)))
789 throw (std::overflow_error("numeric::eval(): division by zero"));
793 return static_cast<const numeric &>((new numeric(::expt(*value,*other.value)))->
794 setflag(status_flags::dynallocated));
797 const numeric & numeric::operator=(int i)
799 return operator=(numeric(i));
802 const numeric & numeric::operator=(unsigned int i)
804 return operator=(numeric(i));
807 const numeric & numeric::operator=(long i)
809 return operator=(numeric(i));
812 const numeric & numeric::operator=(unsigned long i)
814 return operator=(numeric(i));
817 const numeric & numeric::operator=(double d)
819 return operator=(numeric(d));
822 const numeric & numeric::operator=(const char * s)
824 return operator=(numeric(s));
827 /** Return the complex half-plane (left or right) in which the number lies.
828 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
829 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
831 * @see numeric::compare(const numeric & other) */
832 int numeric::csgn(void) const
836 if (!::zerop(::realpart(*value))) {
837 if (::plusp(::realpart(*value)))
842 if (::plusp(::imagpart(*value)))
849 /** This method establishes a canonical order on all numbers. For complex
850 * numbers this is not possible in a mathematically consistent way but we need
851 * to establish some order and it ought to be fast. So we simply define it
852 * to be compatible with our method csgn.
854 * @return csgn(*this-other)
855 * @see numeric::csgn(void) */
856 int numeric::compare(const numeric & other) const
858 // Comparing two real numbers?
859 if (this->is_real() && other.is_real())
860 // Yes, just compare them
861 return ::cl_compare(The(cl_R)(*value), The(cl_R)(*other.value));
863 // No, first compare real parts
864 cl_signean real_cmp = ::cl_compare(::realpart(*value), ::realpart(*other.value));
868 return ::cl_compare(::imagpart(*value), ::imagpart(*other.value));
872 bool numeric::is_equal(const numeric & other) const
874 return (*value == *other.value);
877 /** True if object is zero. */
878 bool numeric::is_zero(void) const
880 return ::zerop(*value); // -> CLN
883 /** True if object is not complex and greater than zero. */
884 bool numeric::is_positive(void) const
887 return ::plusp(The(cl_R)(*value)); // -> CLN
891 /** True if object is not complex and less than zero. */
892 bool numeric::is_negative(void) const
895 return ::minusp(The(cl_R)(*value)); // -> CLN
899 /** True if object is a non-complex integer. */
900 bool numeric::is_integer(void) const
902 return ::instanceof(*value, ::cl_I_ring); // -> CLN
905 /** True if object is an exact integer greater than zero. */
906 bool numeric::is_pos_integer(void) const
908 return (this->is_integer() && ::plusp(The(cl_I)(*value))); // -> CLN
911 /** True if object is an exact integer greater or equal zero. */
912 bool numeric::is_nonneg_integer(void) const
914 return (this->is_integer() && !::minusp(The(cl_I)(*value))); // -> CLN
917 /** True if object is an exact even integer. */
918 bool numeric::is_even(void) const
920 return (this->is_integer() && ::evenp(The(cl_I)(*value))); // -> CLN
923 /** True if object is an exact odd integer. */
924 bool numeric::is_odd(void) const
926 return (this->is_integer() && ::oddp(The(cl_I)(*value))); // -> CLN
929 /** Probabilistic primality test.
