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1#!/usr/bin/env julia
2# -*- julia -*-
3
4# remez.jl - implementation of the Remez algorithm for polynomial approximation
5#
6# Copyright (c) 2015-2018, Arm Limited.
7# SPDX-License-Identifier: MIT
8
9import Base.\
10
11# ----------------------------------------------------------------------
12# Helper functions to cope with different Julia versions.
13if VERSION >= v"0.7.0"
14    array1d(T, d) = Array{T, 1}(undef, d)
15    array2d(T, d1, d2) = Array{T, 2}(undef, d1, d2)
16else
17    array1d(T, d) = Array(T, d)
18    array2d(T, d1, d2) = Array(T, d1, d2)
19end
20if VERSION < v"0.5.0"
21    String = ASCIIString
22end
23if VERSION >= v"0.6.0"
24    # Use Base.invokelatest to run functions made using eval(), to
25    # avoid "world age" error
26    run(f, x...) = Base.invokelatest(f, x...)
27else
28    # Prior to 0.6.0, invokelatest doesn't exist (but fortunately the
29    # world age problem also doesn't seem to exist)
30    run(f, x...) = f(x...)
31end
32
33# ----------------------------------------------------------------------
34# Global variables configured by command-line options.
35floatsuffix = "" # adjusted by --floatsuffix
36xvarname = "x" # adjusted by --variable
37epsbits = 256 # adjusted by --bits
38debug_facilities = Set() # adjusted by --debug
39full_output = false # adjusted by --full
40array_format = false # adjusted by --array
41preliminary_commands = array1d(String, 0) # adjusted by --pre
42
43# ----------------------------------------------------------------------
44# Diagnostic and utility functions.
45
46# Enable debugging printouts from a particular subpart of this
47# program.
48#
49# Arguments:
50#    facility   Name of the facility to debug. For a list of facility names,
51#               look through the code for calls to debug().
52#
53# Return value is a BigFloat.
54function enable_debug(facility)
55    push!(debug_facilities, facility)
56end
57
58# Print a diagnostic.
59#
60# Arguments:
61#    facility   Name of the facility for which this is a debug message.
62#    printargs  Arguments to println() if debugging of that facility is
63#               enabled.
64macro debug(facility, printargs...)
65    printit = quote
66        print("[", $facility, "] ")
67    end
68    for arg in printargs
69        printit = quote
70            $printit
71            print($(esc(arg)))
72        end
73    end
74    return quote
75        if $facility in debug_facilities
76            $printit
77            println()
78        end
79    end
80end
81
82# Evaluate a polynomial.
83
84# Arguments:
85#    coeffs   Array of BigFloats giving the coefficients of the polynomial.
86#             Starts with the constant term, i.e. coeffs[i] is the
87#             coefficient of x^(i-1) (because Julia arrays are 1-based).
88#    x        Point at which to evaluate the polynomial.
89#
90# Return value is a BigFloat.
91function poly_eval(coeffs::Array{BigFloat}, x::BigFloat)
92    n = length(coeffs)
93    if n == 0
94        return BigFloat(0)
95    elseif n == 1
96        return coeffs[1]
97    else
98        return coeffs[1] + x * poly_eval(coeffs[2:n], x)
99    end
100end
101
102# Evaluate a rational function.
103
104# Arguments:
105#    ncoeffs  Array of BigFloats giving the coefficients of the numerator.
106#             Starts with the constant term, and 1-based, as above.
107#    dcoeffs  Array of BigFloats giving the coefficients of the denominator.
108#             Starts with the constant term, and 1-based, as above.
109#    x        Point at which to evaluate the function.
110#
111# Return value is a BigFloat.
112function ratfn_eval(ncoeffs::Array{BigFloat}, dcoeffs::Array{BigFloat},
113                    x::BigFloat)
114    return poly_eval(ncoeffs, x) / poly_eval(dcoeffs, x)
115end
116
117# Format a BigFloat into an appropriate output format.
118# Arguments:
119#    x        BigFloat to format.
120#
121# Return value is a string.
122function float_to_str(x)
123    return string(x) * floatsuffix
124end
125
126# Format a polynomial into an arithmetic expression, for pasting into
127# other tools such as gnuplot.
128
129# Arguments:
130#    coeffs   Array of BigFloats giving the coefficients of the polynomial.
131#             Starts with the constant term, and 1-based, as above.
132#
133# Return value is a string.
134function poly_to_string(coeffs::Array{BigFloat})
135    n = length(coeffs)
136    if n == 0
137        return "0"
138    elseif n == 1
139        return float_to_str(coeffs[1])
140    else
141        return string(float_to_str(coeffs[1]), "+", xvarname, "*(",
142                      poly_to_string(coeffs[2:n]), ")")
143    end
144end
145
146# Format a rational function into a string.
147
148# Arguments:
149#    ncoeffs  Array of BigFloats giving the coefficients of the numerator.
150#             Starts with the constant term, and 1-based, as above.
151#    dcoeffs  Array of BigFloats giving the coefficients of the denominator.
152#             Starts with the constant term, and 1-based, as above.
153#
154# Return value is a string.
155function ratfn_to_string(ncoeffs::Array{BigFloat}, dcoeffs::Array{BigFloat})
156    if length(dcoeffs) == 1 && dcoeffs[1] == 1
157        # Special case: if the denominator is just 1, leave it out.
158        return poly_to_string(ncoeffs)
159    else
160        return string("(", poly_to_string(ncoeffs), ")/(",
161                      poly_to_string(dcoeffs), ")")
162    end
163end
164
165# Format a list of x,y pairs into a string.
166
167# Arguments:
168#    xys      Array of (x,y) pairs of BigFloats.
169#
170# Return value is a string.
171function format_xylist(xys::Array{Tuple{BigFloat,BigFloat}})
172    return ("[\n" *
173            join(["  "*string(x)*" -> "*string(y) for (x,y) in xys], "\n") *
174            "\n]")
175end
176
177# ----------------------------------------------------------------------
178# Matrix-equation solver for matrices of BigFloat.
179#
180# I had hoped that Julia's type-genericity would allow me to solve the
181# matrix equation Mx=V by just writing 'M \ V'. Unfortunately, that
182# works by translating the inputs into double precision and handing
183# off to an optimised library, which misses the point when I have a
184# matrix and vector of BigFloat and want my result in _better_ than
185# double precision. So I have to implement my own specialisation of
186# the \ operator for that case.
