***************************************************************************** MPFUN2020: A thread-safe arbitrary precision package with special functions MPFUN20-Fort version Revision date: 9 Sep 2023 AUTHOR: David H. Bailey Lawrence Berkeley National Lab (retired) and University of California, Davis Email: dhbailey@lbl.gov COPYRIGHT AND DISCLAIMER: All software in this package (c) 2023 David H. Bailey. By downloading or using this software you agree to the copyright, disclaimer and license agreement in the accompanying file DISCLAIMER.txt. FULL DOCUMENTATION David H. Bailey, "MPFUN2020: A new thread-safe arbitrary precision package," https://www.davidhbailey.com/dhbpapers/mpfun2020.pdf INDEX OF THIS README FILE: I. PURPOSE OF PACKAGE II. INSTALLING CODING ENVIRONMENT (FOR MAC OS X SYSTEMS, IF NEEDED) III. INSTALLING FORTRAN COMPILER (IF NEEDED) IV. [Not needed in this version] V. DOWNLOADING AND COMPILING MPFUN20-Fort VI. BRIEF SUMMARY OF CODING INSTRUCTIONS AND USAGE VII. SAMPLE APPLICATION PROGRAMS AND TESTS VIII. RECENTLY TESTED PLATFORMS AND NOTES IX. RECENT UPDATES X. APPENDIX: TRANSCENDENTAL, SPECIAL AND MISCELLANEOUS FUNCTIONS +++++ I. PURPOSE OF PACKAGE This package permits one to perform floating-point computations (real and complex) to arbitrarily high numeric precision, by making only relatively minor changes to existing Fortran-90 programs (mostly changes to type statements). All basic arithmetic operations and transcendental functions are supported, together with numerous special functions. The package comes in two versions: one completely self-contained, all-Fortran version that is simple to install; and one version based on the MPFR package that is more complicated to install but runs somewhat faster on most applications. Both versions are completely thread-safe, which means that user-level applications can be easily converted for parallel execution, say by using a threaded parallel environment such as OpenMP. Both versions also detect, and provide means to overcome, accuracy problems rooted in the usage of inexact double-precision constants and expressions. A high-level Fortran-90 interface, supporting both multiprecision real and complex datatypes, is provided for each, so that most users need only to make minor changes to existing double-precision code. The two versions are "plug-compatible" in the sense that applications written for one also run with the other (provided a simple guideline is followed). The two versions of this package are: MPFUN20-Fort: This is an all-Fortran version based on 8-byte integer arithmetic. It includes support for a medium precision datatype, which results in faster execution on very large problems, and features FFT-based multiplication to accelerate very high precision computations. It compiles in just a few seconds on any system with a Fortran-2008 compliant compiler (examples include the GNU gfortran compiler, the Intel ifort compiler and the NAG Fortran compiler). MPFUN20-MPFR: This is virtually identical to MPFUN20-Fort in its user interface, but it calls the MPFR package for all low-level functions and operations. The MPFUN20-MPFR version is faster than MPFUN20-Fort on most applications, particularly those that involve transcendental functions. However, installation of MPFUN20-MPFR is significantly more complicated (because the GMP and MPFR packages must first be installed, usually requiring administrator privilege). What follows are the instructions for MPFUN20-Fort. II. INSTALLING CODING ENVIRONMENT (FOR MAC OS X SYSTEMS, IF NEEDED) For Apple Mac OS X systems (highly recommended for this software), first install the latest supported version of Xcode, which is available for free from the Apple Developer website: https://developer.apple.com/ Here click on Account, then enter your Apple ID and password, then go to https://developer.apple.com/download/more/?