931 * @return true if object is exact integer and prime. */
932 bool numeric::is_prime(void) const
934 return (this->is_integer() && ::isprobprime(The(cl_I)(*value))); // -> CLN
937 /** True if object is an exact rational number, may even be complex
938 * (denominator may be unity). */
939 bool numeric::is_rational(void) const
941 return ::instanceof(*value, ::cl_RA_ring); // -> CLN
944 /** True if object is a real integer, rational or float (but not complex). */
945 bool numeric::is_real(void) const
947 return ::instanceof(*value, ::cl_R_ring); // -> CLN
950 bool numeric::operator==(const numeric & other) const
952 return (*value == *other.value); // -> CLN
955 bool numeric::operator!=(const numeric & other) const
957 return (*value != *other.value); // -> CLN
960 /** True if object is element of the domain of integers extended by I, i.e. is
961 * of the form a+b*I, where a and b are integers. */
962 bool numeric::is_cinteger(void) const
964 if (::instanceof(*value, ::cl_I_ring))
966 else if (!this->is_real()) { // complex case, handle n+m*I
967 if (::instanceof(::realpart(*value), ::cl_I_ring) &&
968 ::instanceof(::imagpart(*value), ::cl_I_ring))
974 /** True if object is an exact rational number, may even be complex
975 * (denominator may be unity). */
976 bool numeric::is_crational(void) const
978 if (::instanceof(*value, ::cl_RA_ring))
980 else if (!this->is_real()) { // complex case, handle Q(i):
981 if (::instanceof(::realpart(*value), ::cl_RA_ring) &&
982 ::instanceof(::imagpart(*value), ::cl_RA_ring))
988 /** Numerical comparison: less.
990 * @exception invalid_argument (complex inequality) */
991 bool numeric::operator<(const numeric & other) const
993 if (this->is_real() && other.is_real())
994 return (The(cl_R)(*value) < The(cl_R)(*other.value)); // -> CLN
995 throw (std::invalid_argument("numeric::operator<(): complex inequality"));
996 return false; // make compiler shut up
999 /** Numerical comparison: less or equal.
1001 * @exception invalid_argument (complex inequality) */
1002 bool numeric::operator<=(const numeric & other) const
1004 if (this->is_real() && other.is_real())
1005 return (The(cl_R)(*value) <= The(cl_R)(*other.value)); // -> CLN
1006 throw (std::invalid_argument("numeric::operator<=(): complex inequality"));
1007 return false; // make compiler shut up
1010 /** Numerical comparison: greater.
1012 * @exception invalid_argument (complex inequality) */
1013 bool numeric::operator>(const numeric & other) const
1015 if (this->is_real() && other.is_real())
1016 return (The(cl_R)(*value) > The(cl_R)(*other.value)); // -> CLN
1017 throw (std::invalid_argument("numeric::operator>(): complex inequality"));
1018 return false; // make compiler shut up
1021 /** Numerical comparison: greater or equal.
1023 * @exception invalid_argument (complex inequality) */
1024 bool numeric::operator>=(const numeric & other) const
1026 if (this->is_real() && other.is_real())
1027 return (The(cl_R)(*value) >= The(cl_R)(*other.value)); // -> CLN
1028 throw (std::invalid_argument("numeric::operator>=(): complex inequality"));
1029 return false; // make compiler shut up
1032 /** Converts numeric types to machine's int. You should check with
1033 * is_integer() if the number is really an integer before calling this method.
1034 * You may also consider checking the range first. */
1035 int numeric::to_int(void) const
1037 GINAC_ASSERT(this->is_integer());
1038 return ::cl_I_to_int(The(cl_I)(*value)); // -> CLN
1041 /** Converts numeric types to machine's long. You should check with
1042 * is_integer() if the number is really an integer before calling this method.