187#
188# Fortunately, the point of using BigFloats is that we have precision
189# to burn, so I can do completely naïve Gaussian elimination without
190# worrying about instability.
191
192# Arguments:
193#    matrix_in    2-dimensional array of BigFloats, representing a matrix M
194#                 in row-first order, i.e. matrix_in[r,c] represents the
195#                 entry in row r col c.
196#    vector_in    1-dimensional array of BigFloats, representing a vector V.
197#
198# Return value: a 1-dimensional array X of BigFloats, satisfying M X = V.
199#
200# Expects the input to be an invertible square matrix and a vector of
201# the corresponding size, on pain of failing an assertion.
202function \(matrix_in :: Array{BigFloat,2},
203           vector_in :: Array{BigFloat,1})
204    # Copy the inputs, because we'll be mutating them as we go.
205    M = copy(matrix_in)
206    V = copy(vector_in)
207
208    # Input consistency criteria: matrix is square, and vector has
209    # length to match.
210    n = length(V)
211    @assert(n > 0)
212    @assert(size(M) == (n,n))
213
214    @debug("gausselim", "starting, n=", n)
215
216    for i = 1:1:n
217        # Straightforward Gaussian elimination: find the largest
218        # non-zero entry in column i (and in a row we haven't sorted
219        # out already), swap it into row i, scale that row to
220        # normalise it to 1, then zero out the rest of the column by
221        # subtracting a multiple of that row from each other row.
222
223        @debug("gausselim", "matrix=", repr(M))
224        @debug("gausselim", "vector=", repr(V))
225
226        # Find the best pivot.
227        bestrow = 0
228        bestval = 0
229        for j = i:1:n
230            if abs(M[j,i]) > bestval
231                bestrow = j
232                bestval = M[j,i]
233            end
234        end
235        @assert(bestrow > 0) # make sure we did actually find one
236
237        @debug("gausselim", "bestrow=", bestrow)
238
239        # Swap it into row i.
240        if bestrow != i
241            for k = 1:1:n
242                M[bestrow,k],M[i,k] = M[i,k],M[bestrow,k]
243            end
244            V[bestrow],V[i] = V[i],V[bestrow]
245        end
246
247        # Scale that row so that M[i,i] becomes 1.
248        divisor = M[i,i]
249        for k = 1:1:n
250            M[i,k] = M[i,k] / divisor
251        end
252        V[i] = V[i] / divisor
253        @assert(M[i,i] == 1)
254
255        # Zero out all other entries in column i, by subtracting
256        # multiples of this row.
257        for j = 1:1:n
258            if j != i
259                factor = M[j,i]
260                for k = 1:1:n
261                    M[j,k] = M[j,k] - M[i,k] * factor
262                end
263                V[j] = V[j] - V[i] * factor
264                @assert(M[j,i] == 0)
265            end
266        end
267    end
268
269    @debug("gausselim", "matrix=", repr(M))
270    @debug("gausselim", "vector=", repr(V))
271    @debug("gausselim", "done!")
272
273    # Now we're done: M is the identity matrix, so the equation Mx=V
274    # becomes just x=V, i.e. V is already exactly the vector we want
275    # to return.
276    return V
277end
278
279# ----------------------------------------------------------------------
280# Least-squares fitting of a rational function to a set of (x,y)
281# points.
282#
283# We use this to get an initial starting point for the Remez
284# iteration. Therefore, it doesn't really need to be particularly
285# accurate; it only needs to be good enough to wiggle back and forth
286# across the target function the right number of times (so as to give
287# enough error extrema to start optimising from) and not have any
288# poles in the target interval.
289#
290# Least-squares fitting of a _polynomial_ is actually a sensible thing
291# to do, and minimises the rms error. Doing the following trick with a
292# rational function P/Q is less sensible, because it cannot be made to
293# minimise the error function (P/Q-f)^2 that you actually wanted;
294# instead it minimises (P-fQ)^2. But that should be good enough to
295# have the properties described above.
296#
297# Some theory: suppose you're trying to choose a set of parameters a_i
298# so as to minimise the sum of squares of some error function E_i.
299# Basic calculus says, if you do this in one variable, just
300# differentiate and solve for zero. In this case, that works fine even
301# with multiple variables, because you _partially_ differentiate with
302# respect to each a_i, giving a system of equations, and that system
303# turns out to be linear so we just solve it as a matrix.
304#
305# In this case, our parameters are the coefficients of P and Q; to
306# avoid underdetermining the system we'll fix Q's constant term at 1,
307# so that our error function (as described above) is
308#
309# E = \sum (p_0 + p_1 x + ... + p_n x^n - y - y q_1 x - ... - y q_d x^d)^2
310#
311# where the sum is over all (x,y) coordinate pairs. Setting dE/dp_j=0
312# (for each j) gives an equation of the form
313#
314# 0 = \sum 2(p_0 + p_1 x + ... + p_n x^n - y - y q_1 x - ... - y q_d x^d) x^j
315#
316# and setting dE/dq_j=0 gives one of the form
317#
318# 0 = \sum 2(p_0 + p_1 x + ... + p_n x^n - y - y q_1 x - ... - y q_d x^d) y x^j
319#
320# And both of those row types, treated as multivariate linear
321# equations in the p,q values, have each coefficient being a value of
322# the form \sum x^i, \sum y x^i or \sum y^2 x^i, for various i. (Times
323# a factor of 2, but we can throw that away.) So we can go through the
324# list of input coordinates summing all of those things, and then we
325# have enough information to construct our matrix and solve it
326# straight off for the rational function coefficients.
327
328# Arguments:
329#    f        The function to be approximated. Maps BigFloat -> BigFloat.
330#    xvals    Array of BigFloats, giving the list of x-coordinates at which
331#             to evaluate f.
332#    n        Degree of the numerator polynomial of the desired rational
333#             function.
334#    d        Degree of the denominator polynomial of the desired rational
335#             function.
336#    w        Error-weighting function. Takes two BigFloat arguments x,y
337#             and returns a scaling factor for the error at that location.
338#             A larger value indicates that the error should be given
339#             greater weight in the square sum we try to minimise.
340#             If unspecified, defaults to giving everything the same weight.
341#
342# Return values: a pair of arrays of BigFloats (N,D) giving the
343# coefficients of the returned rational function. N has size n+1; D
344# has size d+1. Both start with the constant term, i.e. N[i] is the
345# coefficient of x^(i-1) (because Julia arrays are 1-based). D[1] will
346# be 1.
