🚀 go-pugleaf

In article <7811@june.cs.washington.edu> david@uw-june.UUCP (David Callahan) writes:
>..... < 80-bit stuff > ...

>Another reason involves portable numerical codes.
>Consider a routine such as MCHEPS (?) from LINPACK
>I think. It experimentally determines machine precision with a loop
>which is roughly:
>	COUNT = 1
>	EPS = 1.0
>	WHILE (EPS+1.0 > 1.0) 
>		COUNT = COUNT + 1
>		EPS = EPS / 2
>After execution COUNT will be some number.
>
>Since such a number will influence error estimates it would 
>be nice if it reflected the precision of EPS in memory.

There are better routines than the old one in linpack. 

Here is one:

      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.  The present version is documented in 
C   W. J. Cody, "MACHAR: A subroutine to dynamically determine machine
C   parameters," TOMS 14, December, 1988.
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     - the smallest positive floating-point number such
C                 that  1.0+EPS .NE. 1.0. In particular, if either
C                 IBETA = 2  or  IRND = 0, EPS = FLOAT(IBETA)**MACHEP.
C                 Otherwise,  EPS = (FLOAT(IBETA)**MACHEP)/2
C       EPSNEG  - A small positive floating-point number such that
C                 1.0-EPSNEG .NE. 1.0. In particular, if IBETA = 2
C                 or  IRND = 0, EPSNEG = FLOAT(IBETA)**NEGEPS.
C                 Otherwise,  EPSNEG = (IBETA**NEGEPS)/2.  Because
C                 NEGEPS is bounded below by -(IT+3), EPSNEG may not
C                 be the smallest number that can alter 1.0 by
C                 subtraction.
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 revision - December 4, 1987
C
C     Author - W. J. Cody
C              Argonne National Laboratory
C
C-----------------------------------------------------------------------
      INTEGER I,IBETA,IEXP,IRND,IT,ITEMP,IZ,J,K,MACHEP,MAXEXP,
     1        MINEXP,MX,NEGEP,NGRD,NXRES
#ifdef DP
      DOUBLE PRECISION
#else
      REAL
#endif

     1   A,B,BETA,BETAIN,BETAH,CONV,EPS,EPSNEG,ONE,T,TEMP,TEMPA,
     2   TEMP1,TWO,XMAX,XMIN,Y,Z,ZERO
C-----------------------------------------------------------------------
#ifdef DP
      CONV(I) = DBLE(I)
#else
      CONV(I) = REAL(I)
#endif

      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 CARD OF MACHAR ----------
      END




Keith H. Bierman
It's Not My Fault ---- I Voted for Bill & Opus