1043 * You may also consider checking the range first. */
1044 long numeric::to_long(void) const
1046 GINAC_ASSERT(this->is_integer());
1047 return ::cl_I_to_long(The(cl_I)(*value)); // -> CLN
1050 /** Converts numeric types to machine's double. You should check with is_real()
1051 * if the number is really not complex before calling this method. */
1052 double numeric::to_double(void) const
1054 GINAC_ASSERT(this->is_real());
1055 return ::cl_double_approx(::realpart(*value)); // -> CLN
1058 /** Real part of a number. */
1059 const numeric numeric::real(void) const
1061 return numeric(::realpart(*value)); // -> CLN
1064 /** Imaginary part of a number. */
1065 const numeric numeric::imag(void) const
1067 return numeric(::imagpart(*value)); // -> CLN
1071 // Unfortunately, CLN did not provide an official way to access the numerator
1072 // or denominator of a rational number (cl_RA). Doing some excavations in CLN
1073 // one finds how it works internally in src/rational/cl_RA.h:
1074 struct cl_heap_ratio : cl_heap {
1079 inline cl_heap_ratio* TheRatio (const cl_N& obj)
1080 { return (cl_heap_ratio*)(obj.pointer); }
1081 #endif // ndef SANE_LINKER
1083 /** Numerator. Computes the numerator of rational numbers, rationalized
1084 * numerator of complex if real and imaginary part are both rational numbers
1085 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
1087 const numeric numeric::numer(void) const
1089 if (this->is_integer()) {
1090 return numeric(*this);
1093 else if (::instanceof(*value, ::cl_RA_ring)) {
1094 return numeric(::numerator(The(cl_RA)(*value)));
1096 else if (!this->is_real()) { // complex case, handle Q(i):
1097 cl_R r = ::realpart(*value);
1098 cl_R i = ::imagpart(*value);
1099 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_I_ring))
1100 return numeric(*this);
1101 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_RA_ring))
1102 return numeric(::complex(r*::denominator(The(cl_RA)(i)), ::numerator(The(cl_RA)(i))));
1103 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_I_ring))
1104 return numeric(::complex(::numerator(The(cl_RA)(r)), i*::denominator(The(cl_RA)(r))));
1105 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_RA_ring)) {
1106 cl_I s = ::lcm(::denominator(The(cl_RA)(r)), ::denominator(The(cl_RA)(i)));
1107 return numeric(::complex(::numerator(The(cl_RA)(r))*(exquo(s,::denominator(The(cl_RA)(r)))),
1108 ::numerator(The(cl_RA)(i))*(exquo(s,::denominator(The(cl_RA)(i))))));
1112 else if (instanceof(*value, ::cl_RA_ring)) {
1113 return numeric(TheRatio(*value)->numerator);
1115 else if (!this->is_real()) { // complex case, handle Q(i):
1116 cl_R r = ::realpart(*value);
1117 cl_R i = ::imagpart(*value);
1118 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_I_ring))
1119 return numeric(*this);
1120 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_RA_ring))
1121 return numeric(::complex(r*TheRatio(i)->denominator, TheRatio(i)->numerator));
1122 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_I_ring))
1123 return numeric(::complex(TheRatio(r)->numerator, i*TheRatio(r)->denominator));
1124 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_RA_ring)) {
1125 cl_I s = ::lcm(TheRatio(r)->denominator, TheRatio(i)->denominator);
1126 return numeric(::complex(TheRatio(r)->numerator*(exquo(s,TheRatio(r)->denominator)),
1127 TheRatio(i)->numerator*(exquo(s,TheRatio(i)->denominator))));
1130 #endif // def SANE_LINKER
1131 // at least one float encountered
1132 return numeric(*this);
1135 /** Denominator. Computes the denominator of rational numbers, common integer
1136 * denominator of complex if real and imaginary part are both rational numbers
1137 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1138 const numeric numeric::denom(void) const
1140 if (this->is_integer()) {
1144 if (instanceof(*value, ::cl_RA_ring)) {
1145 return numeric(::denominator(The(cl_RA)(*value)));
1147 if (!this->is_real()) { // complex case, handle Q(i):
1148 cl_R r = ::realpart(*value);
1149 cl_R i = ::imagpart(*value);
1150 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_I_ring))
1152 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_RA_ring))
1153 return numeric(::denominator(The(cl_RA)(i)));
1154 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_I_ring))
1155 return numeric(::denominator(The(cl_RA)(r)));
1156 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_RA_ring))
1157 return numeric(::lcm(::denominator(The(cl_RA)(r)), ::denominator(The(cl_RA)(i))));
1160 if (instanceof(*value, ::cl_RA_ring)) {
1161 return numeric(TheRatio(*value)->denominator);
1163 if (!this->is_real()) { // complex case, handle Q(i):
1164 cl_R r = ::realpart(*value);
1165 cl_R i = ::imagpart(*value);
1166 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_I_ring))
1168 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_RA_ring))
1169 return numeric(TheRatio(i)->denominator);
1170 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_I_ring))
1171 return numeric(TheRatio(r)->denominator);
1172 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_RA_ring))
1173 return numeric(::lcm(TheRatio(r)->denominator, TheRatio(i)->denominator));
1175 #endif // def SANE_LINKER
1176 // at least one float encountered
1180 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1181 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1182 * 2^(n-1) <= x < 2^n.