347function ratfn_leastsquares(f::Function, xvals::Array{BigFloat}, n, d,
348                            w = (x,y)->BigFloat(1))
349    # Accumulate sums of x^i y^j, for j={0,1,2} and a range of x.
350    # Again because Julia arrays are 1-based, we'll have sums[i,j]
351    # being the sum of x^(i-1) y^(j-1).
352    maxpow = max(n,d) * 2 + 1
353    sums = zeros(BigFloat, maxpow, 3)
354    for x = xvals
355        y = f(x)
356        weight = w(x,y)
357        for i = 1:1:maxpow
358            for j = 1:1:3
359                sums[i,j] += x^(i-1) * y^(j-1) * weight
360            end
361        end
362    end
363
364    @debug("leastsquares", "sums=", repr(sums))
365
366    # Build the matrix. We're solving n+d+1 equations in n+d+1
367    # unknowns. (We actually have to return n+d+2 coefficients, but
368    # one of them is hardwired to 1.)
369    matrix = array2d(BigFloat, n+d+1, n+d+1)
370    vector = array1d(BigFloat, n+d+1)
371    for i = 0:1:n
372        # Equation obtained by differentiating with respect to p_i,
373        # i.e. the numerator coefficient of x^i.
374        row = 1+i
375        for j = 0:1:n
376            matrix[row, 1+j] = sums[1+i+j, 1]
377        end
378        for j = 1:1:d
379            matrix[row, 1+n+j] = -sums[1+i+j, 2]
380        end
381        vector[row] = sums[1+i, 2]
382    end
383    for i = 1:1:d
384        # Equation obtained by differentiating with respect to q_i,
385        # i.e. the denominator coefficient of x^i.
386        row = 1+n+i
387        for j = 0:1:n
388            matrix[row, 1+j] = sums[1+i+j, 2]
389        end
390        for j = 1:1:d
391            matrix[row, 1+n+j] = -sums[1+i+j, 3]
392        end
393        vector[row] = sums[1+i, 3]
394    end
395
396    @debug("leastsquares", "matrix=", repr(matrix))
397    @debug("leastsquares", "vector=", repr(vector))
398
399    # Solve the matrix equation.
400    all_coeffs = matrix \ vector
401
402    @debug("leastsquares", "all_coeffs=", repr(all_coeffs))
403
404    # And marshal the results into two separate polynomial vectors to
405    # return.
406    ncoeffs = all_coeffs[1:n+1]
407    dcoeffs = vcat([1], all_coeffs[n+2:n+d+1])
408    return (ncoeffs, dcoeffs)
409end
410
411# ----------------------------------------------------------------------
412# Golden-section search to find a maximum of a function.
413
414# Arguments:
415#    f        Function to be maximised/minimised. Maps BigFloat -> BigFloat.
416#    a,b,c    BigFloats bracketing a maximum of the function.
417#
418# Expects:
419#    a,b,c are in order (either a<=b<=c or c<=b<=a)
420#    a != c             (but b can equal one or the other if it wants to)
421#    f(a) <= f(b) >= f(c)
422#
423# Return value is an (x,y) pair of BigFloats giving the extremal input
424# and output. (That is, y=f(x).)
425function goldensection(f::Function, a::BigFloat, b::BigFloat, c::BigFloat)
426    # Decide on a 'good enough' threshold.
427    threshold = abs(c-a) * 2^(-epsbits/2)
428
429    # We'll need the golden ratio phi, of course. Or rather, in this
430    # case, we need 1/phi = 0.618...
431    one_over_phi = 2 / (1 + sqrt(BigFloat(5)))
432
433    # Flip round the interval endpoints so that the interval [a,b] is
434    # at least as large as [b,c]. (Then we can always pick our new
435    # point in [a,b] without having to handle lots of special cases.)
436    if abs(b-a) < abs(c-a)
437        a,  c  = c,  a
438    end
439
440    # Evaluate the function at the initial points.
441    fa = f(a)
442    fb = f(b)
443    fc = f(c)
444
445    @debug("goldensection", "starting")
446
447    while abs(c-a) > threshold
448        @debug("goldensection", "a: ", a, " -> ", fa)
449        @debug("goldensection", "b: ", b, " -> ", fb)
450        @debug("goldensection", "c: ", c, " -> ", fc)
451
452        # Check invariants.
453        @assert(a <= b <= c || c <= b <= a)
454        @assert(fa <= fb >= fc)
455
456        # Subdivide the larger of the intervals [a,b] and [b,c]. We've
457        # arranged that this is always [a,b], for simplicity.
458        d = a + (b-a) * one_over_phi
459
460        # Now we have an interval looking like this (possibly
461        # reversed):
462        #
463        #    a            d       b            c
464        #
465        # and we know f(b) is bigger than either f(a) or f(c). We have
466        # two cases: either f(d) > f(b), or vice versa. In either
467        # case, we can narrow to an interval of 1/phi the size, and
468        # still satisfy all our invariants (three ordered points,
469        # [a,b] at least the width of [b,c], f(a)<=f(b)>=f(c)).
470        fd = f(d)
471        @debug("goldensection", "d: ", d, " -> ", fd)
472        if fd > fb
473            a,  b,  c  = a,  d,  b
474            fa, fb, fc = fa, fd, fb
475            @debug("goldensection", "adb case")
476        else
477            a,  b,  c  = c,  b,  d
478            fa, fb, fc = fc, fb, fd
479            @debug("goldensection", "cbd case")
480        end
481    end
482
483    @debug("goldensection", "done: ", b, " -> ", fb)
484    return (b, fb)
485end
486
487# ----------------------------------------------------------------------
488# Find the extrema of a function within a given interval.
489
490# Arguments:
491#    f         The function to be approximated. Maps BigFloat -> BigFloat.
492#    grid      A set of points at which to evaluate f. Must be high enough
493#              resolution to make extrema obvious.
494#
495# Returns an array of (x,y) pairs of BigFloats, with each x,y giving
496# the extremum location and its value (i.e. y=f(x)).
497function find_extrema(f::Function, grid::Array{BigFloat})
498    len = length(grid)
499    extrema = array1d(Tuple{BigFloat, BigFloat}, 0)
500    for i = 1:1:len
501        # We have to provide goldensection() with three points
502        # bracketing the extremum. If the extremum is at one end of
503        # the interval, then the only way we can do that is to set two
504        # of the points equal (which goldensection() will cope with).
505        prev = max(1, i-1)
506        next = min(i+1, len)
507
508        # Find our three pairs of (x,y) coordinates.