=xcode From this list, select the latest non-beta version of Xcode; as of the above date, this is 14.3.1. Download this package on your system, double click to decompress, and then install the Xcode app in the Applications folder. Double-click on the app to run Xcode, allowing it install additional components, then quit Xcode. Open a terminal window, using the Terminal application in the Utilities folder, and type xcode-select --install which installs various command-line tools (in the latest software release, this last step may not be necessary). The entire process of downloading Xcode and installing command-line tools typically takes 20 minutes. When this is completed, you should be ready to continue with the installation. III. INSTALLING FORTRAN COMPILER (IF NEEDED) Running this software is relatively straightforward, provided that one has a Unix-based system, such as Linux or Apple OS X, and a Fortran-2008 compliant compiler. These requirements are met by the GNU gfortran compiler, the Intel ifort compiler, the NAG nagfor compiler, IBM's xlf, PGI's pgf90 and others. The gfortran compiler (highly recommended for this software) is available free for a variety of systems at this website: https://gcc.gnu.org/wiki/GFortranBinaries For Apple Mac OS X systems, download the installer file here: https://github.com/fxcoudert/gfortran-for-macOS/releases The gfortran compiler is normally placed in /usr/local/lib and /usr/local/bin. Thus before one uses gfortran, one must insert a line in one's shell initialization file (if the Z shell is used, as on most Apple OS X systems, the shell initialization file is ~/.zshrc). The line to be included is: PATH=/usr/local/lib:/usr/local/bin:$PATH The following line is also recommended for gfortran compiler users: GFORTRAN_UNBUFFERED_ALL=yes; export GFORTRAN_UNBUFFERED_ALL The following line is recommended for inclusion in the shell initialization file, no matter what compiler is used (it prevents stack overflow system errors): ulimit -s unlimited On most Unix systems (including Apple Mac OS X systems), the shell initialization file must be manually executed upon initiating a terminal shell, typically by typing "source .zshrc". IV. [Not needed in this version] V. DOWNLOADING AND COMPILING MPFUN20-Fort From the website https://www.davidhbailey.com/dhbsoftware, download the file "mpfun20-fort-vnn.tar.gz" (replace "vnn" by whatever is the current version on the website, such as "v30"). If the file is not decompressed by your browser, use gunzip at the shell level to do this. Some browsers (such as the Apple Safari browser) do not drop the ".gz" suffix after decompression; if so, remove this suffix manually at the shell level. Then type tar xfv mpfun20-fort-vnn.tar (where again "vnn" is replaced by the downloaded version). This should create the directory and unpack all files. The MPFUN20-Fort software comes in two variants, which are in directories fortran-var1 and fortran-var2, respectively: Variant 1: This is recommended for beginning users and for basic applications that do not dynamically change the working precision level (or do so only rarely). Variant 2: This is recommended for more sophisticated applications that dynamically change the working precision level. It does not allow some mixed-mode combinations, and requires one to explicitly specify a working precision parameter for some functions. However, in the present author's experience, these restrictions result in less overall effort to produce a debugged, efficient application code. See documentation paper for additional details on the differences between these two variants. The Fortran source files and scripts required for each of these variants are in the respective directories fortran-var1 and fortran-var2. Compile/link scripts are available for the GNU gfortran, the Intel ifort and NAG Fortran compilers. These scripts automatically select the proper source files from the package for compilation and employ the appropriate compiler flags. For example, to compile Variant 1 of the library using the GNU gfortran compiler, go to the fortran-var1 directory and type ./gnu-complib1.scr The first time this script is executed will likely produce errors; just repeat the script. Then to compile and link the application program tpslq1.f90 for variant 1, using the GNU gfortran compiler, producing the executable file tpslq1, type ./gnu-complink1.scr tpslq1 To execute the program, with output to tpslq1.txt, type ./tpslq1 > tpslq1.txt These scripts assume that the user program is in the same directory as the library files; this can easily be changed by editing the script files. Several sample test programs, together