/* Copyright (C) 1989, 2000 Aladdin Enterprises.  All rights reserved.
  
  This software is provided AS-IS with no warranty, either express or
  implied.
  
  This software is distributed under license and may not be copied,
  modified or distributed except as expressly authorized under the terms
  of the license contained in the file LICENSE in this distribution.
  
  For more information about licensing, please refer to
  http://www.ghostscript.com/licensing/. For information on
  commercial licensing, go to http://www.artifex.com/licensing/ or
  contact Artifex Software, Inc., 101 Lucas Valley Road #110,
  San Rafael, CA  94903, U.S.A., +1(415)492-9861.
*/

/* $Id: gsmisc.c,v 1.22 2004/12/08 21:35:13 stefan Exp $ */
/* Miscellaneous utilities for Ghostscript library */

/*
 * In order to capture the original definition of sqrt, which might be
 * either a procedure or a macro and might not have an ANSI-compliant
 * prototype (!), we need to do the following:
 */
#include "std.h"
#if defined(VMS) && defined(__GNUC__)
/*  DEC VAX/VMS C comes with a math.h file, but GNU VAX/VMS C does not. */
#  include "vmsmath.h"
#else
#  include <math.h>
#endif
inline private double
orig_sqrt(double x)
{
    return sqrt(x);
}

/* Here is the real #include section. */
#include "ctype_.h"
#include "malloc_.h"
#include "math_.h"
#include "memory_.h"
#include "string_.h"
#include "gx.h"
#include "gpcheck.h"		/* for gs_return_check_interrupt */
#include "gserror.h"		/* for prototype */
#include "gserrors.h"
#include "gconfigv.h"		/* for USE_ASM */
#include "gxfarith.h"
#include "gxfixed.h"

/* ------ Redirected stdout and stderr  ------ */

#include <stdarg.h>
#define PRINTF_BUF_LENGTH 1024

int outprintf(const gs_memory_t *mem, const char *fmt, ...)
{
    int count;
    char buf[PRINTF_BUF_LENGTH];
    va_list args;

    va_start(args, fmt);

    count = vsprintf(buf, fmt, args);
    outwrite(mem, buf, count);
    if (count >= PRINTF_BUF_LENGTH) {
	count = sprintf(buf, 
	    "PANIC: printf exceeded %d bytes.  Stack has been corrupted.\n", 
	    PRINTF_BUF_LENGTH);
	outwrite(mem, buf, count);
    }
    va_end(args);
    return count;
}

int errprintf(const char *fmt, ...)
{
    int count;
    char buf[PRINTF_BUF_LENGTH];
    va_list args;

    va_start(args, fmt);

    count = vsprintf(buf, fmt, args);
    errwrite(buf, count);
    if (count >= PRINTF_BUF_LENGTH) {
	count = sprintf(buf, 
	    "PANIC: printf exceeded %d bytes.  Stack has been corrupted.\n", 
	    PRINTF_BUF_LENGTH);
	errwrite(buf, count);
    }
    va_end(args);
    return count;
}

/* ------ Debugging ------ */

/* Ghostscript writes debugging output to gs_debug_out. */
/* We define gs_debug and gs_debug_out even if DEBUG isn't defined, */
/* so that we can compile individual modules with DEBUG set. */
char gs_debug[128];
FILE *gs_debug_out;

/* Test whether a given debugging option is selected. */
/* Upper-case letters automatically include their lower-case counterpart. */
bool
gs_debug_c(int c)
{
    return
	(c >= 'a' && c <= 'z' ? gs_debug[c] | gs_debug[c ^ 32] : gs_debug[c]);
}

/* Define the formats for debugging printout. */
const char *const dprintf_file_and_line_format = "%10s(%4d): ";
const char *const dprintf_file_only_format = "%10s(unkn): ";

/*
 * Define the trace printout procedures.  We always include these, in case
 * other modules were compiled with DEBUG set.  Note that they must use
 * out/errprintf, not fprintf nor fput[cs], because of the way that 
 * stdout/stderr are implemented on DLL/shared library builds.
 */
void
dflush(void)
{
    errflush();
}
private const char *
dprintf_file_tail(const char *file)
{
    const char *tail = file + strlen(file);

    while (tail > file &&
	   (isalnum(tail[-1]) || tail[-1] == '.' || tail[-1] == '_')
	)
	--tail;
    return tail;
}
#if __LINE__			/* compiler provides it */
void
dprintf_file_and_line(const char *file, int line)
{
    if (gs_debug['/'])
	dpf(dprintf_file_and_line_format,
		dprintf_file_tail(file), line);
}
#else
void
dprintf_file_only(const char *file)
{
    if (gs_debug['/'])
	dpf(dprintf_file_only_format, dprintf_file_tail(file));
}
#endif
void
printf_program_ident(const gs_memory_t *mem, const char *program_name, long revision_number)
{
    if (program_name)
        outprintf(mem, (revision_number ? "%s " : "%s"), program_name);
    if (revision_number) {
	int fpart = revision_number % 100;

	outprintf(mem, "%d.%02d", (int)(revision_number / 100), fpart);
    }
}
void
eprintf_program_ident(const char *program_name,
		      long revision_number)
{
    if (program_name) {
	epf((revision_number ? "%s " : "%s"), program_name);
	if (revision_number) {
	    int fpart = revision_number % 100;

	    epf("%d.%02d", (int)(revision_number / 100), fpart);
	}
	epf(": ");
    }
}
#if __LINE__			/* compiler provides it */
void
lprintf_file_and_line(const char *file, int line)
{
    epf("%s(%d): ", file, line);
}
#else
void
lprintf_file_only(FILE * f, const char *file)
{
    epf("%s(?): ", file);
}
#endif