509        xp, xi, xn = grid[prev], grid[i], grid[next]
510        yp, yi, yn = f(xp), f(xi), f(xn)
511
512        # See if they look like an extremum, and if so, ask
513        # goldensection() to give a more exact location for it.
514        if yp <= yi >= yn
515            push!(extrema, goldensection(f, xp, xi, xn))
516        elseif yp >= yi <= yn
517            x, y = goldensection(x->-f(x), xp, xi, xn)
518            push!(extrema, (x, -y))
519        end
520    end
521    return extrema
522end
523
524# ----------------------------------------------------------------------
525# Winnow a list of a function's extrema to give a subsequence of a
526# specified length, with the extrema in the subsequence alternating
527# signs, and with the smallest absolute value of an extremum in the
528# subsequence as large as possible.
529#
530# We do this using a dynamic-programming approach. We work along the
531# provided array of extrema, and at all times, we track the best set
532# of extrema we have so far seen for each possible (length, sign of
533# last extremum) pair. Each new extremum is evaluated to see whether
534# it can be added to any previously seen best subsequence to make a
535# new subsequence that beats the previous record holder in its slot.
536
537# Arguments:
538#    extrema   An array of (x,y) pairs of BigFloats giving the input extrema.
539#    n         Number of extrema required as output.
540#
541# Returns a new array of (x,y) pairs which is a subsequence of the
542# original sequence. (So, in particular, if the input was sorted by x
543# then so will the output be.)
544function winnow_extrema(extrema::Array{Tuple{BigFloat,BigFloat}}, n)
545    # best[i,j] gives the best sequence so far of length i and with
546    # sign j (where signs are coded as 1=positive, 2=negative), in the
547    # form of a tuple (cost, actual array of x,y pairs).
548    best = fill((BigFloat(0), array1d(Tuple{BigFloat,BigFloat}, 0)), n, 2)
549
550    for (x,y) = extrema
551        if y > 0
552            sign = 1
553        elseif y < 0
554            sign = 2
555        else
556            # A zero-valued extremum cannot possibly contribute to any
557            # optimal sequence, so we simply ignore it!
558            continue
559        end
560
561        for i = 1:1:n
562            # See if we can create a new entry for best[i,sign] by
563            # appending our current (x,y) to some previous thing.
564            if i == 1
565                # Special case: we don't store a best zero-length
566                # sequence :-)
567                candidate = (abs(y), [(x,y)])
568            else
569                othersign = 3-sign # map 1->2 and 2->1
570                oldscore, oldlist = best[i-1, othersign]
571                newscore = min(abs(y), oldscore)
572                newlist = vcat(oldlist, [(x,y)])
573                candidate = (newscore, newlist)
574            end
575            # If our new candidate improves on the previous value of
576            # best[i,sign], then replace it.
577            if candidate[1] > best[i,sign][1]
578                best[i,sign] = candidate
579            end
580        end
581    end
582
583    # Our ultimate return value has to be either best[n,1] or
584    # best[n,2], but it could be either. See which one has the higher
585    # score.
586    if best[n,1][1] > best[n,2][1]
587        ret = best[n,1][2]
588    else
589        ret = best[n,2][2]
590    end
591    # Make sure we did actually _find_ a good answer.
592    @assert(length(ret) == n)
593    return ret
594end
595
596# ----------------------------------------------------------------------
597# Construct a rational-function approximation with equal and
598# alternating weighted deviation at a specific set of x-coordinates.
599
600# Arguments:
601#    f         The function to be approximated. Maps BigFloat -> BigFloat.
602#    coords    An array of BigFloats giving the x-coordinates. There should
603#              be n+d+2 of them.
604#    n, d      The degrees of the numerator and denominator of the desired
605#              approximation.
606#    prev_err  A plausible value for the alternating weighted deviation.
607#              (Required to kickstart a binary search in the nonlinear case;
608#              see comments below.)
609#    w         Error-weighting function. Takes two BigFloat arguments x,y
610#              and returns a scaling factor for the error at that location.
611#              The returned approximation R should have the minimum possible
612#              maximum value of abs((f(x)-R(x)) * w(x,f(x))). Optional
613#              parameter, defaulting to the always-return-1 function.
614#
615# Return values: a pair of arrays of BigFloats (N,D) giving the
616# coefficients of the returned rational function. N has size n+1; D
617# has size d+1. Both start with the constant term, i.e. N[i] is the
618# coefficient of x^(i-1) (because Julia arrays are 1-based). D[1] will
619# be 1.
620function ratfn_equal_deviation(f::Function, coords::Array{BigFloat},
621                               n, d, prev_err::BigFloat,
622                               w = (x,y)->BigFloat(1))
623    @debug("equaldev", "n=", n, " d=", d, " coords=", repr(coords))
624    @assert(length(coords) == n+d+2)
625
626    if d == 0
627        # Special case: we're after a polynomial. In this case, we
628        # have the particularly easy job of just constructing and
629        # solving a system of n+2 linear equations, to find the n+1
630        # coefficients of the polynomial and also the amount of
631        # deviation at the specified coordinates. Each equation is of
632        # the form
633        #
634        #   p_0 x^0 + p_1 x^1 + ... + p_n x^n ± e/w(x) = f(x)
635        #
636        # in which the p_i and e are the variables, and the powers of
637        # x and calls to w and f are the coefficients.
638
639        matrix = array2d(BigFloat, n+2, n+2)
640        vector = array1d(BigFloat, n+2)
641        currsign = +1
642        for i = 1:1:n+2
643            x = coords[i]
644            for j = 0:1:n
645                matrix[i,1+j] = x^j
646            end
647            y = f(x)
648            vector[i] = y
649            matrix[i, n+2] = currsign / w(x,y)
650            currsign = -currsign
651        end
652
653        @debug("equaldev", "matrix=", repr(matrix))
654        @debug("equaldev", "vector=", repr(vector))
655
656        outvector = matrix \ vector
657
658        @debug("equaldev", "outvector=", repr(outvector))
659
660        ncoeffs = outvector[1:n+1]
661        dcoeffs = [BigFloat(1)]
662        return ncoeffs, dcoeffs
663    else
664        # For a nontrivial rational function, the system of equations
665        # we need to solve becomes nonlinear, because each equation
666        # now takes the form
667        #
668        #   p_0 x^0 + p_1 x^1 + ... + p_n x^n
669        #   --------------------------------- ± e/w(x) = f(x)
670        #     x^0 + q_1 x^1 + ... + q_d x^d
671        #
672        # and multiplying up by the denominator gives you a lot of
673        # terms containing e × q_i. So we can't do this the really
674        # easy way using a matrix equation as above.