with reference output files, are included in the fortran-var1 and fortran-var2 directories -- see Section VIII below. VI. BRIEF SUMMARY OF CODING INSTRUCTIONS AND USAGE What follows is a brief summary of Fortran coding instructions. For full details, see the documentation paper: David H. Bailey, "MPFUN2020: A new thread-safe arbitrary precision package," https://www.davidhbailey.com/dhbpapers/mpfun2020.pdf First set the parameter mpipl, the default standard precision level in digits, which is the maximum precision level to be used for subsequent computation, and is used to specify the amount of storage required for multiprecision data. mpipl is set in a parameter statement in file mpfunf.f90 in the fortran-var1 or fortran-var2 directory of the software. In the code as distributed, mpipl is set to 2500 digits (sufficient to run each of the test programs), but it can be set to any level greater than 50 digits. mpipl is automatically converted to mantissa words by the formula mpwds = int (mpipl / mpdpw + 2) where mpdpw (digits per word) is a system parameter (approx. 18.0617997398 for MPFUN20-Fort and 19.2659197224 in MPFUN-MPFR) set in file mpfuna.f90. The resulting parameter mpwds is the internal default precision level, in words. All subsequent computations are performed to mpwds words precision unless the user, within an application code, specifies a lower precision. After setting the value of mpipl, compile the library, using one of the scripts mentioned above (e.g., gnu-complib1.scr if using the GNU gfortran compiler or intel-complib1.scr if using the Intel compiler). Next, place the following line in every subprogram of the user's application code that contains a multiprecision variable or array, at the beginning of the declaration section, before any implicit or type statements: use mpmodule To designate a variable or array as multiprecision real (MPR) in an application code, use the Fortran-90 type statement with the type "mp_real", as in this example: type (mp_real) a, b(m), c(m,n) Similarly, to designate a variable or array as multiprecision complex (MPC), use a type statement with "mp_complex", as in: type (mp_complex) x, y(m), z(m,n) Thereafter when one of these variables or arrays appears in code, e.g., d = a + b(i) * sqrt(3.d0 - c(i,j)) the proper multiprecision routines are automatically called by the Fortran compiler. Most common mixed-mode combinations (arithmetic operations, comparisons and assignments) involving MPR, MPC, double precision (DP) and integer arguments are supported, although restrictions apply if one uses Variant 2 of the MPFUN20-Fort software. A complete list of supported mixed-mode operations is given in the documentation paper. Users should be aware, however, that there are some hazards in this type of programming, inherent in conventions adopted by all Fortran compilers. For example, the code r1 = 3.14159d0, where r1 is MPR, does NOT produce the true multiprecision equivalent of 3.14159. In fact, the software will flag such usage with a run-time error. To obtain the full MPR converted value, write this as r1 = '3.14159', or, if using variant 2, as r1 = mpreal ('3.14159', nwds), where nwds is the level of working precision to be assigned to r1. Similarly, the code r2 = r1 + 3.d0 * sqrt (2.d0), where r1 and r2 are MPR, does NOT produce the true multiprecision value one might expect, since the expression 3.d0 * sqrt (2.d0) will be performed in double precision, according to Fortran-90 precedence rules. In fact, the above line of code will result in a run-time error. To obtain the fully accurate result, write this as r2 = r1 + 3.d0 * sqrt (mpreal (2.d0)), or, if using variant 2, as r2 = r1 + 3.d0 * sqrt (mpreal (2.d0, nwds)), where nwds is the level of working precision. See documentation paper for details. Input and output of MPR and MPC data are performed using the subroutines mpread and mpwrite. For example, to output the variable r1 in E format to Fortran unit 6 (standard output), to 100-digit accuracy, in a field of width 120 characters, use the line of code call mpwrite (6, 120, 100, r1) The second argument (120 in the above example) must be at least 20 larger than the third argument (100 in the above example). To read the variable r1 from Fortran unit 5 (standard