/* Log an error return.  We always include this, in case other */
/* modules were compiled with DEBUG set. */
#undef gs_log_error		/* in case DEBUG isn't set */
int
gs_log_error(int err, const char *file, int line)
{
    if (gs_log_errors) {
	if (file == NULL)
	    dprintf1("Returning error %d.\n", err);
	else
	    dprintf3("%s(%d): Returning error %d.\n",
		     (const char *)file, line, err);
    }
    return err;
}

/* Check for interrupts before a return. */
int
gs_return_check_interrupt(const gs_memory_t *mem, int code)
{
    if (code < 0)
	return code;
    {
	int icode = gp_check_interrupts(mem);

	return (icode == 0 ? code :
		gs_note_error((icode > 0 ? gs_error_interrupt : icode)));
    }
}

/* ------ Substitutes for missing C library functions ------ */

#ifdef MEMORY__NEED_MEMMOVE	/* see memory_.h */
/* Copy bytes like memcpy, guaranteed to handle overlap correctly. */
/* ANSI C defines the returned value as being the src argument, */
/* but with the const restriction removed! */
void *
gs_memmove(void *dest, const void *src, size_t len)
{
    if (!len)
	return (void *)src;
#define bdest ((byte *)dest)
#define bsrc ((const byte *)src)
    /* We use len-1 for comparisons because adding len */
    /* might produce an offset overflow on segmented systems. */
    if (PTR_LE(bdest, bsrc)) {
	register byte *end = bdest + (len - 1);

	if (PTR_LE(bsrc, end)) {
	    /* Source overlaps destination from above. */
	    register const byte *from = bsrc;
	    register byte *to = bdest;

	    for (;;) {
		*to = *from;
		if (to >= end)	/* faster than = */
		    return (void *)src;
		to++;
		from++;
	    }
	}
    } else {
	register const byte *from = bsrc + (len - 1);

	if (PTR_LE(bdest, from)) {
	    /* Source overlaps destination from below. */
	    register const byte *end = bsrc;
	    register byte *to = bdest + (len - 1);

	    for (;;) {
		*to = *from;
		if (from <= end)	/* faster than = */
		    return (void *)src;
		to--;
		from--;
	    }
	}
    }
#undef bdest
#undef bsrc
    /* No overlap, it's safe to use memcpy. */
    memcpy(dest, src, len);
    return (void *)src;
}
#endif

#ifdef MEMORY__NEED_MEMCPY	/* see memory_.h */
void *
gs_memcpy(void *dest, const void *src, size_t len)
{
    if (len > 0) {
#define bdest ((byte *)dest)
#define bsrc ((const byte *)src)
	/* We can optimize this much better later on. */
	register byte *end = bdest + (len - 1);
	register const byte *from = bsrc;
	register byte *to = bdest;

	for (;;) {
	    *to = *from;
	    if (to >= end)	/* faster than = */
		break;
	    to++;
	    from++;
	}
    }
#undef bdest
#undef bsrc
    return (void *)src;
}
#endif

#ifdef MEMORY__NEED_MEMCHR	/* see memory_.h */
/* ch should obviously be char rather than int, */
/* but the ANSI standard declaration uses int. */
void *
gs_memchr(const void *ptr, int ch, size_t len)
{
    if (len > 0) {
	register const char *p = ptr;
	register uint count = len;

	do {
	    if (*p == (char)ch)
		return (void *)p;
	    p++;
	} while (--count);
    }
    return 0;
}
#endif

#ifdef MEMORY__NEED_MEMSET	/* see memory_.h */
/* ch should obviously be char rather than int, */
/* but the ANSI standard declaration uses int. */
void *
gs_memset(void *dest, register int ch, size_t len)
{
    /*
     * This procedure is used a lot to fill large regions of images,
     * so we take some trouble to optimize it.
     */
    register char *p = dest;
    register size_t count = len;

    ch &= 255;
    if (len >= sizeof(long) * 3) {
	long wd = (ch << 24) | (ch << 16) | (ch << 8) | ch;

	while (ALIGNMENT_MOD(p, sizeof(long)))
	    *p++ = (char)ch, --count;
	for (; count >= sizeof(long) * 4;
	     p += sizeof(long) * 4, count -= sizeof(long) * 4
	     )
	    ((long *)p)[3] = ((long *)p)[2] = ((long *)p)[1] =
		((long *)p)[0] = wd;
	switch (count >> ARCH_LOG2_SIZEOF_LONG) {
	case 3:
	    *((long *)p) = wd; p += sizeof(long);
	case 2:
	    *((long *)p) = wd; p += sizeof(long);
	case 1:
	    *((long *)p) = wd; p += sizeof(long);
	    count &= sizeof(long) - 1;
	case 0:
	default:		/* can't happen */
	    DO_NOTHING;
	}
    }
    /* Do any leftover bytes. */
    for (; count > 0; --count)
	*p++ = (char)ch;
    return dest;
}
#endif

