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Classes | Functions
Gaudi::Math Namespace Reference

## Classes
class | Lomont
| The equality comparison of double numbers using as the metric the maximal number of Units in the Last Place (ULP). More...
class | Lomont< double >
| the specialization for double numbers More...
class | Lomont< float >
| the specialization for float numbers More...

## Functions
GAUDI_API bool | lomont_compare_float (const float af, const float bf, const unsigned short maxULPs)
| equality comparison of float numbers using as the metric the maximal number of Units in the Last Place (ULP). More...
GAUDI_API bool | lomont_compare_double (const double af, const double bf, const unsigned int maxULPs)
| equality comparison of double numbers using as the metric the maximal number of Units in the Last Place (ULP). More...
GAUDI_API float | next_float (const float af, const short ulps)
| Get the floating number that representation is different with respect to the argument for the certain number of "Units in the Last Position". More...
GAUDI_API double | next_double (const double af, const short ulps)
| Get the floating number that representation is different with respect to the argument for the certain number of "Units in the Last Position". More...

## Function Documentation

bool Gaudi::Math::lomont_compare_double | ( | const double | af,
|  | const double | bf,
|  | const unsigned int | maxULPs
| ) |  |

equality comparison of double numbers using as the metric the maximal number of Units in the Last Place (ULP).

It is a slightly modified version of very efficient implementation of the initial Bruce Dawson's algorithm by Chris Lomont.

See also
  www.lomont.org
  http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm

C.Lomont claims the algorithm is factor 2-10 more efficient with respect to classical Knuth's algorithm for comparison of the floating number using the relative precision.

The effective relative difference depends on the choice of `maxULPS`:

- For the case of maxULPs=1, (of course it is totally unphysical case!!!) the effective relative precision r = |a-b|/(|a|+|b|)is ~6e-16 for |a|,|b|>1.e-304, and then it quickly goes to ~1

const double a = ... ;
const double b = ... ;
const bool equal = Gaudi::Math::lomont_compare_double ( a , b ) ;

Parameters

  af | the first number
  bf | the second number
  maxULPs | the maximal metric deviation in the terms of maximal number of units in the last place

Author
  Vanya BELYAEV Ivan..nosp@m.Bely.nosp@m.aev@n.nosp@m.ikhe.nosp@m.f.nl

Date
  2008-11-08

the final check

Definition at line 289 of file Lomont.cpp.

292 {
293 // ==========================================================================
295 static_assert( std::numeric_limits<double> ::is_specialized &&
296 std::numeric_limits<Long> ::is_specialized &&
297 std::numeric_limits<ULong> ::is_specialized &&
298 boost::integer_traits<ULong> ::is_specialized &&
299 boost::integer_traits<Long> ::is_specialized &&
300 sizeof(double)==sizeof(Long) &&
301 sizeof(double)==sizeof(ULong) &&
302 64 == std::numeric_limits<ULong>::digits , "FAILED ASSUMPTIONS") ;
303 // ==========================================================================
304 Cast_D caster ;
305
306 //Long ai = *reinterpret_cast<const Long*>( &af ) ;
307 //Long bi = *reinterpret_cast<const Long*>( &bf ) ;
308 Long ai = caster.d2l ( af ) ;
309 Long bi = caster.d2l ( bf ) ;
310
311 Long test = (((ULong)(ai^bi))>>63)-1;
312
313 // assert ( (0==test) || ( boost::integer_traits<ULong>::const_max == test ) ) ;
314
315 Long diff = ((( boost::integer_traits<Long>::const_min - ai ) & (~test)) | ( ai& test )) - bi ;
316
317 Long maxDiff_ = maxULPs ;
318
319 Long v1 = maxDiff_ + diff ;
320 Long v2 = maxDiff_ - diff ;
321
322 return 0<=(v1|v2) ;
323 }
compareRootHistos.test
string test
Definition: compareRootHistos.py:15
std::numeric_limits

bool Gaudi::Math::lomont_compare_float | ( | const float | af,
|  | const float | bf,
|  | const unsigned short | maxULPs
| ) |  |

equality comparison of float numbers using as the metric the maximal number of Units in the Last Place (ULP).

It is a slightly modified version of very efficient implementation of the initial Bruce Dawson's algorithm by Chris Lomont.

See also
  www.lomont.org
  http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm

C.Lomont claims the algorithm is factor 2-10 more efficient with respect to classical Knuth's algorithm for comparison of the floating number using the relative precision.

