C------------------------------------------------------------------
C FORTRAN 77 program to test DAW
C
C Method:
C
C Accuracy test compare function values against a local
C Taylor's series expansion. Derivatives are generated
C from the recurrence relation.
C
C Data required
C
C None
C
C Subprograms required from this package
C
C MACHAR - An environmental inquiry program providing
C information on the floating-point arithmetic
C system. Note that the call to MACHAR can
C be deleted provided the following five
C parameters are assigned the values indicated
C
C IBETA - the radix of the floating-point system
C IT - the number of base-IBETA digits in the
C significant of a floating-point number
C XMIN - the smallest positive floating-point number
C XMAX - the largest floating-point number
C
C REN(K) - a function subprogram returning random real
C numbers uniformly distributed over (0,1)
C
C
C Intrinsic functions required are:
C
C ABS, DBLE, LOG, MAX, REAL, SQRT
C
C Reference: "The use of Taylor series to test accuracy of
C function programs", W. J. Cody and L. Stoltz,
C submitted for publication.
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,II,IOUT,IRND,IT,J,JT,K1,
1 K2,K3,MACHEP,MAXEXP,MINEXP,N,NDUM,NEGEP,NGRD
CS REAL
CD DOUBLE PRECISION
1 A,AIT,ALBETA,B,BETA,CONV,DAW,DEL,DELTA,EPS,EPSNEG,
2 FORTEN,HALF,ONE,P,REN,R6,R7,SIXTEN,TWO,T1,W,X,
3 XBIG,XKAY,XL,XMAX,XMIN,XN,X1,X99,Y,Z,ZERO,ZZ
DIMENSION P(0:14)
C------------------------------------------------------------------
CS DATA ZERO,HALF,ONE,TWO/0.0E0,0.5E0,1.0E0,2.0E0/,
CS 1 FORTEN,SIXTEN,X99,DELTA/14.0E0,1.6E1,-999.0E0,6.25E-2/
CD DATA ZERO,HALF,ONE,TWO/0.0D0,0.5D0,1.0D0,2.0D0/,
CD 1 FORTEN,SIXTEN,X99,DELTA/14.0D0,1.6D1,-999.0D0,6.25D-2/
DATA IOUT/6/
C------------------------------------------------------------------
C Define statement functions for conversions
C------------------------------------------------------------------
CS CONV(NDUM) = REAL(NDUM)
CD CONV(NDUM) = DBLE(NDUM)
C------------------------------------------------------------------
C Determine machine parameters and set constants
C------------------------------------------------------------------
CALL MACHAR(IBETA,IT,IRND,NGRD,MACHEP,NEGEP,IEXP,MINEXP,
1 MAXEXP,EPS,EPSNEG,XMIN,XMAX)
BETA = CONV(IBETA)
AIT = CONV(IT)
ALBETA = LOG(BETA)
A = DELTA
B = ONE
JT = 0
C-----------------------------------------------------------------
C Random argument accuracy tests based on local Taylor expansion.
C-----------------------------------------------------------------
DO 300 J = 1, 4
N = 2000
XN = CONV(N)
K1 = 0
K2 = 0
K3 = 0
X1 = ZERO
R6 = ZERO
R7 = ZERO
DEL = (B - A) / XN
XL = A
DO 200 I = 1, N
X = DEL * REN(JT) + XL
C------------------------------------------------------------------
C Purify arguments
C------------------------------------------------------------------
Y = X - DELTA
W = SIXTEN * Y
T1 = W + Y
Y = T1 - W
X = Y + DELTA
C------------------------------------------------------------------
C Use Taylor's Series Expansion
C------------------------------------------------------------------
P(0) = DAW(Y)
Z = Y + Y
P(1) = ONE - Z * P(0)
XKAY = TWO
DO 100 II = 2, 14
P(II) = -(Z*P(II-1)+XKAY*P(II-2))
XKAY = XKAY + TWO
100 CONTINUE
ZZ = P(14)
XKAY = FORTEN
DO 110 II = 1, 14
ZZ = ZZ*DELTA/XKAY + P(14-II)
XKAY = XKAY - ONE
110 CONTINUE
Z = DAW(X)
C------------------------------------------------------------------
C Accumulate Results
C------------------------------------------------------------------
W = (Z - ZZ) / Z
IF (W .GT. ZERO) THEN
K1 = K1 + 1
ELSE IF (W .LT. ZERO) THEN
K3 = K3 + 1
END IF
W = ABS(W)
IF (W .GT. R6) THEN
R6 = W
X1 = X
END IF
R7 = R7 + W * W
XL = XL + DEL
200 CONTINUE
C------------------------------------------------------------------
C Gather and print statistics for test
C------------------------------------------------------------------
K2 = N - K1 - K3
R7 = SQRT(R7/XN)
WRITE (IOUT,1000)
WRITE (IOUT,1010) N,A,B
WRITE (IOUT,1011) K1,K2,K3
WRITE (IOUT,1020) IT,IBETA
W = X99
IF (R6 .NE. ZERO) W = LOG(R6)/ALBETA
WRITE (IOUT,1021) R6,IBETA,W,X1
W = MAX(AIT+W,ZERO)
WRITE (IOUT,1022) IBETA,W
W = X99
IF (R7 .NE. ZERO) W = LOG(R7)/ALBETA
WRITE (IOUT,1023) R7,IBETA,W
W = MAX(AIT+W,ZERO)
WRITE (IOUT,1022) IBETA,W
C------------------------------------------------------------------
C Initialize for next test
C------------------------------------------------------------------
A = B
B = B + B
IF (J .EQ. 1) B = B + HALF
300 CONTINUE
C-----------------------------------------------------------------
C Special tests. First check values for negative arguments.
