SUBROUTINE MACHAR(IBETA,IT,IRND,NGRD,MACHEP,NEGEP,IEXP,MINEXP, 1 MAXEXP,EPS,EPSNEG,XMIN,XMAX) C---------------------------------------------------------------------- C This Fortran 77 subroutine is intended to determine the parameters C of the floating-point arithmetic system specified below. The C determination of the first three uses an extension of an algorithm C due to M. Malcolm, CACM 15 (1972), pp. 949-951, incorporating some, C but not all, of the improvements suggested by M. Gentleman and S. C Marovich, CACM 17 (1974), pp. 276-277. An earlier version of this C program was published in the book Software Manual for the C Elementary Functions by W. J. Cody and W. Waite, Prentice-Hall, C Englewood Cliffs, NJ, 1980. C C The program as given here must be modified before compiling. If C a single (double) precision version is desired, change all C occurrences of CS (CD) in columns 1 and 2 to blanks. C C Parameter values reported are as follows: C C IBETA - the radix for the floating-point representation C IT - the number of base IBETA digits in the floating-point C significand C IRND - 0 if floating-point addition chops C 1 if floating-point addition rounds, but not in the C IEEE style C 2 if floating-point addition rounds in the IEEE style C 3 if floating-point addition chops, and there is C partial underflow C 4 if floating-point addition rounds, but not in the C IEEE style, and there is partial underflow C 5 if floating-point addition rounds in the IEEE style, C and there is partial underflow C NGRD - the number of guard digits for multiplication with C truncating arithmetic. It is C 0 if floating-point arithmetic rounds, or if it C truncates and only IT base IBETA digits C participate in the post-normalization shift of the C floating-point significand in multiplication; C 1 if floating-point arithmetic truncates and more C than IT base IBETA digits participate in the C post-normalization shift of the floating-point C significand in multiplication. C MACHEP - the largest negative integer such that C 1.0+FLOAT(IBETA)**MACHEP .NE. 1.0, except that C MACHEP is bounded below by -(IT+3) C NEGEPS - the largest negative integer such that C 1.0-FLOAT(IBETA)**NEGEPS .NE. 1.0, except that C NEGEPS is bounded below by -(IT+3) C IEXP - the number of bits (decimal places if IBETA = 10) C reserved for the representation of the exponent C (including the bias or sign) of a floating-point C number C MINEXP - the largest in magnitude negative integer such that C FLOAT(IBETA)**MINEXP is positive and normalized C MAXEXP - the smallest positive power of BETA that overflows C EPS - FLOAT(IBETA)**MACHEP. C EPSNEG - FLOAT(IBETA)**NEGEPS. C XMIN - the smallest non-vanishing normalized floating-point C power of the radix, i.e., XMIN = FLOAT(IBETA)**MINEXP C XMAX - the largest finite floating-point number. In C particular XMAX = (1.0-EPSNEG)*FLOAT(IBETA)**MAXEXP C Note - on some machines XMAX will be only the C second, or perhaps third, largest number, being C too small by 1 or 2 units in the last digit of C the significand. C C Latest modification: May 30, 1989 C C Author: W. J. Cody C Mathematics and Computer Science Division C Argonne National Laboratory C Argonne, IL 60439 C C---------------------------------------------------------------------- INTEGER I,IBETA,IEXP,IRND,IT,ITEMP,IZ,J,K,MACHEP,MAXEXP, 1 MINEXP,MX,NEGEP,NGRD,NXRES CS REAL CD DOUBLE PRECISION 1 A,B,BETA,BETAIN,BETAH,CONV,EPS,EPSNEG,ONE,T,TEMP,TEMPA, 2 TEMP1,TWO,XMAX,XMIN,Y,Z,ZERO C---------------------------------------------------------------------- CS CONV(I) = REAL(I) CD CONV(I) = DBLE(I) ONE = CONV(1) TWO = ONE + ONE ZERO = ONE - ONE C---------------------------------------------------------------------- C Determine IBETA, BETA ala Malcolm. C---------------------------------------------------------------------- A = ONE 10 A = A + A TEMP = A+ONE TEMP1 = TEMP-A IF (TEMP1-ONE .EQ. ZERO) GO TO 10 B = ONE 20 B = B + B TEMP = A+B ITEMP = INT(TEMP-A) IF (ITEMP .EQ. 0) GO TO 20 IBETA = ITEMP BETA = CONV(IBETA) C---------------------------------------------------------------------- C Determine IT, IRND. C---------------------------------------------------------------------- IT = 0 B = ONE 100 IT = IT + 1 B = B * BETA TEMP = B+ONE TEMP1 = TEMP-B IF (TEMP1-ONE .EQ. ZERO) GO