The effective relative difference depends on the choice of `maxULPS`:

- For the case of maxULPs=1, (of course it is totally unphysical case!!!) the effective relative precision r = |a-b|/(|a|+|b|)is between 3.5e-8 and 5.5e-8 for |a|,|b|>1.e-37, and then it quickly goes to ~1
- For the case of maxULPS=10 the effective relative precision is between 3e-8 and 6e-7 for |a|,|b|>1.e-37, and then it quickly goes to ~1
- For the case of maxULPS=100 the effective relative precision is around ~6e-6 for |a|,|b|>1.e-37, and then it quickly goes to ~1
- For the case of maxULPS=1000 the effective relative precision is around ~6e-5 for |a|,|b|>1.e-37, and then it quickly goes to ~1

const float a = ... ;
const float b = ... ;
const bool equal = Gaudi::Math::lomont_compare_float ( a , b ) ;

Parameters

  af | the first number
  bf | the second number
  maxULPs | the maximal metric deviation in the terms of maximal number of units in the last place

Author
  Vanya BELYAEV Ivan..nosp@m.Bely.nosp@m.aev@n.nosp@m.ikhe.nosp@m.f.nl

Date
  2008-11-08

Definition at line 193 of file Lomont.cpp.

196 {
197 // ==========================================================================
198 // prerequisites:
199 static_assert( std::numeric_limits<float> ::is_specialized &&
200 boost::integer_traits<int> ::is_specialized &&
201 boost::integer_traits<unsigned int> ::is_specialized &&
202 sizeof(float)==sizeof(int) &&
203 sizeof(float)==sizeof(unsigned int) &&
204 32 == boost::integer_traits<unsigned int>::digits , "FAILED ASSUMPTIONS") ;
205 // ==========================================================================
206
207 Cast_F caster ;
208
209 //int ai = *reinterpret_cast<const int*>( &af ) ;
210 //int bi = *reinterpret_cast<const int*>( &bf ) ;
211 int ai = caster.f2i ( af ) ;
212 int bi = caster.f2i ( bf ) ;
213
214 int test = (((unsigned int)(ai^bi))>>31)-1;
215
216 // assert ( (0==test) || ( boost::integer_traits<unsigned int>::const_max == test ) ) ;
217
218 int diff = ((( boost::integer_traits<int>::const_min - ai ) & (~test)) | ( ai& test )) - bi ;
219
220 int maxDiff_ = maxULPs ;
221
222 int v1 = maxDiff_ + diff ;
223 int v2 = maxDiff_ - diff ;
224
225 return 0<=(v1|v2) ;
226 }
compareRootHistos.test
string test
Definition: compareRootHistos.py:15
std::numeric_limits

double Gaudi::Math::next_double | ( | const double | af,
|  | const short | ulps
| ) |  |

Get the floating number that representation is different with respect to the argument for the certain number of "Units in the Last Position".

For ulps=1, it is just next float number, for ulps=-1 is is the previous one.

This routine is very convenient to test the parameter maxULPS for the routine Gaudi::Math::lomont_compare_double

See also
  Gaudi::Math::lomont_compare_double

Parameters

  af | the reference number
  ulps | the bias

Returns
  the biased float number (on distance "ulps")

Author
  Vanya BELYAEV Ivan..nosp@m.Bely.nosp@m.aev@n.nosp@m.ikhe.nosp@m.f.nl

Date
  2008-11-08

the final check

Definition at line 342 of file Lomont.cpp.

343 {
344 // ==========================================================================
346 static_assert( std::numeric_limits<double> ::is_specialized &&
347 std::numeric_limits<Long> ::is_specialized &&
348 sizeof(double)==sizeof(Long) , "FAILED ASSUMPTIONS") ;
349 // ==========================================================================
350 Cast_D caster ;
351
352 //Long al = *reinterpret_cast<const Long*>( &ad ) ;
353 Long al = caster.d2l ( ad ) ;
354 al += ulps ;
355 //return *reinterpret_cast<double*>(&al) ;
356 return caster.l2d ( al ) ;
357 }
std::numeric_limits

float Gaudi::Math::next_float | ( | const float | af,
|  | const short | ulps
| ) |  |

Get the floating number that representation is different with respect to the argument for the certain number of "Units in the Last Position".

For ulps=1, it is just next float number, for ulps=-1 is is the previous one.

This routine is very convenient to test the parameter maxULPS for the routine Gaudi::Math::lomont_compare_float

See also
  Gaudi:Math::lomont_compare_float

Parameters

  af | the reference number
  ulps | the bias

Returns
  the biased float number (on distance "ulps")

Author
  Vanya BELYAEV Ivan..nosp@m.Bely.nosp@m.aev@n.nosp@m.ikhe.nosp@m.f.nl

Date
  2008-11-08

the final check

Definition at line 245 of file Lomont.cpp.

246 {
248 static_assert( std::numeric_limits<float> ::is_specialized &&
249 std::numeric_limits<int> ::is_specialized &&
250 sizeof(float)==sizeof(int) , "FAILED ASSUMPTIONS") ;
251 // ==========================================================================
252 Cast_F caster ;
253
254 // int ai = *reinterpret_cast<const int*>( &af ) ;
255 int ai = caster.f2i ( af ) ;
256 ai += ulps ;
257 // return *reinterpret_cast<float*>(&ai) ;
258 return caster.i2f ( ai ) ;
259 }
std::numeric_limits