#ifdef malloc__need_realloc	/* see malloc_.h */
/* Some systems have non-working implementations of realloc. */
void *
gs_realloc(void *old_ptr, size_t old_size, size_t new_size)
{
    void *new_ptr;

    if (new_size) {
	new_ptr = malloc(new_size);
	if (new_ptr == NULL)
	    return NULL;
    } else
	new_ptr = NULL;
    /* We have to pass in the old size, since we have no way to */
    /* determine it otherwise. */
    if (old_ptr != NULL) {
	if (new_ptr != NULL)
	    memcpy(new_ptr, old_ptr, min(old_size, new_size));
	free(old_ptr);
    }
    return new_ptr;
}
#endif

/* ------ Debugging support ------ */

/* Dump a region of memory. */
void
debug_dump_bytes(const byte * from, const byte * to, const char *msg)
{
    const byte *p = from;

    if (from < to && msg)
	dprintf1("%s:\n", msg);
    while (p != to) {
	const byte *q = min(p + 16, to);

	dprintf1("0x%lx:", (ulong) p);
	while (p != q)
	    dprintf1(" %02x", *p++);
	dputc('\n');
    }
}

/* Dump a bitmap. */
void
debug_dump_bitmap(const byte * bits, uint raster, uint height, const char *msg)
{
    uint y;
    const byte *data = bits;

    for (y = 0; y < height; ++y, data += raster)
	debug_dump_bytes(data, data + raster, (y == 0 ? msg : NULL));
}

/* Print a string. */
void
debug_print_string(const byte * chrs, uint len)
{
    uint i;

    for (i = 0; i < len; i++)
	dputc(chrs[i]);
    dflush();
}

/* Print a string in hexdump format. */
void
debug_print_string_hex(const byte * chrs, uint len)
{
    uint i;

    for (i = 0; i < len; i++)
        dprintf1("%02x", chrs[i]);
    dflush();
}

/*
 * The following code prints a hex stack backtrace on Linux/Intel systems.
 * It is here to be patched into places where we need to print such a trace
 * because of gdb's inability to put breakpoints in dynamically created
 * threads.
 *
 * first_arg is the first argument of the procedure into which this code
 * is patched.
 */
#define BACKTRACE(first_arg)\
  BEGIN\
    ulong *fp_ = (ulong *)&first_arg - 2;\
    for (; fp_ && (fp_[1] & 0xff000000) == 0x08000000; fp_ = (ulong *)*fp_)\
	dprintf2("  fp=0x%lx ip=0x%lx\n", (ulong)fp_, fp_[1]);\
  END

/* ------ Arithmetic ------ */

/* Compute M modulo N.  Requires N > 0; guarantees 0 <= imod(M,N) < N, */
/* regardless of the whims of the % operator for negative operands. */
int
imod(int m, int n)
{
    if (n <= 0)
	return 0;		/* sanity check */
    if (m >= 0)
	return m % n;
    {
	int r = -m % n;

	return (r == 0 ? 0 : n - r);
    }
}

/* Compute the GCD of two integers. */
int
igcd(int x, int y)
{
    int c = x, d = y;

    if (c < 0)
	c = -c;
    if (d < 0)
	d = -d;
    while (c != 0 && d != 0)
	if (c > d)
	    c %= d;
	else
	    d %= c;
    return d + c;		/* at most one is non-zero */
}

/* Compute X such that A*X = B mod M.  See gxarith.h for details. */
int
idivmod(int a, int b, int m)
{
    /*
     * Use the approach indicated in Knuth vol. 2, section 4.5.2, Algorithm
     * X (p. 302) and exercise 15 (p. 315, solution p. 523).
     */
    int u1 = 0, u3 = m;
    int v1 = 1, v3 = a;
    /*
     * The following loop will terminate with a * u1 = gcd(a, m) mod m.
     * Then x = u1 * b / gcd(a, m) mod m.  Since we require that
     * gcd(a, m) | gcd(a, b), it follows that gcd(a, m) | b, so the
     * division is exact.
     */
    while (v3) {
	int q = u3 / v3, t;

	t = u1 - v1 * q, u1 = v1, v1 = t;
	t = u3 - v3 * q, u3 = v3, v3 = t;
    }
    return imod(u1 * b / igcd(a, m), m);
}

/* Compute floor(log2(N)).  Requires N > 0. */
int
ilog2(int n)
{
    int m = n, l = 0;

    while (m >= 16)
	m >>= 4, l += 4;
    return
	(m <= 1 ? l :
	 "\000\000\001\001\002\002\002\002\003\003\003\003\003\003\003\003"[m] + l);
}