C-----------------------------------------------------------------
WRITE (IOUT,1025)
WRITE (IOUT,1030) IBETA
DO 350 I = 1, 10
X = REN(J)*(TWO+TWO)
B = DAW(X)
A = B + DAW(-X)
IF (A*B .NE. ZERO) A = AIT + LOG(ABS(A/B))/ALBETA
WRITE (IOUT,1031) X,A
X = X + DEL
350 CONTINUE
C-----------------------------------------------------------------
C Next, test with special arguments
C-----------------------------------------------------------------
WRITE (IOUT,1040)
Z = XMIN
ZZ = DAW(Z)
WRITE (IOUT,1041) ZZ
C-----------------------------------------------------------------
C Test of error return for arguments > xmax. First, determine
C xmax
C-----------------------------------------------------------------
IF (HALF .LT. XMIN*XMAX ) THEN
XBIG = HALF/XMIN
ELSE
XBIG = XMAX
END IF
WRITE (IOUT,1050)
Z = XBIG*(ONE-DELTA*DELTA)
WRITE (IOUT,1052) Z
ZZ = DAW(Z)
WRITE (IOUT,1062) ZZ
Z = XBIG
WRITE (IOUT,1053) Z
ZZ = DAW(Z)
WRITE (IOUT,1062) ZZ
W = ONE + DELTA*DELTA
IF (W .LT. XMAX/XBIG ) THEN
Z = XBIG*W
WRITE (IOUT,1053) Z
ZZ = DAW(Z)
WRITE (IOUT,1062) ZZ
END IF
WRITE (IOUT,1100)
STOP
C-----------------------------------------------------------------
1000 FORMAT('1Test of Dawson''s Integral vs Taylor expansion'//)
1010 FORMAT(I7,' Random arguments were tested from the interval ',
1 '(',F5.2,',',F5.2,')'//)
1011 FORMAT(' F(X) was larger',I6,' times,'/
1 10X,' agreed',I6,' times, and'/
1 6X,'was smaller',I6,' times.'//)
1020 FORMAT(' There are',I4,' base',I4,
1 ' significant digits in a floating-point number'//)
1021 FORMAT(' The maximum relative error of',E15.4,' = ',I4,' **',
1 F7.2/4X,'occurred for X =',E13.6)
1022 FORMAT(' The estimated loss of base',I4,
1 ' significant digits is',F7.2//)
1023 FORMAT(' The root mean square relative error was',E15.4,
1 ' = ',I4,' **',F7.2)
1025 FORMAT('1Special Tests'//)
1030 FORMAT(7X,'Estimated loss of base',i3,' significant digits in'//
1 8X'X',10X,'F(x)+F(-x)'/)
1031 FORMAT(3XF7.3,F16.2)
1040 FORMAT(//' Test of special arguments'//)
1041 FORMAT(' F(XMIN) = ',E24.17/)
1050 FORMAT(' Test of Error Returns'///)
1052 FORMAT(' DAW will be called with the argument',E13.6,/
1 ' This should not underflow'//)
1053 FORMAT(' DAW will be called with the argument',E13.6,/
1 ' This may underflow'//)
1062 FORMAT(' DAW returned the value',E13.6///)
1100 FORMAT(' This concludes the tests')
C---------- Last line of DAW test program ----------
END
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
FUNCTION REN(K)
C---------------------------------------------------------------------
C Random number generator - based on Algorithm 266 by Pike and
C Hill (modified by Hansson), Communications of the ACM,
C Vol. 8, No. 10, October 1965.
C
C This subprogram is intended for use on computers with
C fixed point wordlength of at least 29 bits. It is
C best if the floating-point significand has at most
C 29 bits.
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 IY,J,K
CS REAL CONV,C1,C2,C3,ONE,REN
CD DOUBLE PRECISION CONV,C1,C2,C3,ONE,REN
DATA IY/100001/
CS DATA ONE,C1,C2,C3/1.0E0,2796203.0E0,1.0E-6,1.0E-12/
CD DATA ONE,C1,C2,C3/1.0D0,2796203.0D0,1.0D-6,1.0D-12/
C---------------------------------------------------------------------
C Statement functions for conversion between integer and float
C---------------------------------------------------------------------
CS CONV(J) = REAL(J)
CD CONV(J) = DBLE(J)
C---------------------------------------------------------------------
J = K
IY = IY * 125
IY = IY - (IY/2796203) * 2796203
REN = CONV(IY) / C1 * (ONE + C2 + C3)
RETURN
C---------- Last card of REN ----------
END