TO 100 IRND = 0 BETAH = BETA / TWO TEMP = A+BETAH IF (TEMP-A .NE. ZERO) IRND = 1 TEMPA = A + BETA TEMP = TEMPA+BETAH IF ((IRND .EQ. 0) .AND. (TEMP-TEMPA .NE. ZERO)) IRND = 2 C---------------------------------------------------------------------- C Determine NEGEP, EPSNEG. C---------------------------------------------------------------------- NEGEP = IT + 3 BETAIN = ONE / BETA A = ONE DO 200 I = 1, NEGEP A = A * BETAIN 200 CONTINUE B = A 210 TEMP = ONE-A IF (TEMP-ONE .NE. ZERO) GO TO 220 A = A * BETA NEGEP = NEGEP - 1 GO TO 210 220 NEGEP = -NEGEP EPSNEG = A C---------------------------------------------------------------------- C Determine MACHEP, EPS. C---------------------------------------------------------------------- MACHEP = -IT - 3 A = B 300 TEMP = ONE+A IF (TEMP-ONE .NE. ZERO) GO TO 320 A = A * BETA MACHEP = MACHEP + 1 GO TO 300 320 EPS = A C---------------------------------------------------------------------- C Determine NGRD. C---------------------------------------------------------------------- NGRD = 0 TEMP = ONE+EPS IF ((IRND .EQ. 0) .AND. (TEMP*ONE-ONE .NE. ZERO)) NGRD = 1 C---------------------------------------------------------------------- C Determine IEXP, MINEXP, XMIN. C C Loop to determine largest I and K = 2**I such that C (1/BETA) ** (2**(I)) C does not underflow. C Exit from loop is signaled by an underflow. C---------------------------------------------------------------------- I = 0 K = 1 Z = BETAIN T = ONE + EPS NXRES = 0 400 Y = Z Z = Y * Y C---------------------------------------------------------------------- C Check for underflow here. C---------------------------------------------------------------------- A = Z * ONE TEMP = Z * T IF ((A+A .EQ. ZERO) .OR. (ABS(Z) .GE. Y)) GO TO 410 TEMP1 = TEMP * BETAIN IF (TEMP1*BETA .EQ. Z) GO TO 410 I = I + 1 K = K + K GO TO 400 410 IF (IBETA .EQ. 10) GO TO 420 IEXP = I + 1 MX = K + K GO TO 450 C---------------------------------------------------------------------- C This segment is for decimal machines only. C---------------------------------------------------------------------- 420 IEXP = 2 IZ = IBETA 430 IF (K .LT. IZ) GO TO 440 IZ = IZ * IBETA IEXP = IEXP + 1 GO TO 430 440 MX = IZ + IZ - 1 C---------------------------------------------------------------------- C Loop to determine MINEXP, XMIN. C Exit from loop is signaled by an underflow. C---------------------------------------------------------------------- 450 XMIN = Y Y = Y * BETAIN C---------------------------------------------------------------------- C Check for underflow here. C---------------------------------------------------------------------- A = Y * ONE TEMP = Y * T IF (((A+A) .EQ. ZERO) .OR. (ABS(Y) .GE. XMIN)) GO TO 460 K = K + 1 TEMP1 = TEMP * BETAIN IF ((TEMP1*BETA .NE. Y) .OR. (TEMP .EQ. Y)) THEN GO TO 450 ELSE NXRES = 3 XMIN = Y END IF 460 MINEXP = -K C---------------------------------------------------------------------- C Determine MAXEXP, XMAX. C---------------------------------------------------------------------- IF ((MX .GT. K+K-3) .OR. (IBETA .EQ. 10)) GO TO 500 MX = MX + MX IEXP = IEXP + 1 500 MAXEXP = MX + MINEXP C---------------------------------------------------------------------- C Adjust IRND to reflect partial underflow. C---------------------------------------------------------------------- IRND = IRND + NXRES C---------------------------------------------------------------------- C Adjust for IEEE-style machines. C---------------------------------------------------------------------- IF (IRND .GE. 2) MAXEXP = MAXEXP - 2 C---------------------------------------------------------------------- C Adjust for machines with implicit leading bit in binary C significand, and machines with radix point at extreme C right of significand. C---------------------------------------------------------------------- I = MAXEXP + MINEXP IF ((IBETA .EQ. 2) .AND. (I .EQ. 0)) MAXEXP = MAXEXP - 1 IF (I .GT. 20) MAXEXP = MAXEXP - 1 IF (A .NE. Y) MAXEXP = MAXEXP - 2 XMAX = ONE - EPSNEG IF (XMAX*ONE .NE. XMAX) XMAX = ONE - BETA * EPSNEG XMAX = XMAX / (BETA * BETA * BETA * XMIN) I = MAXEXP + MINEXP + 3 IF (I .LE. 0) GO TO 520 DO 510 J = 1, I IF (IBETA .EQ. 2) XMAX = XMAX + XMAX IF (IBETA .NE. 2) XMAX = XMAX * BETA 510 CONTINUE 520 RETURN C---------- Last line of MACHAR ---------- END