input), use the line of code call mpread (5, r1) or, if using variant 2, as call mpread (5, r1, nwds) where nwds is the level of working precision to be assigned to r1. See documentation paper for details such as formatting. Most Fortran-2008 intrinsic functions are supported with MPR and MPC arguments, as appropriate, and numerous special functions are also supported. A complete list of supported functions and subroutines is summarized in the Appendix below (section X), and, also in the documentation paper. VII. SAMPLE APPLICATION PROGRAMS AND TESTS The current release of the software includes a set of sample application programs in the fortran-var1 and fortran-var2 directories (the files are identical between directories): testmpfun.f90 Tests most arithmetic, transcendental and special functions. tpslq1.f90 Performs the standard 1-level PSLQ integer relation algorithm. tpslqm1.f90 Performs the 1-level multipair PSLQ integer relation algorithm. tpslqm2.f90 Performs the 2-level multipair PSLQ integer relation algorithm. tpslqm3.f90 Performs the 3-level multipair PSLQ integer relation algorithm. tpphix3.f90 Performs a Poisson polynomial application, using 3-level multipair PSLQ. tquad.f90 Evaluates a set of definite integrals, using tanh-sinh, exp-sinh and sinh-sinh algorithms. tquadgs.f90 Evaluates a set of definite integrals, using Gaussian quadrature. Corresponding reference output files (e.g., tpphix3.ref.txt) are also included for each of the above programs. In addition, the fortran-var1 and fortran-var2 directories include test scripts that compile the library and run each of the above sample programs above (except tquadgs.f90, which takes considerably more run time). In directory fortran-var1, these scripts are: gnu-mpfun-tests1.scr intel-mpfun-tests1.scr nag-mpfun-tests1.scr and the same scripts in directory fortran-var2, except for 2 instead of 1 in the filenames. For each test program, the script outputs either TEST PASSED or TEST FAILED. If all tests pass, then one can be fairly confident that the MPFUN2020 software and underlying compilers are working properly. Full descriptions of these application programs are included in the documentation paper: David H. Bailey, "MPFUN2020: A new thread-safe arbitrary precision package," https://www.davidhbailey.com/dhbpapers/mpfun2020.pdf VIII. RECENTLY TESTED PLATFORMS AND NOTES: 1. gfortran compiler, version 12.0.0, on an Apple MacBook Pro, OS X version 12.3.1, with an Apple Silicon (ARM) M1 Pro processor. 2. gfortran compiler, version 12.0.0, on an Apple Mac Studio, OS X version 12.3.1, with an Apple Silicon (ARM) M1 Max processor. 3. gfortran compiler, version 10.2.1, on a Debian Linux system, version 4.19.152-1, with an Intel processor. 4. NAG nagfor compiler, version 7.0, on a Debian Linux system, version 4.19.152-1, with an Intel processor. 5. Intel ifort compiler, version 2021.4.0, on a Debian Linux system, version 4.19.152-1, with an Intel processor. NOTE for Intel ifort: Due to what is evidently a compiler bug, one must make these changes to mpfunh1.f90 and mpfunh2.f90 before compiling: replace "mp_polylog_ini" (3 instances) with "mp_polylog_inim"; and replace "mp_polylog_neg" (6 instances) with "mp_polylog_negm". IX. RECENT UPDATES: 3 Mar 2022: Revised mpfune.f90, mpfung1.f90, mpfung2.f90, mpfunh1.f90 and mpfunh2.f90 to implement the hypergeom_pfq function; updated testmpfun.f90. 18 Apr 2022: Fixed a bug in mpfung1.f90, mpfung2.f90, mpfunh1.f90 and mpfunh2.f90. 14 May 2022: Implemented the special functions hurwitz_zetan_be and polygamma_be in mpfune.f90, with corresponding changes in mpfung1.f90, mpfung2.f90, mpfunh1.f90, mpfunh2.f90 and testmpfun.f90. 19 May 2022: Implemented the special functions bessel_i, bessel_j, bessel_k, bessel_y in mpfune.f90 with corresponding changes in mpfung1.f90, mpfung2.f90, mpfunh1.f90, mpfunh2.f90, mpmodule.f90 and testmpfun.f90. 7 Jan 2023: Fixed a problem with mpeformat and mpfformat; inserted intent statements in all routines that did not already have them; changed all parameter statements to object oriented style. 20 Mar 2023: Made minor changes to the testmpfun.f90 program. 10 Jun 2023: Produced a new version of mpfune.f90 by means of a conversion program (convmp.f90) from a high-level code. This included minor changes to all of the library files. 