1184 * @return number of bits (excluding sign) needed to represent that number
1185 * in two's complement if it is an integer, 0 otherwise. */
1186 int numeric::int_length(void) const
1188 if (this->is_integer())
1189 return ::integer_length(The(cl_I)(*value)); // -> CLN
1196 // static member variables
1201 unsigned numeric::precedence = 30;
1207 const numeric some_numeric;
1208 const type_info & typeid_numeric=typeid(some_numeric);
1209 /** Imaginary unit. This is not a constant but a numeric since we are
1210 * natively handing complex numbers anyways. */
1211 const numeric I = numeric(::complex(cl_I(0),cl_I(1)));
1214 /** Exponential function.
1216 * @return arbitrary precision numerical exp(x). */
1217 const numeric exp(const numeric & x)
1219 return ::exp(*x.value); // -> CLN
1223 /** Natural logarithm.
1225 * @param z complex number
1226 * @return arbitrary precision numerical log(x).
1227 * @exception overflow_error (logarithmic singularity) */
1228 const numeric log(const numeric & z)
1231 throw (std::overflow_error("log(): logarithmic singularity"));
1232 return ::log(*z.value); // -> CLN
1236 /** Numeric sine (trigonometric function).
1238 * @return arbitrary precision numerical sin(x). */
1239 const numeric sin(const numeric & x)
1241 return ::sin(*x.value); // -> CLN
1245 /** Numeric cosine (trigonometric function).
1247 * @return arbitrary precision numerical cos(x). */
1248 const numeric cos(const numeric & x)
1250 return ::cos(*x.value); // -> CLN
1254 /** Numeric tangent (trigonometric function).
1256 * @return arbitrary precision numerical tan(x). */
1257 const numeric tan(const numeric & x)
1259 return ::tan(*x.value); // -> CLN
1263 /** Numeric inverse sine (trigonometric function).
1265 * @return arbitrary precision numerical asin(x). */
1266 const numeric asin(const numeric & x)
1268 return ::asin(*x.value); // -> CLN
1272 /** Numeric inverse cosine (trigonometric function).
1274 * @return arbitrary precision numerical acos(x). */
1275 const numeric acos(const numeric & x)
1277 return ::acos(*x.value); // -> CLN
1283 * @param z complex number
1285 * @exception overflow_error (logarithmic singularity) */
1286 const numeric atan(const numeric & x)
1289 x.real().is_zero() &&
1290 !abs(x.imag()).is_equal(_num1()))
1291 throw (std::overflow_error("atan(): logarithmic singularity"));
1292 return ::atan(*x.value); // -> CLN
1298 * @param x real number
1299 * @param y real number
1300 * @return atan(y/x) */
1301 const numeric atan(const numeric & y, const numeric & x)
1303 if (x.is_real() && y.is_real())
1304 return ::atan(::realpart(*x.value), ::realpart(*y.value)); // -> CLN
1306 throw (std::invalid_argument("numeric::atan(): complex argument"));
1310 /** Numeric hyperbolic sine (trigonometric function).
1312 * @return arbitrary precision numerical sinh(x). */
1313 const numeric sinh(const numeric & x)
1315 return ::sinh(*x.value); // -> CLN
1319 /** Numeric hyperbolic cosine (trigonometric function).
1321 * @return arbitrary precision numerical cosh(x). */
1322 const numeric cosh(const numeric & x)
1324 return ::cosh(*x.value); // -> CLN
1328 /** Numeric hyperbolic tangent (trigonometric function).
1330 * @return arbitrary precision numerical tanh(x). */
1331 const numeric tanh(const numeric & x)
1333 return ::tanh(*x.value); // -> CLN
1337 /** Numeric inverse hyperbolic sine (trigonometric function).
1339 * @return arbitrary precision numerical asinh(x). */
1340 const numeric asinh(const numeric & x)
1342 return ::asinh(*x.value); // -> CLN
1346 /** Numeric inverse hyperbolic cosine (trigonometric function).
