C------------------------------------------------------------------ C FORTRAN 77 program to test ERF and related functions. C C Method: C C Accuracy test compare function values against local Taylor's C series expansions. Derivatives for erfc(x) are expressed as C repeated integrals of erfc(x). These are generated from the C recurrence relation using a technique due to Gautschi (see C references). 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, DBLE, LOG, MAX, REAL, SQRT C C References: "Performance evaluation of programs for the error C and complementary error functions", W. J. Cody, C submitted for publication. C C "Evaluation of the repeated integrals of the coerror C function", W. Gautschi, TOMS 3, 1977, pp. 240-252. 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,IOUT,IRND,IT,J,JT,K,K1,K2,K3, 1 MACHEP,MAXEXP,MINEXP,N,NEGEP,NGRD,N0,N1 CS REAL ERF,ERFC,ERFCX, CD DOUBLE PRECISION DERF,DERFC,DERFCX, 1 A,AIT,ALBETA,B,BETA,C,CONV,C1,C2,DEL,EPS,EPSCON,EPSNEG,FF, 2 FUNC1,FUNC2,FUNC3,F0,HALF,ONE,R,REN,R1,R6,R7,SC,SIXTEN, 3 THRESH,TEN,TWO,U,V,W,X,XBIG,XC,XL,XMAX,XMIN,XN,XN1,X99, 4 Z,ZERO,ZZ DIMENSION R1(500) C------------------------------------------------------------------ C C1 = 1/sqrt(pi) C------------------------------------------------------------------ CS DATA ZERO,HALF,ONE,TWO,TEN/0.0E0,0.5E0,1.0E0,2.0E0,10.0E0/, CS 1 SIXTEN,THRESH,X99/16.0E0,0.46875E0,-999.0E0/, CS 2 C1/5.6418958354775628695E-1/ CD DATA ZERO,HALF,ONE,TWO,TEN/0.0D0,0.5D0,1.0D0,2.0D0,10.0D0/, CD 1 SIXTEN,THRESH,X99/16.0D0,0.46875D0,-999.0D0/, CD 2 C1/5.6418958354775628695D-1/ DATA IOUT/6/ C------------------------------------------------------------------ C Statement functions for conversion C------------------------------------------------------------------ CS CONV(I) = REAL(I) CS FUNC1(X) = ERF(X) CS FUNC2(X) = ERFC(X) CS FUNC3(X) = ERFCX(X) CD CONV(I) = DBLE(I) CD FUNC1(X) = DERF(X) CD FUNC2(X) = DERFC(X) CD FUNC3(X) = DERFCX(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) C = ABS(AIT*ALBETA) + TEN A = ZERO B = THRESH N = 2000 XN = CONV(N) JT = 0 N0 = (IBETA/2)*(IT+5)/6+4 C----------------------------------------------------------------- C Determine largest argument for ERFC test by Newton iteration C----------------------------------------------------------------- C2 = LOG(XMIN) + LOG(ONE/C1) XBIG = SQRT(-C2) 50 X = XBIG F0 = X*X FF = F0 + HALF/F0 + LOG(X) + C2 F0 = X+X + ONE/X - ONE/(X*F0) XBIG = X - FF/F0 IF (ABS(X-XBIG)/X .GT. TEN*EPS) GO TO 50 C----------------------------------------------------------------- C Random argument accuracy tests C----------------------------------------------------------------- DO 300 J = 1, 5 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 IF (J .EQ. 1) THEN C----------------------------------------------------------------- C Test erf against double series expansion C----------------------------------------------------------------- F0 = CONV(N0) FF = F0+F0+ONE Z = X*X W = Z+Z U = ZERO V = ZERO DO 60 K = 1, N0 U = -Z/F0*(ONE+U) V = W/FF*(ONE+V) F0 = F0 - ONE FF = FF - TWO 60 CONTINUE V = C1*(X+X)*(((U*V+(U+V))+HALF)+HALF) U = FUNC1(X) ELSE C----------------------------------------------------------------- C Test erfc or scaled erfc against expansion in repeated C integrals of the coerror function. C----------------------------------------------------------------- Z = X + HALF X = Z - HALF R = ZERO IF (X .LE. ONE) THEN N0 = 499 ELSE N0 = MIN(499,INT(C/(ABS(LOG(Z))))) END IF N1 = N0 XN1 = CONV(N1+1) DO 100 K=1,N0 R = HALF/(Z+XN1*R) R1(N1) = R N1 = N1 - 1 XN1 = XN1 - ONE 100 CONTINUE FF = C1/(Z+R1(1)) IF ((J/2)*2 .EQ. J) THEN F0 = FUNC2(Z)*EXP(X+HALF*HALF) U = FUNC2(X) ELSE F0 = FUNC3(Z) U = FUNC3(X) END IF SC = F0/FF C----------------------------------------------------------------- C Scale things to avoid premature underflow C----------------------------------------------------------------- EPSCON = F0 FF = SIXTEN*FF/EPS DO 110 N1=1,N0 FF = R1(N1)*FF R1(N1) = FF * SC IF (R1(N1) .LT. EPSCON ) THEN K = N1 GO TO 111 END IF 110 CONTINUE 111 V = R1(K) DO 120 N1 = 1, K-1 V = V + R1(K-N1) 120 CONTINUE C----------------------------------------------------------------- C