9 Sep 2023: Revised the special function gamma, with more accurate handling of values closes to 1; made several other minor improvements. X. APPENDIX: TRANSCENDENTAL, SPECIAL AND MISCELLANEOUS FUNCTIONS: As mentioned above, this software supports most Fortran-2008 intrinsics, including all the well-known transcendentals (e.g., sin, exp, log, etc.), and, in addition, a set of 30 special functions (e.g., Bessel functions, gamma function, zeta function, etc.). Further, the package includes a set of I/O and conversion functions, such as functions to convert between double precision and multiprecision real or between double complex and multiprecision complex. These functions and subroutines are listed below (a few are listed in more than one table). Some additional functions to work with the medium precision datatype are listed in the documentation paper. In these tables, "F" denotes function, "S" denotes subroutine, "MPR" denotes multiprecision real, "MPC" denotes multiprecision complex, "DP" denotes double precision, "DC" denotes double complex, "Int" denotes integer and "QP" denotes IEEE quad precision (if supported by the compiler). The variable names r1,r2,r3 are MPR; z1 is MPC; d1 is DP; dc1 is DC; q1 is QP; i1,i2,i3,n,nb,np,nq are integers; s1 is character(1); sn is character(n); rb is an MPR vector of length nb; ss is an MPR vector of length n; aa is an MPR vector of length np; and bb is an MPR vector of length nq. 1. Standard Fortran-2008 transcendental functions: Type Name Description MPR abs(r1) Absolute value MPR abs(z1) Absolute value of complex arg MPR acos(r1) Inverse cosine MPR acosh(r1) Inverse hyperbolic cosine MPR aimag(z1) Imaginary part of complex arg MPR aint(r1) Truncates to integer MPR anint(r1) Rounds to closest integer MPR asin(r1) Inverse sine MPR asinh(r1) Inverse hyperbolic sine MPR atan(r1) Inverse tangent MPR atan2(r1,r2) Arctangent with two args MPR atanh(r1) Inverse hyperbolic tangent MPR bessel_j0(r1) Bessel function of the first kind, order 0 MPR bessel_j1(r1) Bessel function of the first kind, order 1 MPR bessel_jn(n,r1) Besel function of the first kind, order n MPR bessel_y0(r1) Bessel function of the second kind, order 0 MPR bessel_y1(r1) Bessel function of the second kind, order 1 MPR bessel_yn(n,r1) Besel function of the second kind, order n MPC conjg(z1) Complex conjugate MPR cos(r1) Cosine of real arg MPC cos(z1) Cosine of complex arg MPR cosh(r1) Hyperbolic cosine DP dble(r1) Converts MPR argument to DP DC dcmplx(z1) Converts MPC argument to DC MPR erf(r1) Error function MPR erfc(r1) Complementary error function MPR exp(r1) Exponential function of real arg MPC exp(z1) Exponential function of complex arg MPR gamma(r1) Gamma function MPR hypot(r1,r2) Hypotenuse of two args MPR log(r1) Natural logarithm of real arg MPC log(z1) Natural logarithm of complex arg MPR log10(r1) Base-10 logarithm of real arg MPR log_gamma(r1) Log gamma function MPR max(r1,r2) Maximum of two (or three) args MPR min(r1,r2) Minimum of two (or three) args MPR mod(r1,r2) Mod function = r1 - r2*aint(r1/r2) MPR sign(r1,r2) Transfers sign from r2 to r1 MPR sin(r1) Sine function of real arg MPC sin(z1) Sine function of complex arg MPR sinh(r1) Hyperbolic sine MPR sqrt(r1) Square root of real arg MPC sqrt(z1) Square root of complex arg MPR tan(r1) Tangent function MPR tanh(r1) Hyperbolic tangent function 2. Special functions: Type Name Description F:MPR agm(r1,r2) Arithmetic-geometric mean F:MPR airy(r1) Airy function [1] S mpberne(nb,rb) Initialize array rb of length nb with first nb even Bernoulli numbers [2][3] F:MPR bessel_i(r1,r2) BesselI function, order r1, of r2 F:MPR bessel_in(n,r1) BesselI function, order n, of r1 F:MPR bessel_j(r1,r2) BesselJ function, order r1, of r2 F:MPR bessel_jn(n,r1) BesselJ function, order n, of r1 F:MPR bessel_i(r1,r2) BesselK function, order r1, of r2 F:MPR bessel_kn(n,r1) BesselK function, order n, of r1 F:MPR bessel_y(r1,r2) BesselY function, order r1, of r2 F:MPR bessel_yn(n,r1) BesselY function, order n, of r1 F:MPR digamma(r1) Digamma function of r1 [1] F:MPR digamma_be(nb,rb,r1) Digamma function of r1, using nb even Bernoulli numbers in rb [3] F:MPR erf(r1) Error function F:MPR erfc(r1) Complementary error function