#if defined(NEED_SET_FMUL2FIXED) && !USE_ASM

/*
 * Floating multiply with fixed result, for avoiding floating point in
 * common coordinate transformations.  Assumes IEEE representation,
 * 16-bit short, 32-bit long.  Optimized for the case where the first
 * operand has no more than 16 mantissa bits, e.g., where it is a user space
 * coordinate (which are often integers).
 *
 * The assembly language version of this code is actually faster than
 * the FPU, if the code is compiled with FPU_TYPE=0 (which requires taking
 * a trap on every FPU operation).  If there is no FPU, the assembly
 * language version of this code is over 10 times as fast as the emulated FPU.
 */
int
set_fmul2fixed_(fixed * pr, long /*float */ a, long /*float */ b)
{
    ulong ma = (ushort)(a >> 8) | 0x8000;
    ulong mb = (ushort)(b >> 8) | 0x8000;
    int e = 260 + _fixed_shift - ( ((byte)(a >> 23)) + ((byte)(b >> 23)) );
    ulong p1 = ma * (b & 0xff);
    ulong p = ma * mb;

#define p_bits (size_of(p) * 8)

    if ((byte) a) {		/* >16 mantissa bits */
	ulong p2 = (a & 0xff) * mb;

	p += ((((uint) (byte) a * (uint) (byte) b) >> 8) + p1 + p2) >> 8;
    } else
	p += p1 >> 8;
    if ((uint) e < p_bits)	/* e = -1 is possible */
	p >>= e;
    else if (e >= p_bits) {	/* also detects a=0 or b=0 */
	*pr = fixed_0;
	return 0;
    } else if (e >= -(p_bits - 1) || p >= 1L << (p_bits - 1 + e))
	return_error(gs_error_limitcheck);
    else
	p <<= -e;
    *pr = ((a ^ b) < 0 ? -p : p);
    return 0;
}
int
set_dfmul2fixed_(fixed * pr, ulong /*double lo */ xalo, long /*float */ b, long /*double hi */ xahi)
{
    return set_fmul2fixed_(pr,
			   (xahi & (3L << 30)) +
			   ((xahi << 3) & 0x3ffffff8) +
			   (xalo >> 29),
			   b);
}

#endif /* NEED_SET_FMUL2FIXED */

#if USE_FPU_FIXED

/*
 * Convert from floating point to fixed point with scaling.
 * These are efficient algorithms for FPU-less machines.
 */
#define mbits_float 23
#define mbits_double 20
int
set_float2fixed_(fixed * pr, long /*float */ vf, int frac_bits)
{
    fixed mantissa;
    int shift;

    if (!(vf & 0x7f800000)) {
	*pr = fixed_0;
	return 0;
    }
    mantissa = (fixed) ((vf & 0x7fffff) | 0x800000);
    shift = ((vf >> 23) & 255) - (127 + 23) + frac_bits;
    if (shift >= 0) {
	if (shift >= sizeof(fixed) * 8 - 24)
	    return_error(gs_error_limitcheck);
	if (vf < 0)
	    mantissa = -mantissa;
	*pr = (fixed) (mantissa << shift);
    } else
	*pr = (shift < -24 ? fixed_0 :
	       vf < 0 ? -(fixed) (mantissa >> -shift) :		/* truncate */
	       (fixed) (mantissa >> -shift));
    return 0;
}
int
set_double2fixed_(fixed * pr, ulong /*double lo */ lo,
		  long /*double hi */ hi, int frac_bits)
{
    fixed mantissa;
    int shift;

    if (!(hi & 0x7ff00000)) {
	*pr = fixed_0;
	return 0;
    }
    /* We only use 31 bits of mantissa even if sizeof(long) > 4. */
    mantissa = (fixed) (((hi & 0xfffff) << 10) | (lo >> 22) | 0x40000000);
    shift = ((hi >> 20) & 2047) - (1023 + 30) + frac_bits;
    if (shift > 0)
	return_error(gs_error_limitcheck);
    *pr = (shift < -30 ? fixed_0 :
	   hi < 0 ? -(fixed) (mantissa >> -shift) :	/* truncate */
	   (fixed) (mantissa >> -shift));
    return 0;
}
/*
 * Given a fixed value x with fbits bits of fraction, set v to the mantissa
 * (left-justified in 32 bits) and f to the exponent word of the
 * corresponding floating-point value with mbits bits of mantissa in the
 * first word.  (The exponent part of f is biased by -1, because we add the
 * top 1-bit of the mantissa to it.)
 */
static const byte f2f_shifts[] =
{4, 3, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0};

#define f2f_declare(v, f)\
	ulong v;\
	long f
#define f2f(x, v, f, mbits, fbits)\
	if ( x < 0 )\
	  f = 0xc0000000 + (29 << mbits) - ((long)fbits << mbits), v = -x;\
	else\
	  f = 0x40000000 + (29 << mbits) - ((long)fbits << mbits), v = x;\
	if ( v < 0x8000 )\
	  v <<= 15, f -= 15 << mbits;\
	if ( v < 0x800000 )\
	  v <<= 8, f -= 8 << mbits;\
	if ( v < 0x8000000 )\
	  v <<= 4, f -= 4 << mbits;\
	{ int shift = f2f_shifts[v >> 28];\
	  v <<= shift, f -= shift << mbits;\
	}
long
fixed2float_(fixed x, int frac_bits)
{
    f2f_declare(v, f);

    if (x == 0)
	return 0;
    f2f(x, v, f, mbits_float, frac_bits);
    return f + (((v >> 7) + 1) >> 1);
}
void
set_fixed2double_(double *pd, fixed x, int frac_bits)
{
    f2f_declare(v, f);

    if (x == 0) {
	((long *)pd)[1 - arch_is_big_endian] = 0;
	((ulong *) pd)[arch_is_big_endian] = 0;
    } else {
	f2f(x, v, f, mbits_double, frac_bits);
	((long *)pd)[1 - arch_is_big_endian] = f + (v >> 11);
	((ulong *) pd)[arch_is_big_endian] = v << 21;
    }
}