675        #
676        # Fortunately, this is a fairly easy kind of nonlinear system.
677        # The equations all become linear if you switch to treating e
678        # as a constant, so a reasonably sensible approach is to pick
679        # a candidate value of e, solve all but one of the equations
680        # for the remaining unknowns, and then see what the error
681        # turns out to be in the final equation. The Chebyshev
682        # alternation theorem guarantees that that error in the last
683        # equation will be anti-monotonic in the input e, so we can
684        # just binary-search until we get the two as close to equal as
685        # we need them.
686
687        function try_e(e)
688            # Try a given value of e, derive the coefficients of the
689            # resulting rational function by setting up equations
690            # based on the first n+d+1 of the n+d+2 coordinates, and
691            # see what the error turns out to be at the final
692            # coordinate.
693            matrix = array2d(BigFloat, n+d+1, n+d+1)
694            vector = array1d(BigFloat, n+d+1)
695            currsign = +1
696            for i = 1:1:n+d+1
697                x = coords[i]
698                y = f(x)
699                y_adj = y - currsign * e / w(x,y)
700                for j = 0:1:n
701                    matrix[i,1+j] = x^j
702                end
703                for j = 1:1:d
704                    matrix[i,1+n+j] = -x^j * y_adj
705                end
706                vector[i] = y_adj
707                currsign = -currsign
708            end
709
710            @debug("equaldev", "trying e=", e)
711            @debug("equaldev", "matrix=", repr(matrix))
712            @debug("equaldev", "vector=", repr(vector))
713
714            outvector = matrix \ vector
715
716            @debug("equaldev", "outvector=", repr(outvector))
717
718            ncoeffs = outvector[1:n+1]
719            dcoeffs = vcat([BigFloat(1)], outvector[n+2:n+d+1])
720
721            x = coords[n+d+2]
722            y = f(x)
723            last_e = (ratfn_eval(ncoeffs, dcoeffs, x) - y) * w(x,y) * -currsign
724
725            @debug("equaldev", "last e=", last_e)
726
727            return ncoeffs, dcoeffs, last_e
728        end
729
730        threshold = 2^(-epsbits/2) # convergence threshold
731
732        # Start by trying our previous iteration's error value. This
733        # value (e0) will be one end of our binary-search interval,
734        # and whatever it caused the last point's error to be, that
735        # (e1) will be the other end.
736        e0 = prev_err
737        @debug("equaldev", "e0 = ", e0)
738        nc, dc, e1 = try_e(e0)
739        @debug("equaldev", "e1 = ", e1)
740        if abs(e1-e0) <= threshold
741            # If we're _really_ lucky, we hit the error right on the
742            # nose just by doing that!
743            return nc, dc
744        end
745        s = sign(e1-e0)
746        @debug("equaldev", "s = ", s)
747
748        # Verify by assertion that trying our other interval endpoint
749        # e1 gives a value that's wrong in the other direction.
750        # (Otherwise our binary search won't get a sensible answer at
751        # all.)
752        nc, dc, e2 = try_e(e1)
753        @debug("equaldev", "e2 = ", e2)
754        @assert(sign(e2-e1) == -s)
755
756        # Now binary-search until our two endpoints narrow enough.
757        local emid
758        while abs(e1-e0) > threshold
759            emid = (e1+e0)/2
760            nc, dc, enew = try_e(emid)
761            if sign(enew-emid) == s
762                e0 = emid
763            else
764                e1 = emid
765            end
766        end
767
768        @debug("equaldev", "final e=", emid)
769        return nc, dc
770    end
771end
772
773# ----------------------------------------------------------------------
774# Top-level function to find a minimax rational-function approximation.
775
776# Arguments:
777#    f         The function to be approximated. Maps BigFloat -> BigFloat.
778#    interval  A pair of BigFloats giving the endpoints of the interval
779#              (in either order) on which to approximate f.
780#    n, d      The degrees of the numerator and denominator of the desired
781#              approximation.
782#    w         Error-weighting function. Takes two BigFloat arguments x,y
783#              and returns a scaling factor for the error at that location.
784#              The returned approximation R should have the minimum possible
785#              maximum value of abs((f(x)-R(x)) * w(x,f(x))). Optional
786#              parameter, defaulting to the always-return-1 function.
787#
788# Return values: a tuple (N,D,E,X), where
789
790#    N,D       A pair of arrays of BigFloats giving the coefficients
791#              of the returned rational function. N has size n+1; D
792#              has size d+1. Both start with the constant term, i.e.
793#              N[i] is the coefficient of x^(i-1) (because Julia
794#              arrays are 1-based). D[1] will be 1.
795#    E         The maximum weighted error (BigFloat).
796#    X         An array of pairs of BigFloats giving the locations of n+2
797#              points and the weighted error at each of those points. The
798#              weighted error values will have alternating signs, which
799#              means that the Chebyshev alternation theorem guarantees
800#              that any other function of the same degree must exceed
801#              the error of this one at at least one of those points.
802function ratfn_minimax(f::Function, interval::Tuple{BigFloat,BigFloat}, n, d,
803                       w = (x,y)->BigFloat(1))
804    # We start off by finding a least-squares approximation. This
805    # doesn't need to be perfect, but if we can get it reasonably good
806    # then it'll save iterations in the refining stage.
807    #
808    # Least-squares approximations tend to look nicer in a minimax
809    # sense if you evaluate the function at a big pile of Chebyshev
810    # nodes rather than uniformly spaced points. These values will
811    # also make a good grid to use for the initial search for error
812    # extrema, so we'll keep them around for that reason too.
813
814    # Construct the grid.
815    lo, hi = minimum(interval), maximum(interval)
816    local grid
817    let
818        mid = (hi+lo)/2
819        halfwid = (hi-lo)/2
820        nnodes = 16 * (n+d+1)
821        pi = 2*asin(BigFloat(1))
822        grid = [ mid - halfwid * cos(pi*i/nnodes) for i=0:1:nnodes ]
823    end
824
825    # Find the initial least-squares approximation.
826    (nc, dc) = ratfn_leastsquares(f, grid, n, d, w)
827    @debug("minimax", "initial leastsquares approx = ",
828           ratfn_to_string(nc, dc))
829
830    # Threshold of convergence. We stop when the relative difference
831    # between the min and max (winnowed) error extrema is less than
832    # this.