F:MPR expint(r1) Exponential integral function F:MPR gamma(r1) Gamma function F:MPR hurwitz_zetan(k,r1) Hurwitz zeta function, order n >= 2, of r1 [4] F:MPR hurwitz_zetan_be (nb,rb,n,r1) Hurwitz zeta function, order n >= 2, of r1, using nb even Bernoulli numbers in rb [3] F:MPR hypergeom_pfq(np,nq,aa,bb,r1) Hypergeometric pFq function of aa, bb and r1; dimensions are aa(np) and bb(nq) [5] F:MPR incgamma(r1,r2) Incomplete gamma function [6] F:MPR polygamma(k,r1) Polygamma function, order k >= 1, of r1 [4] F:MPR polygamma_be(nb,rb,k,r1) Polygamma function, order k >= 1, of r1, using nb even Bernoulli numbers in rb [3] S polylog_ini(n,ss) Initialize array ss, of size |n|, for computing polylogarithms of order n when n < 0 [2][6] F:MPR polylog_neg(n,ss,r1) Polylogarithm function of r1, for n < 0, using precomputed data in ss [7] F:MPR polylog_pos(n,r1) Polylogarithm function, order n >= 0, of r1 [8] F:MPR struve_hn(n,r1) StruveH function, order n >= 0, of r1 [9] F:MPR zeta(r1) Zeta function of r1 F:MPR zeta_be(nb,rb,r1) Zeta function of r1, using nb even Bernoulli numbers in rb [3] F:MPR zeta_int(n) Zeta function of integer argument n [2] Notes: [1]: Only available with MPFUN20-MPFR. [2]: In variant 1, an integer precision level argument (mantissa words) may optionally be added as the final argument; this argument is required in variant 2. [3]: For most applications, set nb > 2X precision in decimal digits; see mpberne above. [4]: For hurwitz_zetan and polygamma, the argument r1 is limited to the range (0, 1). [5]: For hypergeom_pfq, the integers np and nq must not exceed 10. [6]: For incgamma, r1 must not be zero, and must not be negative unless r2 = 0. [7]: For polylog_ini and polylog_neg, the integer n is limited to the range [-1000, -1]. [8]: For polylog_pos, the argument r1 is limited to the range (-1,1). [9]: For struve_hn, the argument r1 is limited to the range [-1000, 1000]. 3. Miscellaneous I/O, conversion and transcendental functions: Type Name Description F:MPC mpcmplx(r1,r2) Converts (r1,r2) to MPC [1] F:MPC mpcmplx(dc1) Converts DC arg to MPC [1] F:MPC mpcmplx(z1) Converts MPC arg to MPC [1] F:MPC mpcmplxdc(dc1) Converts DC to MPC, without checking [1][2] S mpcssh(r1,r2,r3) Returns both cosh and sinh of r1, in the same time as calling just cosh or just sinh S mpcssn(r1,r2,r3) Returns both cos and sin of r1, in the same time as calling just cos or just sin S mpdecmd(r1,d1,i1) Converts r1 to the form d1*10^i1 S mpeform(r1,i1,i2,s1) Converts r1 to char(1) string in Ei1.i2 suitable for output S mpfform(r1,i1,i2,s1) Converts r1 to char(1) string in Fi1.i2 suitable for output F:MPR mpegamma() Returns Euler's gamma constant [1] S mpinit() Initializes for extra-high precision [1] F:MPR mplog2() Returns log(2) [1] F:MPR mpnrt(r1,i1) Returns the i1-th root of r1 F:MPR mppi() Returns pi [1] F:MPR mpprod(r1,d1) Returns r1*d1, without checking [2] F:MPR mpquot(r1,d1) Returns r1/d1, without checking [2] F:MPR mprand(r1) Returns pseudorandom number, based on r1 Start with an irrational, say r1 = mplog2() Typical iterated usage: r1 = mprand(r1) S mpread(i1,r1) Inputs r1 from Fortran unit i1; up to five MPR args may be listed [1] S mpread(i1,z1) Inputs z1 from Fortran unit i1; up to five MPC args may be listed [1] F:MPR mpreal(r1) Converts MPR arg to MPR [1] F:MPR mpreal(z1) Converts MPC arg to MPR [1] F:MPR mpreal(d1) Converts DP arg to MPR [1] F:MPR mpreal(q1) Converts QP arg to MPR [1] F:MPR mpreal(s1,i1) Converts char(1) string of length i1 to MPR [1] F:MPR mpreal(sn) Converts char(n) string to MPR [1] F:MPR mpreald(d1) Converts DP to MPR, without checking [1][2] F:MPR mprealq(q1) Converts QP to MPR, without checking [1][2] F:Int mpwprec(r1) Returns precision in words assigned to r1 F:Int mpwprec(z1) Returns precision in words assigned to z1 S mpwrite(i1,i2,i3,r1) Outputs r1 in Ei2.i3 format to unit i1; up to five MPR args may be listed S mpwrite(i1,i2,i3,z1) Outputs z1 in Ei2.i3 format to unit i1; up to five MPC args may be listed F:QP qreal(r1) Converts MPR arg to quad precision (if supported) Notes: [1]: In variant 1, an integer precision level argument (mantissa words) may optionally be added as the final argument; this argument is required in variant 2. [2]: These do not check DP, DC or QP values.