#endif				/* USE_FPU_FIXED */

/*
 * Compute A * B / C when 0 <= B < C and A * B exceeds (or might exceed)
 * the capacity of a long.
 * Note that this procedure takes the floor, rather than truncating
 * towards zero, if A < 0.  This ensures that 0 <= R < C.
 */

#define num_bits (sizeof(fixed) * 8)
#define half_bits (num_bits / 2)
#define half_mask ((1L << half_bits) - 1)

/*
 * If doubles aren't wide enough, we lose too much precision by using double
 * arithmetic: we have to use the slower, accurate fixed-point algorithm.
 * See the simpler implementation below for more information.
 */
#define MAX_OTHER_FACTOR_BITS\
  (ARCH_DOUBLE_MANTISSA_BITS - ARCH_SIZEOF_FIXED * 8)
#define ROUND_BITS\
  (ARCH_SIZEOF_FIXED * 8 * 2 - ARCH_DOUBLE_MANTISSA_BITS)

#if USE_FPU_FIXED || ROUND_BITS >= MAX_OTHER_FACTOR_BITS - 1

#ifdef DEBUG
struct {
    long mnanb, mnab, manb, mab, mnc, mdq, mde, mds, mqh, mql;
} fmq_stat;
#  define mincr(x) ++fmq_stat.x
#else
#  define mincr(x) DO_NOTHING
#endif
fixed
fixed_mult_quo(fixed signed_A, fixed B, fixed C)
{
    /* First compute A * B in double-fixed precision. */
    ulong A = (signed_A < 0 ? -signed_A : signed_A);
    long msw;
    ulong lsw;
    ulong p1;

    if (B <= half_mask) {
	if (A <= half_mask) {
	    ulong P = A * B;
	    fixed Q = P / (ulong)C;

	    mincr(mnanb);
	    /* If A < 0 and the division isn't exact, take the floor. */
	    return (signed_A >= 0 ? Q : Q * C == P ? -Q : ~Q /* -Q - 1 */);
	}
	/*
	 * We might still have C <= half_mask, which we can
	 * handle with a simpler algorithm.
	 */
	lsw = (A & half_mask) * B;
	p1 = (A >> half_bits) * B;
	if (C <= half_mask) {
	    fixed q0 = (p1 += lsw >> half_bits) / C;
	    ulong rem = ((p1 - C * q0) << half_bits) + (lsw & half_mask);
	    ulong q1 = rem / (ulong)C;
	    fixed Q = (q0 << half_bits) + q1;

	    mincr(mnc);
	    /* If A < 0 and the division isn't exact, take the floor. */
	    return (signed_A >= 0 ? Q : q1 * C == rem ? -Q : ~Q);
	}
	msw = p1 >> half_bits;
	mincr(manb);
    } else if (A <= half_mask) {
	p1 = A * (B >> half_bits);
	msw = p1 >> half_bits;
	lsw = A * (B & half_mask);
	mincr(mnab);
    } else {			/* We have to compute all 4 products.  :-( */
	ulong lo_A = A & half_mask;
	ulong hi_A = A >> half_bits;
	ulong lo_B = B & half_mask;
	ulong hi_B = B >> half_bits;
	ulong p1x = hi_A * lo_B;

	msw = hi_A * hi_B;
	lsw = lo_A * lo_B;
	p1 = lo_A * hi_B;
	if (p1 > max_ulong - p1x)
	    msw += 1L << half_bits;
	p1 += p1x;
	msw += p1 >> half_bits;
	mincr(mab);
    }
    /* Finish up by adding the low half of p1 to the high half of lsw. */
#if max_fixed < max_long
    p1 &= half_mask;
#endif
    p1 <<= half_bits;
    if (p1 > max_ulong - lsw)
	msw++;
    lsw += p1;
    /*
     * Now divide the double-length product by C.  Note that we know msw
     * < C (otherwise the quotient would overflow).  Start by shifting
     * (msw,lsw) and C left until C >= 1 << (num_bits - 1).
     */
    {
	ulong denom = C;
	int shift = 0;

#define bits_4th (num_bits / 4)
	if (denom < 1L << (num_bits - bits_4th)) {
	    mincr(mdq);
	    denom <<= bits_4th, shift += bits_4th;
	}
#undef bits_4th
#define bits_8th (num_bits / 8)
	if (denom < 1L << (num_bits - bits_8th)) {
	    mincr(mde);
	    denom <<= bits_8th, shift += bits_8th;
	}
#undef bits_8th
	while (!(denom & (-1L << (num_bits - 1)))) {
	    mincr(mds);
	    denom <<= 1, ++shift;
	}
	msw = (msw << shift) + (lsw >> (num_bits - shift));
	lsw <<= shift;
#if max_fixed < max_long
	lsw &= (1L << (sizeof(fixed) * 8)) - 1;
#endif
	/* Compute a trial upper-half quotient. */
	{
	    ulong hi_D = denom >> half_bits;
	    ulong lo_D = denom & half_mask;
	    ulong hi_Q = (ulong) msw / hi_D;

