Min and max operations for multiplication and/or division under the simple interval system

Abstract

One embodiment of the present invention provides a system for performing a minimum computation for an interval multiplication operation. This system receives four floating-point numbers, including a first floating-point number, a second floating-point number, a third floating-point number and a fourth floating-point number. The system then computes a minimum of the four floating-point numbers, wherein if the four floating-point numbers include one or more default NaN (not-a-number) values, the system sets the minimum to negative infinity. One embodiment of the present invention provides a system for performing a minimum computation for an interval division operation. This system receives four floating-point numbers, including a first floating-point number, a second floating-point number, a third floating-point number and a fourth floating-point number. The system then computes a minimum of the four floating-point numbers, wherein if the four floating-point numbers include one or more default NaN (not-a-number) values, the default NaN values are ignored in computing the minimum.

1 - 10  (Canceled).  
     
     
         11 . A method for performing extremum computation for an interval division operation, comprising: 
 receiving at least four floating-point numbers, including a first floating-point number, a second floating-point number, a third floating-point number and a fourth floating-point number; and    computing a minimum of the at least four floating-point numbers, wherein computing the minimum includes computing the minimum in an arithmetic unit for intervals;    wherein if the at least four floating-point numbers include one or more default NaN (not-a-number) values, the default NaN values are ignored in computing the minimum.    
     
     
         12 . The method of  claim 11 , 
 wherein the minimum is a left endpoint of a resulting interval from the interval division operation;    wherein the first floating-point number is the result of a division operation between the left endpoint of a first interval and the left endpoint of a second interval;    wherein the second floating-point number is the result of a division operation between the left endpoint of the first interval and the right endpoint of the second interval;    wherein the third floating-point number is the result of a division operation between the right endpoint of the first interval and the left endpoint of the second interval; and    wherein the fourth floating-point number is the result of a division operation between the right endpoint of the first interval and the right endpoint of the second interval.    
     
     
         13 . The method of  claim 12 , wherein computing the minimum involves setting the minimum to negative infinity, if the second interval contains zero.  
     
     
         14 . The method of  claim 11 , wherein computing the minimum involves setting the minimum to a value representing the empty interval, if any of the at least four floating-point numbers contain the value representing the empty interval.  
     
     
         15 . The method of  claim 14 , wherein the value representing the empty interval is a non-default NaN value.  
     
     
         16 . The method of  claim 11 , wherein if none of the at least four floating-point numbers is a default NaN value or a value representing the empty interval, computing the minimum involves selecting the minimum of the at least four floating-point numbers.  
     
     
         17 . The method of  claim 11 , further comprising: 
 computing a maximum of the at least four floating-point numbers;    wherein if the at least four floating-point numbers include one or more default NaN (not-a-number) values, the default NaN values are ignored in computing the maximum.    
     
     
         18 . The method of  claim 17 , 
 wherein the maximum is a right endpoint of a resulting interval from the interval division operation;    wherein the first floating-point number is the result of a division operation between the left endpoint of a first interval and the left endpoint of a second interval;    wherein the second floating-point number is the result of a division operation between the left endpoint of the first interval and the right endpoint of the second interval;    wherein the third floating-point number is the result of a division operation between the right endpoint of the first interval and the left endpoint of the second interval; and    wherein the fourth floating-point number is the result of a division operation between the right endpoint of the first interval and the right endpoint of the second interval.    
     
     
         19 . The method of  claim 18 , wherein computing the maximum involves setting the maximum to positive infinity, if the second interval contains zero.  
     
     
         20 . The method of  claim 17 , wherein computing the maximum involves setting the maximum to a value representing the empty interval, if any of the at least four floating-point numbers contain the value representing the empty interval.  
     
     
         21 . The method of  claim 20 , wherein the value representing the empty interval is a non-default NaN value.  
     
     
         22 . The method of  claim 17 , wherein if none of the at least four floating-point numbers is a default NaN value or a value representing the empty interval, computing the maximum involves selecting the maximum of the at least four floating-point numbers.  
     
     
         23 - 32  (Canceled).  
     
     
         33 . A computer-readable storage medium storing instructions that when executed by a computer cause the computer to perform a method for performing extremum computation for an interval division operation, the method comprising: 
 receiving at least four floating-point numbers, including a first floating-point number, a second floating-point number, a third floating-point number and a fourth floating-point number; and    computing a minimum of the at least four floating-point numbers;    wherein if the at least four floating-point numbers include one or more default NaN (not-a-number) values, the default NaN values are ignored in computing the minimum.    
     
     
         34 . The computer-readable storage medium of  claim 33 , 
 wherein the minimum is a left endpoint of a resulting interval from the interval division operation;    wherein the first floating-point number is the result of a division operation between the left endpoint of a first interval and the left endpoint of a second interval;    wherein the second floating-point number is the result of a division operation between the left endpoint of the first interval and the right endpoint of the second interval;    wherein the third floating-point number is the result of a division operation between the right endpoint of the first interval and the left endpoint of the second interval; and    wherein the fourth floating-point number is the result of a division operation between the right endpoint of the first interval and the right endpoint of the second interval.    
     
     
         35 . The computer-readable storage medium of  claim 34 , wherein computing the minimum involves setting the minimum to negative infinity, if the second interval contains zero.  
     
     
         36 . The computer-readable storage medium of  claim 33 , wherein computing the minimum involves setting the minimum to a value representing the empty interval, if any of the at least four floating-point numbers contain the value representing the empty interval.  
     