1348 * @return arbitrary precision numerical acosh(x). */
1349 const numeric acosh(const numeric & x)
1351 return ::acosh(*x.value); // -> CLN
1355 /** Numeric inverse hyperbolic tangent (trigonometric function).
1357 * @return arbitrary precision numerical atanh(x). */
1358 const numeric atanh(const numeric & x)
1360 return ::atanh(*x.value); // -> CLN
1364 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1365 * integer arguments. */
1366 const numeric zeta(const numeric & x)
1368 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1369 // only knows about integer arguments and zeta(3).evalf() automatically
1370 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1371 // being an exact zero for CLN, which can be tested and then we can just
1372 // pass the number casted to an int:
1374 int aux = (int)(::cl_double_approx(::realpart(*x.value)));
1375 if (zerop(*x.value-aux))
1376 return ::cl_zeta(aux); // -> CLN
1378 clog << "zeta(" << x
1379 << "): Does anybody know good way to calculate this numerically?"
1385 /** The gamma function.
1386 * This is only a stub! */
1387 const numeric gamma(const numeric & x)
1389 clog << "gamma(" << x
1390 << "): Does anybody know good way to calculate this numerically?"
1396 /** The psi function (aka polygamma function).
1397 * This is only a stub! */
1398 const numeric psi(const numeric & x)
1401 << "): Does anybody know good way to calculate this numerically?"
1407 /** The psi functions (aka polygamma functions).
1408 * This is only a stub! */
1409 const numeric psi(const numeric & n, const numeric & x)
1411 clog << "psi(" << n << "," << x
1412 << "): Does anybody know good way to calculate this numerically?"
1418 /** Factorial combinatorial function.
1420 * @param n integer argument >= 0
1421 * @exception range_error (argument must be integer >= 0) */
1422 const numeric factorial(const numeric & n)
1424 if (!n.is_nonneg_integer())
1425 throw (std::range_error("numeric::factorial(): argument must be integer >= 0"));
1426 return numeric(::factorial(n.to_int())); // -> CLN
1430 /** The double factorial combinatorial function. (Scarcely used, but still
1431 * useful in cases, like for exact results of Gamma(n+1/2) for instance.)
1433 * @param n integer argument >= -1
1434 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1435 * @exception range_error (argument must be integer >= -1) */
1436 const numeric doublefactorial(const numeric & n)
1438 if (n == numeric(-1)) {
1441 if (!n.is_nonneg_integer()) {
1442 throw (std::range_error("numeric::doublefactorial(): argument must be integer >= -1"));
1444 return numeric(::doublefactorial(n.to_int())); // -> CLN
1448 /** The Binomial coefficients. It computes the binomial coefficients. For
1449 * integer n and k and positive n this is the number of ways of choosing k
1450 * objects from n distinct objects. If n is negative, the formula
1451 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1452 const numeric binomial(const numeric & n, const numeric & k)
1454 if (n.is_integer() && k.is_integer()) {
1455 if (n.is_nonneg_integer()) {
1456 if (k.compare(n)!=1 && k.compare(_num0())!=-1)
1457 return numeric(::binomial(n.to_int(),k.to_int())); // -> CLN
1461 return _num_1().power(k)*binomial(k-n-_num1(),k);
1465 // should really be gamma(n+1)/(gamma(r+1)/gamma(n-r+1) or a suitable limit
1466 throw (std::range_error("numeric::binomial(): don´t know how to evaluate that."));
1470 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1471 * in the expansion of the function x/(e^x-1).
1473 * @return the nth Bernoulli number (a rational number).
1474 * @exception range_error (argument must be integer >= 0) */
1475 const numeric bernoulli(const numeric & nn)
1477 if (!nn.is_integer() || nn.is_negative())
1478 throw (std::range_error("numeric::bernoulli(): argument must be integer >= 0"));
1481 if (!nn.compare(_num1()))
1482 return numeric(-1,2);
1485 // Until somebody has the Blues and comes up with a much better idea and
1486 // codes it (preferably in CLN) we make this a remembering function which
1487 // computes its results using the defining formula
1488 // B(nn) == - 1/(nn+1) * sum_{k=0}^{nn-1}(binomial(nn+1,k)*B(k))
1490 // Be warned, though: the Bernoulli numbers are probably computationally
1491 // very expensive anyhow and you shouldn't expect miracles to happen.