	    /* hi_Q might be too high by 1 or 2, but it isn't too low. */
	    ulong p0 = hi_Q * hi_D;
	    ulong p1 = hi_Q * lo_D;
	    ulong hi_P;

	    while ((hi_P = p0 + (p1 >> half_bits)) > msw ||
		   (hi_P == msw && ((p1 & half_mask) << half_bits) > lsw)
		) {		/* hi_Q was too high by 1. */
		--hi_Q;
		p0 -= hi_D;
		p1 -= lo_D;
		mincr(mqh);
	    }
	    p1 = (p1 & half_mask) << half_bits;
	    if (p1 > lsw)
		msw--;
	    lsw -= p1;
	    msw -= hi_P;
	    /* Now repeat to get the lower-half quotient. */
	    msw = (msw << half_bits) + (lsw >> half_bits);
#if max_fixed < max_long
	    lsw &= half_mask;
#endif
	    lsw <<= half_bits;
	    {
		ulong lo_Q = (ulong) msw / hi_D;
		long Q;

		p1 = lo_Q * lo_D;
		p0 = lo_Q * hi_D;
		while ((hi_P = p0 + (p1 >> half_bits)) > msw ||
		       (hi_P == msw && ((p1 & half_mask) << half_bits) > lsw)
		    ) {		/* lo_Q was too high by 1. */
		    --lo_Q;
		    p0 -= hi_D;
		    p1 -= lo_D;
		    mincr(mql);
		}
		Q = (hi_Q << half_bits) + lo_Q;
		return (signed_A >= 0 ? Q : p0 | p1 ? ~Q /* -Q - 1 */ : -Q);
	    }
	}
    }
}

#else				/* use doubles */

/*
 * Compute A * B / C as above using doubles.  If floating point is
 * reasonably fast, this is much faster than the fixed-point algorithm.
 */
fixed
fixed_mult_quo(fixed signed_A, fixed B, fixed C)
{
    /*
     * Check whether A * B will fit in the mantissa of a double.
     */
#define MAX_OTHER_FACTOR (1L << MAX_OTHER_FACTOR_BITS)
    if (B < MAX_OTHER_FACTOR || any_abs(signed_A) < MAX_OTHER_FACTOR) {
#undef MAX_OTHER_FACTOR
	/*
	 * The product fits, so a straightforward double computation
	 * will be exact.
	 */
	return (fixed)floor((double)signed_A * B / C);
    } else {
	/*
	 * The product won't fit.  However, the approximate product will
	 * only be off by at most +/- 1/2 * (1 << ROUND_BITS) because of
	 * rounding.  If we add 1 << ROUND_BITS to the value of the product
	 * (i.e., 1 in the least significant bit of the mantissa), the
	 * result is always greater than the correct product by between 1/2
	 * and 3/2 * (1 << ROUND_BITS).  We know this is less than C:
	 * because of the 'if' just above, we know that B >=
	 * MAX_OTHER_FACTOR; since B <= C, we know C >= MAX_OTHER_FACTOR;
	 * and because of the #if that chose between the two
	 * implementations, we know that C >= 2 * (1 << ROUND_BITS).  Hence,
	 * the quotient after dividing by C will be at most 1 too large.
	 */
	fixed q =
	    (fixed)floor(((double)signed_A * B + (1L << ROUND_BITS)) / C);

	/*
	 * Compute the remainder R.  If the quotient was correct,
	 * 0 <= R < C.  If the quotient was too high, -C <= R < 0.
	 */
	if (signed_A * B - q * C < 0)
	    --q;
	return q;
    }
}

#endif

#undef MAX_OTHER_FACTOR_BITS
#undef ROUND_BITS

#undef num_bits
#undef half_bits
#undef half_mask

/* Trace calls on sqrt when debugging. */
double
gs_sqrt(double x, const char *file, int line)
{
    if (gs_debug_c('~')) {
	dprintf3("[~]sqrt(%g) at %s:%d\n", x, (const char *)file, line);
	dflush();
    }
    return orig_sqrt(x);
}

/*
 * Define sine and cosine functions that take angles in degrees rather than
 * radians, and that are implemented efficiently on machines with slow
 * (or no) floating point.
 */
#if USE_FPU < 0			/****** maybe should be <= 0 ? ***** */