Remove scaling here C----------------------------------------------------------------- V = V*EPS/SIXTEN + F0 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) WRITE (IOUT,1010) N,A,B WRITE (IOUT,1011) K1 ELSE IF ((J/2)*2 .EQ. J) THEN WRITE (IOUT,1001) WRITE (IOUT,1010) N,A,B WRITE (IOUT,1012) K1 ELSE WRITE (IOUT,1002) WRITE (IOUT,1010) N,A,B WRITE (IOUT,1013) K1 END IF WRITE (IOUT,1015) K2,K3 WRITE (IOUT,1020) IT,IBETA IF (R6 .NE. ZERO) THEN W = LOG(ABS(R6))/ALBETA ELSE W = X99 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 = X99 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------------------------------------------------------------------ IF (J .EQ. 1) THEN A = B B = TWO ELSE IF (J .EQ. 3) THEN A = B B = AINT(XBIG*SIXTEN)/SIXTEN-HALF ELSE IF (J .EQ. 4) THEN B = TEN + TEN END IF 300 CONTINUE C----------------------------------------------------------------- C Special tests C First check values for negative arguments. C----------------------------------------------------------------- WRITE (IOUT,1025) WRITE (IOUT,1030) IBETA X = ZERO DEL = -HALF DO 350 I = 1, 10 U = FUNC1(X) A = U + FUNC1(-X) IF (A*U .NE. ZERO) A = AIT + LOG(ABS(A/U))/ALBETA V = FUNC2(X) B = U + V - ONE IF (B .NE. ZERO) B = AIT + LOG(ABS(B))/ALBETA W = FUNC3(X) C = AINT(X*SIXTEN)/SIXTEN R = (X-C)*(X+C) C = (EXP(C*C)*EXP(R)*V-W)/W IF (C .NE. ZERO) C = MAX(ZERO,AIT + LOG(ABS(C))/ALBETA) WRITE (IOUT,1031) X,A,B,C X = X + DEL 350 CONTINUE C----------------------------------------------------------------- C Next, test with special arguments C----------------------------------------------------------------- WRITE (IOUT,1040) Z = XMAX ZZ = FUNC1(Z) WRITE (IOUT,1041) Z,ZZ Z = ZERO ZZ = FUNC1(Z) WRITE (IOUT,1041) Z,ZZ ZZ = FUNC2(Z) WRITE (IOUT,1042) Z,ZZ Z = -XMAX ZZ = FUNC2(Z) WRITE (IOUT,1042) Z,ZZ C----------------------------------------------------------------- C Test of error returns C----------------------------------------------------------------- WRITE (IOUT,1050) W = XBIG Z = W*(ONE-HALF*HALF) WRITE (IOUT,1052) Z ZZ = FUNC2(Z) WRITE (IOUT,1062) ZZ Z = W*(ONE+TEN*EPS) WRITE (IOUT,1053) Z ZZ = FUNC2(Z) WRITE (IOUT,1062) ZZ W = XMAX IF (C1 .LT. XMAX*XMIN) W = C1/XMIN Z = W*(ONE-ONE/SIXTEN) WRITE (IOUT,1054) Z ZZ = FUNC3(Z) WRITE (IOUT,1064) ZZ W = -SQRT(LOG(XMAX/TWO)) Z = W*(ONE-ONE/TEN) WRITE (IOUT,1055) Z ZZ = FUNC3(Z) WRITE (IOUT,1064) ZZ Z = W*(ONE+TEN*EPS) WRITE (IOUT,1056) Z ZZ = FUNC3(Z) WRITE (IOUT,1064) ZZ WRITE (IOUT,1100) STOP C----------------------------------------------------------------- 1000 FORMAT('1Test of erf(x) vs double series expansion'//) 1001 FORMAT(///' Test of erfc(x) vs exp(x+1/4) SUM i^n erfc(x+1/2)'//) 1002 FORMAT('1Test of exp(x*x) erfc(x) vs SUM i^n erfc(x+1/2)'//) 1010 FORMAT(I7,' Random arguments were tested from the interval (', 1 F7.3,',',F7.3,')'//) 1011 FORMAT(' ERF(X) was larger',I6,' times,') 1012 FORMAT(' ERFC(X) was larger',I6,' times,') 1013 FORMAT(' ERFCX(X) was larger',I6,' times,') 1015 FORMAT(14X,' agreed',I6,' times, and'/ 1 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'//) 1030 FORMAT(7X,'Estimated loss of base',i3,'significant digits in'// 1 3X'X',5X,'Erf(x)+Erf(-x)',3X,'Erf(x)+Erfc(x)-1', 1 3X,'Erfcx(x)-exp(x*x)*erfc(x)'/) 1031 FORMAT(F7.3,3F16.2) 1040 FORMAT(//' Test of special arguments'//) 1041 FORMAT(' ERF (',E13.6,') = ',E13.6//) 1042 FORMAT(' ERFC (',E13.6,') = ',E13.6//) 1050 FORMAT(' Test of Error Returns'///) 1052 FORMAT(' ERFC will be called with the argument',E13.6,/ 1 ' This should not underflow'//) 1053 FORMAT(' ERFC will be called with the argument',E13.6,/ 1 ' This may underflow'//) 1054 FORMAT(' ERFCX will be called with the argument',E13.6,/ 1 ' This should not underflow'//) 1055 FORMAT(' ERFCX will be called with the argument',E13.6,/ 1 ' This should not overflow'//) 1056 FORMAT(' ERFCX will be called with the argument',E13.6,/ 1 ' This may overflow'//) 1062 FORMAT(' ERFC returned the value',E13.6///) 1064 FORMAT(' ERFCX returned the value',E13.6///) 1100 FORMAT(' This concludes the tests') C---------- Last line of ERF 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