1492 static vector<numeric> results;
1493 static int highest_result = -1;
1494 int n = nn.sub(_num2()).div(_num2()).to_int();
1495 if (n <= highest_result)
1497 if (results.capacity() < (unsigned)(n+1))
1498 results.reserve(n+1);
1500 numeric tmp; // used to store the sum
1501 for (int i=highest_result+1; i<=n; ++i) {
1502 // the first two elements:
1503 tmp = numeric(-2*i-1,2);
1504 // accumulate the remaining elements:
1505 for (int j=0; j<i; ++j)
1506 tmp += binomial(numeric(2*i+3),numeric(j*2+2))*results[j];
1507 // divide by -(nn+1) and store result:
1508 results.push_back(-tmp/numeric(2*i+3));
1515 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1516 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1518 * @param n an integer
1519 * @return the nth Fibonacci number F(n) (an integer number)
1520 * @exception range_error (argument must be an integer) */
1521 const numeric fibonacci(const numeric & n)
1523 if (!n.is_integer())
1524 throw (std::range_error("numeric::fibonacci(): argument must be integer"));
1525 // The following addition formula holds:
1526 // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
1527 // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
1528 // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
1530 // Replace m by m+1:
1531 // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
1532 // Now put in m = n, to get
1533 // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
1534 // F(2n+1) = F(n)^2 + F(n+1)^2
1536 // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
1539 if (n.is_negative())
1541 return -fibonacci(-n);
1543 return fibonacci(-n);
1547 cl_I m = The(cl_I)(*n.value) >> 1L; // floor(n/2);
1548 for (uintL bit=::integer_length(m); bit>0; --bit) {
1549 // Since a squaring is cheaper than a multiplication, better use
1550 // three squarings instead of one multiplication and two squarings.
1551 cl_I u2 = ::square(u);
1552 cl_I v2 = ::square(v);
1553 if (::logbitp(bit-1, m)) {
1554 v = ::square(u + v) - u2;
1557 u = v2 - ::square(v - u);
1562 // Here we don't use the squaring formula because one multiplication
1563 // is cheaper than two squarings.
1564 return u * ((v << 1) - u);
1566 return ::square(u) + ::square(v);
1570 /** Absolute value. */
1571 numeric abs(const numeric & x)
1573 return ::abs(*x.value); // -> CLN
1577 /** Modulus (in positive representation).
1578 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1579 * sign of a or is zero. This is different from Maple's modp, where the sign
1580 * of b is ignored. It is in agreement with Mathematica's Mod.
1582 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1583 * integer, 0 otherwise. */
1584 numeric mod(const numeric & a, const numeric & b)
1586 if (a.is_integer() && b.is_integer())
1587 return ::mod(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1589 return _num0(); // Throw?
1593 /** Modulus (in symmetric representation).
1594 * Equivalent to Maple's mods.
1596 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1597 numeric smod(const numeric & a, const numeric & b)
1599 if (a.is_integer() && b.is_integer()) {
1600 cl_I b2 = The(cl_I)(ceiling1(The(cl_I)(*b.value) / 2)) - 1;
1601 return ::mod(The(cl_I)(*a.value) + b2, The(cl_I)(*b.value)) - b2;
1603 return _num0(); // Throw?
1607 /** Numeric integer remainder.
1608 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1609 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1610 * sign of a or is zero.
1612 * @return remainder of a/b if both are integer, 0 otherwise. */
1613 numeric irem(const numeric & a, const numeric & b)
1615 if (a.is_integer() && b.is_integer())
1616 return ::rem(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1618 return _num0(); // Throw?
1622 /** Numeric integer remainder.
1623 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1624 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1625 * and irem(a,b) has the sign of a or is zero.