833    #
834    # This is set to the cube root of machine epsilon on a more or
835    # less empirical basis, because the rational-function case will
836    # not converge reliably if you set it to only the square root.
837    # (Repeatable by using the --test mode.) On the assumption that
838    # input and output error in each iteration can be expected to be
839    # related by a simple power law (because it'll just be down to how
840    # many leading terms of a Taylor series are zero), the cube root
841    # was the next thing to try.
842    threshold = 2^(-epsbits/3)
843
844    # Main loop.
845    while true
846        # Find all the error extrema we can.
847        function compute_error(x)
848            real_y = f(x)
849            approx_y = ratfn_eval(nc, dc, x)
850            return (approx_y - real_y) * w(x, real_y)
851        end
852        extrema = find_extrema(compute_error, grid)
853        @debug("minimax", "all extrema = ", format_xylist(extrema))
854
855        # Winnow the extrema down to the right number, and ensure they
856        # have alternating sign.
857        extrema = winnow_extrema(extrema, n+d+2)
858        @debug("minimax", "winnowed extrema = ", format_xylist(extrema))
859
860        # See if we've finished.
861        min_err = minimum([abs(y) for (x,y) = extrema])
862        max_err = maximum([abs(y) for (x,y) = extrema])
863        variation = (max_err - min_err) / max_err
864        @debug("minimax", "extremum variation = ", variation)
865        if variation < threshold
866            @debug("minimax", "done!")
867            return nc, dc, max_err, extrema
868        end
869
870        # If not, refine our function by equalising the error at the
871        # extrema points, and go round again.
872        (nc, dc) = ratfn_equal_deviation(f, map(x->x[1], extrema),
873                                         n, d, max_err, w)
874        @debug("minimax", "refined approx = ", ratfn_to_string(nc, dc))
875    end
876end
877
878# ----------------------------------------------------------------------
879# Check if a polynomial is well-conditioned for accurate evaluation in
880# a given interval by Horner's rule.
881#
882# This is true if at every step where Horner's rule computes
883# (coefficient + x*value_so_far), the constant coefficient you're
884# adding on is of larger magnitude than the x*value_so_far operand.
885# And this has to be true for every x in the interval.
886#
887# Arguments:
888#    coeffs    The coefficients of the polynomial under test. Starts with
889#              the constant term, i.e. coeffs[i] is the coefficient of
890#              x^(i-1) (because Julia arrays are 1-based).
891#    lo, hi    The bounds of the interval.
892#
893# Return value: the largest ratio (x*value_so_far / coefficient), at
894# any step of evaluation, for any x in the interval. If this is less
895# than 1, the polynomial is at least somewhat well-conditioned;
896# ideally you want it to be more like 1/8 or 1/16 or so, so that the
897# relative rounding error accumulated at each step are reduced by
898# several factors of 2 when the next coefficient is added on.
899
900function wellcond(coeffs, lo, hi)
901    x = max(abs(lo), abs(hi))
902    worst = 0
903    so_far = 0
904    for i = length(coeffs):-1:1
905        coeff = abs(coeffs[i])
906        so_far *= x
907        if coeff != 0
908            thisval = so_far / coeff
909            worst = max(worst, thisval)
910            so_far += coeff
911        end
912    end
913    return worst
914end
915
916# ----------------------------------------------------------------------
917# Small set of unit tests.
918
919function test()
920    passes = 0
921    fails = 0
922
923    function approx_eq(x, y, limit=1e-6)
924        return abs(x - y) < limit
925    end
926
927    function test(condition)
928        if condition
929            passes += 1
930        else
931            println("fail")
932            fails += 1
933        end
934    end
935
936    # Test Gaussian elimination.
937    println("Gaussian test 1:")
938    m = BigFloat[1 1 2; 3 5 8; 13 34 21]
939    v = BigFloat[1, -1, 2]
940    ret = m \ v
941    println("  ",repr(ret))
942    test(approx_eq(ret[1], 109/26))
943    test(approx_eq(ret[2], -105/130))
944    test(approx_eq(ret[3], -31/26))
945
946    # Test leastsquares rational functions.
947    println("Leastsquares test 1:")
948    n = 10000
949    a = array1d(BigFloat, n+1)
950    for i = 0:1:n
951        a[1+i] = i/BigFloat(n)
952    end
953    (nc, dc) = ratfn_leastsquares(x->exp(x), a, 2, 2)
954    println("  ",ratfn_to_string(nc, dc))
955    for x = a
956        test(approx_eq(exp(x), ratfn_eval(nc, dc, x), 1e-4))
957    end
958
959    # Test golden section search.
960    println("Golden section test 1:")
961    x, y = goldensection(x->sin(x),
962                              BigFloat(0), BigFloat(1)/10, BigFloat(4))
963    println("  ", x, " -> ", y)
964    test(approx_eq(x, asin(BigFloat(1))))
965    test(approx_eq(y, 1))
966
967    # Test extrema-winnowing algorithm.
968    println("Winnow test 1:")
969    extrema = [(x, sin(20*x)*sin(197*x))
970               for x in BigFloat(0):BigFloat(1)/1000:BigFloat(1)]
971    winnowed = winnow_extrema(extrema, 12)
972    println("  ret = ", format_xylist(winnowed))
973    prevx, prevy = -1, 0
974    for (x,y) = winnowed
975        test(x > prevx)
976        test(y != 0)
977        test(prevy * y <= 0) # tolerates initial prevx having no sign
978        test(abs(y) > 0.9)
979        prevx, prevy = x, y
980    end
981
982    # Test actual minimax approximation.