#define sin0 0.00000000000000000
#define sin1 0.01745240643728351
#define sin2 0.03489949670250097
#define sin3 0.05233595624294383
#define sin4 0.06975647374412530
#define sin5 0.08715574274765817
#define sin6 0.10452846326765346
#define sin7 0.12186934340514748
#define sin8 0.13917310096006544
#define sin9 0.15643446504023087
#define sin10 0.17364817766693033
#define sin11 0.19080899537654480
#define sin12 0.20791169081775931
#define sin13 0.22495105434386498
#define sin14 0.24192189559966773
#define sin15 0.25881904510252074
#define sin16 0.27563735581699916
#define sin17 0.29237170472273671
#define sin18 0.30901699437494740
#define sin19 0.32556815445715670
#define sin20 0.34202014332566871
#define sin21 0.35836794954530027
#define sin22 0.37460659341591201
#define sin23 0.39073112848927377
#define sin24 0.40673664307580015
#define sin25 0.42261826174069944
#define sin26 0.43837114678907740
#define sin27 0.45399049973954675
#define sin28 0.46947156278589081
#define sin29 0.48480962024633706
#define sin30 0.50000000000000000
#define sin31 0.51503807491005416
#define sin32 0.52991926423320490
#define sin33 0.54463903501502708
#define sin34 0.55919290347074679
#define sin35 0.57357643635104605
#define sin36 0.58778525229247314
#define sin37 0.60181502315204827
#define sin38 0.61566147532565829
#define sin39 0.62932039104983739
#define sin40 0.64278760968653925
#define sin41 0.65605902899050728
#define sin42 0.66913060635885824
#define sin43 0.68199836006249848
#define sin44 0.69465837045899725
#define sin45 0.70710678118654746
#define sin46 0.71933980033865108
#define sin47 0.73135370161917046
#define sin48 0.74314482547739413
#define sin49 0.75470958022277201
#define sin50 0.76604444311897801
#define sin51 0.77714596145697090
#define sin52 0.78801075360672190
#define sin53 0.79863551004729283
#define sin54 0.80901699437494745
#define sin55 0.81915204428899180
#define sin56 0.82903757255504174
#define sin57 0.83867056794542394
#define sin58 0.84804809615642596
#define sin59 0.85716730070211222
#define sin60 0.86602540378443860
#define sin61 0.87461970713939574
#define sin62 0.88294759285892688
#define sin63 0.89100652418836779
#define sin64 0.89879404629916704
#define sin65 0.90630778703664994
#define sin66 0.91354545764260087
#define sin67 0.92050485345244037
#define sin68 0.92718385456678731
#define sin69 0.93358042649720174
#define sin70 0.93969262078590832
#define sin71 0.94551857559931674
#define sin72 0.95105651629515353
#define sin73 0.95630475596303544
#define sin74 0.96126169593831889
#define sin75 0.96592582628906831
#define sin76 0.97029572627599647
#define sin77 0.97437006478523525
#define sin78 0.97814760073380558
#define sin79 0.98162718344766398
#define sin80 0.98480775301220802
#define sin81 0.98768834059513777
#define sin82 0.99026806874157036
#define sin83 0.99254615164132198
#define sin84 0.99452189536827329
#define sin85 0.99619469809174555
#define sin86 0.99756405025982420
#define sin87 0.99862953475457383
#define sin88 0.99939082701909576
#define sin89 0.99984769515639127
#define sin90 1.00000000000000000

private const double sin_table[361] =
{
    sin0,
    sin1, sin2, sin3, sin4, sin5, sin6, sin7, sin8, sin9, sin10,
    sin11, sin12, sin13, sin14, sin15, sin16, sin17, sin18, sin19, sin20,
    sin21, sin22, sin23, sin24, sin25, sin26, sin27, sin28, sin29, sin30,
    sin31, sin32, sin33, sin34, sin35, sin36, sin37, sin38, sin39, sin40,
    sin41, sin42, sin43, sin44, sin45, sin46, sin47, sin48, sin49, sin50,
    sin51, sin52, sin53, sin54, sin55, sin56, sin57, sin58, sin59, sin60,
    sin61, sin62, sin63, sin64, sin65, sin66, sin67, sin68, sin69, sin70,
    sin71, sin72, sin73, sin74, sin75, sin76, sin77, sin78, sin79, sin80,
    sin81, sin82, sin83, sin84, sin85, sin86, sin87, sin88, sin89, sin90,
    sin89, sin88, sin87, sin86, sin85, sin84, sin83, sin82, sin81, sin80,
    sin79, sin78, sin77, sin76, sin75, sin74, sin73, sin72, sin71, sin70,
    sin69, sin68, sin67, sin66, sin65, sin64, sin63, sin62, sin61, sin60,
    sin59, sin58, sin57, sin56, sin55, sin54, sin53, sin52, sin51, sin50,
    sin49, sin48, sin47, sin46, sin45, sin44, sin43, sin42, sin41, sin40,
    sin39, sin38, sin37, sin36, sin35, sin34, sin33, sin32, sin31, sin30,
    sin29, sin28, sin27, sin26, sin25, sin24, sin23, sin22, sin21, sin20,
    sin19, sin18, sin17, sin16, sin15, sin14, sin13, sin12, sin11, sin10,
    sin9, sin8, sin7, sin6, sin5, sin4, sin3, sin2, sin1, sin0,
    -sin1, -sin2, -sin3, -sin4, -sin5, -sin6, -sin7, -sin8, -sin9, -sin10,
    -sin11, -sin12, -sin13, -sin14, -sin15, -sin16, -sin17, -sin18, -sin19, -sin20,
    -sin21, -sin22, -sin23, -sin24, -sin25, -sin26, -sin27, -sin28, -sin29, -sin30,
    -sin31, -sin32, -sin33, -sin34, -sin35, -sin36, -sin37, -sin38, -sin39, -sin40,
    -sin41, -sin42, -sin43, -sin44, -sin45, -sin46, -sin47, -sin48, -sin49, -sin50,
    -sin51, -sin52, -sin53, -sin54, -sin55, -sin56, -sin57, -sin58, -sin59, -sin60,
    -sin61, -sin62, -sin63, -sin64, -sin65, -sin66, -sin67, -sin68, -sin69, -sin70,
    -sin71, -sin72, -sin73, -sin74, -sin75, -sin76, -sin77, -sin78, -sin79, -sin80,
    -sin81, -sin82, -sin83, -sin84, -sin85, -sin86, -sin87, -sin88, -sin89, -sin90,
    -sin89, -sin88, -sin87, -sin86, -sin85, -sin84, -sin83, -sin82, -sin81, -sin80,
    -sin79, -sin78, -sin77, -sin76, -sin75, -sin74, -sin73, -sin72, -sin71, -sin70,
    -sin69, -sin68, -sin67, -sin66, -sin65, -sin64, -sin63, -sin62, -sin61, -sin60,
    -sin59, -sin58, -sin57, -sin56, -sin55, -sin54, -sin53, -sin52, -sin51, -sin50,
    -sin49, -sin48, -sin47, -sin46, -sin45, -sin44, -sin43, -sin42, -sin41, -sin40,
    -sin39, -sin38, -sin37, -sin36, -sin35, -sin34, -sin33, -sin32, -sin31, -sin30,
    -sin29, -sin28, -sin27, -sin26, -sin25, -sin24, -sin23, -sin22, -sin21, -sin20,
    -sin19, -sin18, -sin17, -sin16, -sin15, -sin14, -sin13, -sin12, -sin11, -sin10,
    -sin9, -sin8, -sin7, -sin6, -sin5, -sin4, -sin3, -sin2, -sin1, -sin0
};