1627 * @return remainder of a/b and quotient stored in q if both are integer,
1629 numeric irem(const numeric & a, const numeric & b, numeric & q)
1631 if (a.is_integer() && b.is_integer()) { // -> CLN
1632 cl_I_div_t rem_quo = truncate2(The(cl_I)(*a.value), The(cl_I)(*b.value));
1633 q = rem_quo.quotient;
1634 return rem_quo.remainder;
1638 return _num0(); // Throw?
1643 /** Numeric integer quotient.
1644 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1646 * @return truncated quotient of a/b if both are integer, 0 otherwise. */
1647 numeric iquo(const numeric & a, const numeric & b)
1649 if (a.is_integer() && b.is_integer())
1650 return truncate1(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1652 return _num0(); // Throw?
1656 /** Numeric integer quotient.
1657 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1658 * r == a - iquo(a,b,r)*b.
1660 * @return truncated quotient of a/b and remainder stored in r if both are
1661 * integer, 0 otherwise. */
1662 numeric iquo(const numeric & a, const numeric & b, numeric & r)
1664 if (a.is_integer() && b.is_integer()) { // -> CLN
1665 cl_I_div_t rem_quo = truncate2(The(cl_I)(*a.value), The(cl_I)(*b.value));
1666 r = rem_quo.remainder;
1667 return rem_quo.quotient;
1670 return _num0(); // Throw?
1675 /** Numeric square root.
1676 * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
1677 * should return integer 2.
1679 * @param z numeric argument
1680 * @return square root of z. Branch cut along negative real axis, the negative
1681 * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
1682 * where imag(z)>0. */
1683 numeric sqrt(const numeric & z)
1685 return ::sqrt(*z.value); // -> CLN
1689 /** Integer numeric square root. */
1690 numeric isqrt(const numeric & x)
1692 if (x.is_integer()) {
1694 ::isqrt(The(cl_I)(*x.value), &root); // -> CLN
1697 return _num0(); // Throw?
1701 /** Greatest Common Divisor.
1703 * @return The GCD of two numbers if both are integer, a numerical 1
1704 * if they are not. */
1705 numeric gcd(const numeric & a, const numeric & b)
1707 if (a.is_integer() && b.is_integer())
1708 return ::gcd(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1714 /** Least Common Multiple.
1716 * @return The LCM of two numbers if both are integer, the product of those
1717 * two numbers if they are not. */
1718 numeric lcm(const numeric & a, const numeric & b)
1720 if (a.is_integer() && b.is_integer())
1721 return ::lcm(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1723 return *a.value * *b.value;
1727 /** Floating point evaluation of Archimedes' constant Pi. */
1730 return numeric(::cl_pi(cl_default_float_format)); // -> CLN
1734 /** Floating point evaluation of Euler's constant Gamma. */
1735 ex EulerGammaEvalf(void)
1737 return numeric(::cl_eulerconst(cl_default_float_format)); // -> CLN
1741 /** Floating point evaluation of Catalan's constant. */
1742 ex CatalanEvalf(void)
1744 return numeric(::cl_catalanconst(cl_default_float_format)); // -> CLN
1748 // It initializes to 17 digits, because in CLN cl_float_format(17) turns out to
1749 // be 61 (<64) while cl_float_format(18)=65. We want to have a cl_LF instead
1750 // of cl_SF, cl_FF or cl_DF but everything else is basically arbitrary.
1751 _numeric_digits::_numeric_digits()
1756 cl_default_float_format = ::cl_float_format(17);
1760 _numeric_digits& _numeric_digits::operator=(long prec)
1763 cl_default_float_format = ::cl_float_format(prec);
1768 _numeric_digits::operator long()
1770 return (long)digits;
1774 void _numeric_digits::print(ostream & os) const
1776 debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
1781 ostream& operator<<(ostream& os, const _numeric_digits & e)
1788 // static member variables
1793 bool _numeric_digits::too_late = false;
1796 /** Accuracy in decimal digits. Only object of this type! Can be set using
1797 * assignment from C++ unsigned ints and evaluated like any built-in type. */
1798 _numeric_digits Digits;
1800 #ifndef NO_NAMESPACE_GINAC
1801 } // namespace GiNaC
1802 #endif // ndef NO_NAMESPACE_GINAC