C------------------------------------------------------------------ C FORTRAN 77 program to test ALGAMA or DLGAMA C C Method: C C Accuracy tests compare function values against values C generated with the duplication formula. 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 EPS - The smallest positive floating-point C number such that 1.0+EPS .NE. 1.0 C XMIN - The smallest non-vanishing floating-point C integral power of the radix C XMAX - The largest finite 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, ANINT, DBLE, LOG, MAX, REAL, SQRT C C Reference: "Performance evaluation of programs related C to the real gamma function", W. J. Cody, C submitted for publication. C C Latest modification: June 16, 1988 C C Authors: W. J. Cody and L. Stoltz C Mathematics and Computer Science Division C Argonne National Laboratory C Argonne, IL 60439 C C-------------------------------------------------------------------- INTEGER I,IBETA,IEXP,IOUT,IRND,IT,J,JT,K1,K2,K3, 1 MACHEP,MAXEXP,MINEXP,N,NEGEP,NGRD CS REAL ALGAMA, CD DOUBLE PRECISION DLGAMA, 1 A,AIT,ALBETA,B,BETA,CL,CONV,C1,C2,C3,DEL,EPS,EPSNEG,FUNC, 2 HALF,ONE,P875,P3125,P625,P6875,REN,R6,R7,SIXTEN,TEN, 3 TWO,U,V,W,X,XC,XL,XMAX,XMIN,XN,X99,XP99,Y,Z,ZERO,ZZ C------------------------------------------------------------------ C C1 = 0.5 - LN(SQRT(PI)) C C2 = LN(2) C C3 = LN(2) - 11/16 C------------------------------------------------------------------ CS DATA C1,C2/-7.2364942924700087072E-2,6.9314718055994530942E-1/, CS 1 C3,ZERO,HALF/5.6471805599453094172E-3,0.0E0,0.5E0/, CS 2 ONE,TWO,TEN,SIXTEN/1.0E0,2.0E0,10.0E0,16.0E0/, CS 3 P6875,P875,P3125,P625/0.6875E0,0.875E0,1.3125E0,1.625E0/, CS 4 X99,XP99/-999.0E0,0.99E0/ CD DATA C1,C2/-7.2364942924700087072D-2,6.9314718055994530942D-1/, CD 1 C3,ZERO,HALF/5.6471805599453094172D-3,0.0D0,0.5D0/, CD 2 ONE,TWO,TEN,SIXTEN/1.0D0,2.0D0,10.0D0,16.0D0/, CD 3 P6875,P875,P3125,P625/0.6875D0,0.875D0,1.3125D0,1.625D0/, CD 4 X99,XP99/-999.0D0,0.99D0/ DATA IOUT/6/ C------------------------------------------------------------------ C Statement functions for conversion C------------------------------------------------------------------ CS CONV(I) = REAL(I) CD CONV(I) = DBLE(I) CS FUNC(X) = ALGAMA(X) CD FUNC(X) = DLGAMA(X) 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) ALBETA = LOG(BETA) AIT = CONV(IT) A = ZERO B = P875 N = 2000 XN = CONV(N) JT = 0 C----------------------------------------------------------------- C Determine largest argument for DLGAMA by iteration C----------------------------------------------------------------- CL = XP99 * XMAX Z = -CL / X99 80 ZZ = CL / (LOG(Z)-ONE) IF (ABS(ZZ/Z-ONE) .GT. (TWO*BETA*EPS)) THEN Z = ZZ GO TO 80 END IF CL = ZZ C----------------------------------------------------------------- C Random argument accuracy tests C----------------------------------------------------------------- DO 300 J = 1, 3 K1 = 0 K3 = 0 XC = ZERO R6 = ZERO R7 = ZERO DEL = (B - A) / XN XL = A DO 200 I = 1, N X = DEL * REN(JT) + XL C----------------------------------------------------------------- C Use duplication formula C----------------------------------------------------------------- IF (J .NE. 3) THEN IF (J .EQ. 1) THEN Z = X + HALF X = Z - HALF Y = X + X ELSE X = X + X X = X * HALF Y = (X + X) - ONE Z = X - HALF END IF U = FUNC(X) W = (Y-HALF)-HALF ZZ = ANINT(W*SIXTEN)/SIXTEN W = W - ZZ V = (((HALF-ZZ*P6875) - C1) - W*P6875)-C3*(W+ZZ) V = ((V + FUNC(Y)) - FUNC(Z)) ELSE Z = X * HALF + HALF Y = Z - HALF X = Y + Y U = FUNC(X) V = (C1 + ((X-HALF)-HALF)*C2)+FUNC(Y)+FUNC(Z)-HALF END IF C-------------------------------------------------------------------- C Accumulate results C-------------------------------------------------------------------- W = (U - V) / U 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 XC = X END IF R7 = R7 + W * W XL = XL + DEL 200 CONTINUE C------------------------------------------------------------------ C Gather and print statistics for test C------------------------------------------------------------------ K2 = N - K3 - K1 R7 = SQRT(R7/XN) IF (J .EQ. 1) THEN WRITE (IOUT,1000) ELSE IF (J .EQ. 2) THEN WRITE (IOUT,1001) ELSE WRITE (IOUT,1002) END IF WRITE (IOUT,1010) N,A,B WRITE (IOUT,1011) K1,K2,K3 WRITE (IOUT,1020) IT,IBETA IF (R6 .NE. ZERO) THEN W = LOG(ABS(R6))/ALBETA ELSE W = ALL9 END IF WRITE (IOUT,1021) R6,IBETA,W,XC W = MAX(AIT+W,ZERO) WRITE (IOUT,1022) IBETA,W IF (R7 .NE. ZERO) THEN W = LOG(ABS(R7))/ALBETA ELSE W = ALL9 END IF WRITE (IOUT,1023) R7,IBETA,W W = MAX(AIT+W,ZERO) WRITE (IOUT,1022) IBETA,W C------------------------------------------------------------------ C Initialize for next test C------------------------------------------------------------------ A = P3125 B = P625 IF (J .EQ. 2) THEN A = TWO + TWO B = TEN + TEN END IF 300 CONTINUE C----------------------------------------------------------------- C Special tests C First test with special arguments C----------------------------------------------------------------- WRITE (IOUT,1025) WRITE (IOUT,1040) Z = EPS ZZ = FUNC(Z) WRITE (IOUT,1041) Z,ZZ Z = HALF ZZ = FUNC(Z) WRITE (IOUT,1041) Z,ZZ Z = ONE ZZ = FUNC(Z) WRITE (IOUT,1041) Z,ZZ Z = TWO ZZ = FUNC(Z) WRITE (IOUT,1041) Z,ZZ C----------------------------------------------------------------- C Test of error returns C----------------------------------------------------------------- WRITE (IOUT,1050) Z = XMIN WRITE (IOUT,1053) Z ZZ = FUNC(Z) WRITE (IOUT,1061) ZZ Z = CL WRITE (IOUT,1053) Z ZZ = FUNC(Z) WRITE (IOUT,1061) ZZ Z = -ONE WRITE (IOUT,1052) Z ZZ = FUNC(Z) WRITE (IOUT,1061) ZZ Z = ZERO WRITE (IOUT,1052) Z ZZ = FUNC(Z) WRITE (IOUT,1061) ZZ Z = XP99 * XMAX WRITE (IOUT,1052) Z ZZ = FUNC(Z) WRITE (IOUT,1061) ZZ WRITE (IOUT,1100) STOP C----------------------------------------------------------------- 1000 FORMAT('1Test of LGAMA(X) vs LN(2*SQRT(PI))-2X*LN(2)+', 1 'LGAMA(2X)-LGAMA(X+1/2)'//) 1001 FORMAT('1Test of LGAMA(X) vs LN(2*SQRT(PI))-(2X-1)*LN(2)+', 1 'LGAMA(X-1/2)-LGAMA(2X-1)'//) 1002 FORMAT('1Test of LGAMA(X) vs -LN(2*SQRT(PI))+X*LN(2)+', 1 'LGAMA(X/2)+LGAMA(X/2+1/2)'//) 1010 FORMAT(I7,' Random arguments were tested from the interval (', 1 F5.1,',',F5.1,')'//) 1011 FORMAT(' LGAMA(X) was larger',I6,' times,'/ 1 14X,' agreed',I6,' times, and'/ 2 10X,'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'//) 1040 FORMAT(//' Test of special arguments'//) 1041 FORMAT(' LGAMA (',E13.6,') = ',E13.6//) 1050 FORMAT('1Test of Error Returns'///) 1052 FORMAT(' LGAMA will be called with the argument',E13.6,/ 1 ' This should trigger an error message'//) 1053 FORMAT(' LGAMA will be called with the argument',E13.6,/ 1 ' This should not trigger an error message'//) 1061 FORMAT(' LGAMA returned the value',E13.6///) 1100 FORMAT(' This concludes the tests') C---------- Last line of ALGAMA/DLGAMA 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