983    println("Minimax test 1 (polynomial):")
984    (nc, dc, e, x) = ratfn_minimax(x->exp(x), (BigFloat(0), BigFloat(1)), 4, 0)
985    println("  ",e)
986    println("  ",ratfn_to_string(nc, dc))
987    test(0 < e < 1e-3)
988    for x = 0:BigFloat(1)/1000:1
989        test(abs(ratfn_eval(nc, dc, x) - exp(x)) <= e * 1.0000001)
990    end
991
992    println("Minimax test 2 (rational):")
993    (nc, dc, e, x) = ratfn_minimax(x->exp(x), (BigFloat(0), BigFloat(1)), 2, 2)
994    println("  ",e)
995    println("  ",ratfn_to_string(nc, dc))
996    test(0 < e < 1e-3)
997    for x = 0:BigFloat(1)/1000:1
998        test(abs(ratfn_eval(nc, dc, x) - exp(x)) <= e * 1.0000001)
999    end
1000
1001    println("Minimax test 3 (polynomial, weighted):")
1002    (nc, dc, e, x) = ratfn_minimax(x->exp(x), (BigFloat(0), BigFloat(1)), 4, 0,
1003                                   (x,y)->1/y)
1004    println("  ",e)
1005    println("  ",ratfn_to_string(nc, dc))
1006    test(0 < e < 1e-3)
1007    for x = 0:BigFloat(1)/1000:1
1008        test(abs(ratfn_eval(nc, dc, x) - exp(x))/exp(x) <= e * 1.0000001)
1009    end
1010
1011    println("Minimax test 4 (rational, weighted):")
1012    (nc, dc, e, x) = ratfn_minimax(x->exp(x), (BigFloat(0), BigFloat(1)), 2, 2,
1013                                   (x,y)->1/y)
1014    println("  ",e)
1015    println("  ",ratfn_to_string(nc, dc))
1016    test(0 < e < 1e-3)
1017    for x = 0:BigFloat(1)/1000:1
1018        test(abs(ratfn_eval(nc, dc, x) - exp(x))/exp(x) <= e * 1.0000001)
1019    end
1020
1021    println("Minimax test 5 (rational, weighted, odd degree):")
1022    (nc, dc, e, x) = ratfn_minimax(x->exp(x), (BigFloat(0), BigFloat(1)), 2, 1,
1023                                   (x,y)->1/y)
1024    println("  ",e)
1025    println("  ",ratfn_to_string(nc, dc))
1026    test(0 < e < 1e-3)
1027    for x = 0:BigFloat(1)/1000:1
1028        test(abs(ratfn_eval(nc, dc, x) - exp(x))/exp(x) <= e * 1.0000001)
1029    end
1030
1031    total = passes + fails
1032    println(passes, " passes ", fails, " fails ", total, " total")
1033end
1034
1035# ----------------------------------------------------------------------
1036# Online help.
1037function help()
1038    print("""
1039Usage:
1040
1041    remez.jl [options] <lo> <hi> <n> <d> <expr> [<weight>]
1042
1043Arguments:
1044
1045    <lo>, <hi>
1046
1047        Bounds of the interval on which to approximate the target
1048        function. These are parsed and evaluated as Julia expressions,
1049        so you can write things like '1/BigFloat(6)' to get an
1050        accurate representation of 1/6, or '4*atan(BigFloat(1))' to
1051        get pi. (Unfortunately, the obvious 'BigFloat(pi)' doesn't
1052        work in Julia.)
1053
1054    <n>, <d>
1055
1056        The desired degree of polynomial(s) you want for your
1057        approximation. These should be non-negative integers. If you
1058        want a rational function as output, set <n> to the degree of
1059        the numerator, and <d> the denominator. If you just want an
1060        ordinary polynomial, set <d> to 0, and <n> to the degree of
1061        the polynomial you want.
1062
1063    <expr>
1064
1065        A Julia expression giving the function to be approximated on
1066        the interval. The input value is predefined as 'x' when this
1067        expression is evaluated, so you should write something along
1068        the lines of 'sin(x)' or 'sqrt(1+tan(x)^2)' etc.
1069
1070    <weight>
1071
1072        If provided, a Julia expression giving the weighting factor
1073        for the approximation error. The output polynomial will
1074        minimise the largest absolute value of (P-f) * w at any point
1075        in the interval, where P is the value of the polynomial, f is
1076        the value of the target function given by <expr>, and w is the
1077        weight given by this function.
1078
1079        When this expression is evaluated, the input value to P and f
1080        is predefined as 'x', and also the true output value f(x) is
1081        predefined as 'y'. So you can minimise the relative error by
1082        simply writing '1/y'.
1083
1084        If the <weight> argument is not provided, the default
1085        weighting function always returns 1, so that the polynomial
1086        will minimise the maximum absolute error |P-f|.
1087
1088Computation options:
1089
1090    --pre=<predef_expr>
1091
1092        Evaluate the Julia expression <predef_expr> before starting
1093        the computation. This permits you to pre-define variables or
1094        functions which the Julia expressions in your main arguments
1095        can refer to. All of <lo>, <hi>, <expr> and <weight> can make
1096        use of things defined by <predef_expr>.
1097
1098        One internal remez.jl function that you might sometimes find
1099        useful in this expression is 'goldensection', which finds the
1100        location and value of a maximum of a function. For example,
1101        one implementation strategy for the gamma function involves
1102        translating it to put its unique local minimum at the origin,
1103        in which case you can write something like this
1104
1105            --pre='(m,my) = goldensection(x -> -gamma(x),
1106                  BigFloat(1), BigFloat(1.5), BigFloat(2))'
1107
1108        to predefine 'm' as the location of gamma's minimum, and 'my'
1109        as the (negated) value that gamma actually takes at that
1110        point, i.e. -gamma(m).
1111
1112        (Since 'goldensection' always finds a maximum, we had to
1113        negate gamma in the input function to make it find a minimum
1114        instead. Consult the comments in the source for more details
1115        on the use of this function.)
1116
1117        If you use this option more than once, all the expressions you
1118        provide will be run in sequence.
1119
1120    --bits=<bits>
1121
1122        Specify the accuracy to which you want the output polynomial,
1123        in bits. Default 256, which should be more than enough.
1124
1125    --bigfloatbits=<bits>
1126
1127        Turn up the precision used by Julia for its BigFloat
1128        evaluation. Default is Julia's default (also 256). You might
1129        want to try setting this higher than the --bits value if the
1130        algorithm is failing to converge for some reason.
1131
1132Output options:
1133
1134    --full
1135
1136        Instead of just printing the approximation function itself,
1137        also print auxiliary information:
1138         - the locations of the error extrema, and the actual
1139           (weighted) error at each of those locations
1140         - the overall maximum error of the function
1141         - a 'well-conditioning quotient', giving the worst-case ratio
1142           between any polynomial coefficient and the largest possible
1143           value of the higher-order terms it will be added to.
1144
1145        The well-conditioning quotient should be less than 1, ideally
1146        by several factors of two, for accurate evaluation in the
1147        target precision. If you request a rational function, a
1148        separate well-conditioning quotient will be printed for the
1149        numerator and denominator.
1150
1151        Use this option when deciding how wide an interval to
1152        approximate your function on, and what degree of polynomial
1153        you need.