     
         37 . The computer-readable storage medium of  claim 36 , wherein the value representing the empty interval is a non-default NaN value.  
     
     
         38 . The computer-readable storage medium of  claim 33 , wherein if none of the at least four floating-point numbers is a default NaN value or a value representing the empty interval, computing the minimum involves selecting the minimum of the at least four floating-point numbers.  
     
     
         39 . The computer-readable storage medium of  claim 33 , the method further comprising: 
 computing a maximum of the at least four floating-point numbers;    wherein if the at least four floating-point numbers include one or more default NaN (not-a-number) values, the default NaN values are ignored in computing the maximum.    
     
     
         40 . The computer-readable storage medium of  claim 39 , 
 wherein the maximum is a right endpoint of a resulting interval from the interval division operation;    wherein the first floating-point number is the result of a division operation between the left endpoint of a first interval and the left endpoint of a second interval;    wherein the second floating-point number is the result of a division operation between the left endpoint of the first interval and the right endpoint of the second interval;    wherein the third floating-point number is the result of a division operation between the right endpoint of the first interval and the left endpoint of the second interval; and    wherein the fourth floating-point number is the result of a division operation between the right endpoint of the first interval and the right endpoint of the second interval.    
     
     
         41 . The computer-readable storage medium of  claim 40 , wherein computing the maximum involves setting the maximum to positive infinity, if the second interval contains zero.  
     
     
         42 . The computer-readable storage medium of  claim 39 , wherein computing the maximum involves setting the maximum to a value representing the empty interval, if any of the at least four floating-point numbers contain the value representing the empty interval.  
     
     
         43 . The computer-readable storage medium of  claim 42 , wherein the value representing the empty interval is a non-default NaN value.  
     
     
         44 . The computer-readable storage medium of  claim 39 , wherein if none of the at least four floating-point numbers is a default NaN value or a value representing the empty interval, computing the maximum involves selecting the maximum of the at least four floating-point numbers.  
     
     
         45 - 54  (Canceled).  
     
     
         55 . An apparatus that performs extremum computation for an interval division operation, comprising: 
 an input that is configured to receive at least four floating-point numbers, including a first floating-point number, a second floating-point number, a third floating-point number and a fourth floating-point number; and    a computing mechanism that is configured to compute a minimum of the at least four floating-point numbers;    wherein if the at least four floating-point numbers include one or more default NaN (not-a-number) values, the default NaN values are ignored in computing the minimum.    
     
     
         56 . The apparatus of  claim 55 , 
 wherein the minimum is a left endpoint of a resulting interval from the interval division operation;    wherein the first floating-point number is the result of a division operation between the left endpoint of a first interval and the left endpoint of a second interval;    wherein the second floating-point number is the result of a division operation between the left endpoint of the first interval and the right endpoint of the second interval;    wherein the third floating-point number is the result of a division operation between the right endpoint of the first interval and the left endpoint of the second interval; and    wherein the fourth floating-point number is the result of a division operation between the right endpoint of the first interval and the right endpoint of the second interval.    
     
     
         57 . The apparatus of  claim 56 , wherein if the second interval contains zero, the computing mechanism is configured to set the minimum to negative infinity.  
     
     
         58 . The apparatus of  claim 55 , wherein the computing mechanism is configured to set the minimum to a value representing the empty interval, if any of the at least four floating-point numbers contain the value representing the empty interval.  
     
     
         59 . The apparatus of  claim 58 , wherein the value representing the empty interval is a non-default NaN value.  
     
     
         60 . The apparatus of  claim 55 , wherein if none of the at least four floating-point numbers is a default NaN value or a value representing the empty interval, the computing mechanism is configured to select the minimum of the at least four floating-point numbers.  
     
     
         61 . The apparatus of  claim 55  wherein: 
 the input is further configured to receive at least four floating-point numbers, including a first floating-point number, a second floating-point number, a third floating-point number and a fourth floating-point number;  
 the computing mechanism is further configured to compute a maximum of the at least four floating-point numbers; and  
 if the at least four floating-point numbers include one or more default NaN (not-a-number) values, the default NaN values are ignored in computing the maximum.  
 
     
     
         62 . The apparatus of  claim 61 , 
 wherein the maximum is a right endpoint of a resulting interval from the interval division operation;    wherein the first floating-point number is the result of a division operation between the left endpoint of a first interval and the left endpoint of a second interval;    wherein the second floating-point number is the result of a division operation between the left endpoint of the first interval and the right endpoint of the second interval;    wherein the third floating-point number is the result of a division operation between the right endpoint of the first interval and the left endpoint of the second interval; and    wherein the fourth floating-point number is the result of a division operation between the right endpoint of the first interval and the right endpoint of the second interval.    
     
     
         63 . The apparatus of  claim 62 , wherein if the second interval contains zero, the computing mechanism is configured to set the maximum to positive infinity.  
     
     
         64 . The apparatus of  claim 61 , wherein the computing mechanism is configured to set the maximum to a value representing the empty interval, if any of the at least four floating-point numbers contain the value representing the empty interval.  
     
     
         65 . The apparatus of  claim 64 , wherein the value representing the empty interval is a non-default NaN value.  
     
     
         66 . The apparatus of  claim 61 , wherein if none of the at least four floating-point numbers is a default NaN value or a value representing the empty interval, the computing mechanism is configured to select the maximum of the at least four floating-point numbers.