double
gs_sin_degrees(double ang)
{
    int ipart;

    if (is_fneg(ang))
	ang = 180 - ang;
    ipart = (int)ang;
    if (ipart >= 360) {
	int arem = ipart % 360;

	ang -= (ipart - arem);
	ipart = arem;
    }
    return
	(ang == ipart ? sin_table[ipart] :
	 sin_table[ipart] + (sin_table[ipart + 1] - sin_table[ipart]) *
	 (ang - ipart));
}

double
gs_cos_degrees(double ang)
{
    int ipart;

    if (is_fneg(ang))
	ang = 90 - ang;
    else
	ang += 90;
    ipart = (int)ang;
    if (ipart >= 360) {
	int arem = ipart % 360;

	ang -= (ipart - arem);
	ipart = arem;
    }
    return
	(ang == ipart ? sin_table[ipart] :
	 sin_table[ipart] + (sin_table[ipart + 1] - sin_table[ipart]) *
	 (ang - ipart));
}

void
gs_sincos_degrees(double ang, gs_sincos_t * psincos)
{
    psincos->sin = gs_sin_degrees(ang);
    psincos->cos = gs_cos_degrees(ang);
    psincos->orthogonal =
	(is_fzero(psincos->sin) || is_fzero(psincos->cos));
}

#else /* we have floating point */

static const int isincos[5] =
{0, 1, 0, -1, 0};

/* GCC with -ffast-math compiles ang/90. as ang*(1/90.), losing precission.
 * This doesn't happen when the numeral is replaced with a non-const variable.
 * So we define the variable to work around the GCC problem. 
 */
static double const_90_degrees = 90.;

double
gs_sin_degrees(double ang)
{
    double quot = ang / const_90_degrees;

    if (floor(quot) == quot) {
	/*
	 * We need 4.0, rather than 4, here because of non-ANSI compilers.
	 * The & 3 is because quot might be negative.
	 */
	return isincos[(int)fmod(quot, 4.0) & 3];
    }
    return sin(ang * (M_PI / 180));
}

double
gs_cos_degrees(double ang)
{
    double quot = ang / const_90_degrees;

    if (floor(quot) == quot) {
	/* See above re the following line. */
	return isincos[((int)fmod(quot, 4.0) & 3) + 1];
    }
    return cos(ang * (M_PI / 180));
}

void
gs_sincos_degrees(double ang, gs_sincos_t * psincos)
{
    double quot = ang / const_90_degrees;

    if (floor(quot) == quot) {
	/* See above re the following line. */
	int quads = (int)fmod(quot, 4.0) & 3;

	psincos->sin = isincos[quads];
	psincos->cos = isincos[quads + 1];
	psincos->orthogonal = true;
    } else {
	double arad = ang * (M_PI / 180);

	psincos->sin = sin(arad);
	psincos->cos = cos(arad);
	psincos->orthogonal = false;
    }
}

#endif /* USE_FPU */

/*
 * Define an atan2 function that returns an angle in degrees and uses
 * the PostScript quadrant rules.  Note that it may return
 * gs_error_undefinedresult.
 */
int
gs_atan2_degrees(double y, double x, double *pangle)
{
    if (y == 0) {	/* on X-axis, special case */
	if (x == 0)
	    return_error(gs_error_undefinedresult);
	*pangle = (x < 0 ? 180 : 0);
    } else {
	double result = atan2(y, x) * radians_to_degrees;

	if (result < 0)
	    result += 360;
	*pangle = result;
    }
    return 0;
}