1154
1155    --variable=<identifier>
1156
1157        When writing the output polynomial or rational function in its
1158        usual form as an arithmetic expression, use <identifier> as
1159        the name of the input variable. Default is 'x'.
1160
1161    --suffix=<suffix>
1162
1163        When writing the output polynomial or rational function in its
1164        usual form as an arithmetic expression, write <suffix> after
1165        every floating-point literal. For example, '--suffix=F' will
1166        generate a C expression in which the coefficients are literals
1167        of type 'float' rather than 'double'.
1168
1169    --array
1170
1171        Instead of writing the output polynomial as an arithmetic
1172        expression in Horner's rule form, write out just its
1173        coefficients, one per line, each with a trailing comma.
1174        Suitable for pasting into a C array declaration.
1175
1176        This option is not currently supported if the output is a
1177        rational function, because you'd need two separate arrays for
1178        the numerator and denominator coefficients and there's no
1179        obviously right way to provide both of those together.
1180
1181Debug and test options:
1182
1183    --debug=<facility>
1184
1185        Enable debugging output from various parts of the Remez
1186        calculation. <facility> should be the name of one of the
1187        classes of diagnostic output implemented in the program.
1188        Useful values include 'gausselim', 'leastsquares',
1189        'goldensection', 'equaldev', 'minimax'. This is probably
1190        mostly useful to people debugging problems with the script, so
1191        consult the source code for more information about what the
1192        diagnostic output for each of those facilities will be.
1193
1194        If you want diagnostics from more than one facility, specify
1195        this option multiple times with different arguments.
1196
1197    --test
1198
1199        Run remez.jl's internal test suite. No arguments needed.
1200
1201Miscellaneous options:
1202
1203    --help
1204
1205        Display this text and exit. No arguments needed.
1206
1207""")
1208end
1209
1210# ----------------------------------------------------------------------
1211# Main program.
1212
1213function main()
1214    nargs = length(argwords)
1215    if nargs != 5 && nargs != 6
1216        error("usage: remez.jl <lo> <hi> <n> <d> <expr> [<weight>]\n" *
1217              "       run 'remez.jl --help' for more help")
1218    end
1219
1220    for preliminary_command in preliminary_commands
1221        eval(Meta.parse(preliminary_command))
1222    end
1223
1224    lo = BigFloat(eval(Meta.parse(argwords[1])))
1225    hi = BigFloat(eval(Meta.parse(argwords[2])))
1226    n = parse(Int,argwords[3])
1227    d = parse(Int,argwords[4])
1228    f = eval(Meta.parse("x -> " * argwords[5]))
1229
1230    # Wrap the user-provided function with a function of our own. This
1231    # arranges to detect silly FP values (inf,nan) early and diagnose
1232    # them sensibly, and also lets us log all evaluations of the
1233    # function in case you suspect it's doing the wrong thing at some
1234    # special-case point.
1235    function func(x)
1236        y = run(f,x)
1237        @debug("f", x, " -> ", y)
1238        if !isfinite(y)
1239            error("f(" * string(x) * ") returned non-finite value " * string(y))
1240        end
1241        return y
1242    end
1243
1244    if nargs == 6
1245        # Wrap the user-provided weight function similarly.
1246        w = eval(Meta.parse("(x,y) -> " * argwords[6]))
1247        function wrapped_weight(x,y)
1248            ww = run(w,x,y)
1249            if !isfinite(ww)
1250                error("w(" * string(x) * "," * string(y) *
1251                      ") returned non-finite value " * string(ww))
1252            end
1253            return ww
1254        end
1255        weight = wrapped_weight
1256    else
1257        weight = (x,y)->BigFloat(1)
1258    end
1259
1260    (nc, dc, e, extrema) = ratfn_minimax(func, (lo, hi), n, d, weight)
1261    if array_format
1262        if d == 0
1263            functext = join([string(x)*",\n" for x=nc],"")
1264        else
1265            # It's unclear how you should best format an array of
1266            # coefficients for a rational function, so I'll leave
1267            # implementing this option until I have a use case.
1268            error("--array unsupported for rational functions")
1269        end
1270    else
1271        functext = ratfn_to_string(nc, dc) * "\n"
1272    end
1273    if full_output
1274        # Print everything you might want to know about the function
1275        println("extrema = ", format_xylist(extrema))
1276        println("maxerror = ", string(e))
1277        if length(dc) > 1
1278            println("wellconditioning_numerator = ",
1279                    string(wellcond(nc, lo, hi)))
1280            println("wellconditioning_denominator = ",
1281                    string(wellcond(dc, lo, hi)))
1282        else
1283            println("wellconditioning = ", string(wellcond(nc, lo, hi)))
1284        end
1285        print("function = ", functext)
1286    else
1287        # Just print the text people will want to paste into their code
1288        print(functext)
1289    end
1290end
1291
1292# ----------------------------------------------------------------------
1293# Top-level code: parse the argument list and decide what to do.
1294
1295what_to_do = main
1296
1297doing_opts = true
1298argwords = array1d(String, 0)
1299for arg = ARGS
1300    global doing_opts, what_to_do, argwords
1301    global full_output, array_format, xvarname, floatsuffix, epsbits
1302    if doing_opts && startswith(arg, "-")
1303        if arg == "--"
1304            doing_opts = false
1305        elseif arg == "--help"
1306            what_to_do = help
1307        elseif arg == "--test"
1308            what_to_do = test
1309        elseif arg == "--full"
1310            full_output = true
1311        elseif arg == "--array"
1312            array_format = true
1313        elseif startswith(arg, "--debug=")
1314            enable_debug(arg[length("--debug=")+1:end])
1315        elseif startswith(arg, "--variable=")
1316            xvarname = arg[length("--variable=")+1:end]
1317        elseif startswith(arg, "--suffix=")
1318            floatsuffix = arg[length("--suffix=")+1:end]
1319        elseif startswith(arg, "--bits=")
1320            epsbits = parse(Int,arg[length("--bits=")+1:end])
1321        elseif startswith(arg, "--bigfloatbits=")
1322            set_bigfloat_precision(
1323                parse(Int,arg[length("--bigfloatbits=")+1:end]))
1324        elseif startswith(arg, "--pre=")
1325            push!(preliminary_commands, arg[length("--pre=")+1:end])
1326        else
1327            error("unrecognised option: ", arg)
1328        end
1329    else
1330        push!(argwords, arg)
1331    end
1332end
1333
1334what_to_do()
1335