src/jpeg/jfdctint.c

/*
 * jfdctint.c
 *
 * Copyright (C) 1991-1996, Thomas G. Lane.
 * This file is part of the Independent JPEG Group's software.
 * For conditions of distribution and use, see the accompanying README file.
 *
 * This file contains a slow-but-accurate integer implementation of the
 * forward DCT (Discrete Cosine Transform).
 *
 * A 2-D DCT can be done by 1-D DCT on each row followed by 1-D DCT
 * on each column.  Direct algorithms are also available, but they are
 * much more complex and seem not to be any faster when reduced to code.
 *
 * This implementation is based on an algorithm described in
 *   C. Loeffler, A. Ligtenberg and G. Moschytz, "Practical Fast 1-D DCT
 *   Algorithms with 11 Multiplications", Proc. Int'l. Conf. on Acoustics,
 *   Speech, and Signal Processing 1989 (ICASSP '89), pp. 988-991.
 * The primary algorithm described there uses 11 multiplies and 29 adds.
 * We use their alternate method with 12 multiplies and 32 adds.
 * The advantage of this method is that no data path contains more than one
 * multiplication; this allows a very simple and accurate implementation in
 * scaled fixed-point arithmetic, with a minimal number of shifts.
 */

#define JPEG_INTERNALS
#include "jinclude.h"
#include "jpeglib.h"
#include "jdct.h"		/* Private declarations for DCT subsystem */

#ifdef DCT_ISLOW_SUPPORTED


/*
 * This module is specialized to the case DCTSIZE = 8.
 */

#if DCTSIZE != 8
  Sorry, this code only copes with 8x8 DCTs. /* deliberate syntax err */
#endif


/*
 * The poop on this scaling stuff is as follows:
 *
 * Each 1-D DCT step produces outputs which are a factor of sqrt(N)
 * larger than the true DCT outputs.  The final outputs are therefore
 * a factor of N larger than desired; since N=8 this can be cured by
 * a simple right shift at the end of the algorithm.  The advantage of
 * this arrangement is that we save two multiplications per 1-D DCT,
 * because the y0 and y4 outputs need not be divided by sqrt(N).
 * In the IJG code, this factor of 8 is removed by the quantization step
 * (in jcdctmgr.c), NOT in this module.
 *
 * We have to do addition and subtraction of the integer inputs, which
 * is no problem, and multiplication by fractional constants, which is
 * a problem to do in integer arithmetic.  We multiply all the constants
 * by CONST_SCALE and convert them to integer constants (thus retaining
 * CONST_BITS bits of precision in the constants).  After doing a
 * multiplication we have to divide the product by CONST_SCALE, with proper
 * rounding, to produce the correct output.  This division can be done
 * cheaply as a right shift of CONST_BITS bits.  We postpone shifting
 * as long as possible so that partial sums can be added together with
 * full fractional precision.
 *
 * The outputs of the first pass are scaled up by PASS1_BITS bits so that
 * they are represented to better-than-integral precision.  These outputs
 * require BITS_IN_JSAMPLE + PASS1_BITS + 3 bits; this fits in a 16-bit word
 * with the recommended scaling.  (For 12-bit sample data, the intermediate
 * array is INT32 anyway.)
 *
 * To avoid overflow of the 32-bit intermediate results in pass 2, we must
 * have BITS_IN_JSAMPLE + CONST_BITS + PASS1_BITS <= 26.  Error analysis
 * shows that the values given below are the most effective.
 */

#if BITS_IN_JSAMPLE == 8
#define CONST_BITS  13
#define PASS1_BITS  2
#else
#define CONST_BITS  13
#define PASS1_BITS  1		/* lose a little precision to avoid overflow */
#endif

/* Some C compilers fail to reduce "FIX(constant)" at compile time, thus
 * causing a lot of useless floating-point operations at run time.
 * To get around this we use the following pre-calculated constants.
 * If you change CONST_BITS you may want to add appropriate values.
 * (With a reasonable C compiler, you can just rely on the FIX() macro...)
 */

#if CONST_BITS == 13
#define FIX_0_298631336  ((JPEG_INT32)  2446)	/* FIX(0.298631336) */
#define FIX_0_390180644  ((JPEG_INT32)  3196)	/* FIX(0.390180644) */
#define FIX_0_541196100  ((JPEG_INT32)  4433)	/* FIX(0.541196100) */
#define FIX_0_765366865  ((JPEG_INT32)  6270)	/* FIX(0.765366865) */
#define FIX_0_899976223  ((JPEG_INT32)  7373)	/* FIX(0.899976223) */
#define FIX_1_175875602  ((JPEG_INT32)  9633)	/* FIX(1.175875602) */
#define FIX_1_501321110  ((JPEG_INT32)  12299)	/* FIX(1.501321110) */
#define FIX_1_847759065  ((JPEG_INT32)  15137)	/* FIX(1.847759065) */
#define FIX_1_961570560  ((JPEG_INT32)  16069)	/* FIX(1.961570560) */
#define FIX_2_053119869  ((JPEG_INT32)  16819)	/* FIX(2.053119869) */
#define FIX_2_562915447  ((JPEG_INT32)  20995)	/* FIX(2.562915447) */
#define FIX_3_072711026  ((JPEG_INT32)  25172)	/* FIX(3.072711026) */
#else
#define FIX_0_298631336  FIX(0.298631336)
#define FIX_0_390180644  FIX(0.390180644)
#define FIX_0_541196100  FIX(0.541196100)
#define FIX_0_765366865  FIX(0.765366865)
#define FIX_0_899976223  FIX(0.899976223)
#define FIX_1_175875602  FIX(1.175875602)
#define FIX_1_501321110  FIX(1.501321110)
#define FIX_1_847759065  FIX(1.847759065)
#define FIX_1_961570560  FIX(1.961570560)
#define FIX_2_053119869  FIX(2.053119869)
#define FIX_2_562915447  FIX(2.562915447)
#define FIX_3_072711026  FIX(3.072711026)
#endif


/* Multiply an INT32 variable by an INT32 constant to yield an INT32 result.
 * For 8-bit samples with the recommended scaling, all the variable
 * and constant values involved are no more than 16 bits wide, so a
 * 16x16->32 bit multiply can be used instead of a full 32x32 multiply.
 * For 12-bit samples, a full 32-bit multiplication will be needed.
 */

#if BITS_IN_JSAMPLE == 8
#define MULTIPLY(var,const)  MULTIPLY16C16(var,const)
#else
#define MULTIPLY(var,const)  ((var) * (const))
#endif


/*
 * Perform the forward DCT on one block of samples.
 */

GLOBAL(void)
jpeg_fdct_islow (DCTELEM * data)
{
  JPEG_INT32 tmp0, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7;
  JPEG_INT32 tmp10, tmp11, tmp12, tmp13;
  JPEG_INT32 z1, z2, z3, z4, z5;
  DCTELEM *dataptr;
  int ctr;
  SHIFT_TEMPS

  /* Pass 1: process rows. */
  /* Note results are scaled up by sqrt(8) compared to a true DCT; */
  /* furthermore, we scale the results by 2**PASS1_BITS. */

  dataptr = data;
  for (ctr = DCTSIZE-1; ctr >= 0; ctr--) {
    tmp0 = dataptr[0] + dataptr[7];
    tmp7 = dataptr[0] - dataptr[7];
    tmp1 = dataptr[1] + dataptr[6];
    tmp6 = dataptr[1] - dataptr[6];
    tmp2 = dataptr[2] + dataptr[5];
    tmp5 = dataptr[2] - dataptr[5];
    tmp3 = dataptr[3] + dataptr[4];
    tmp4 = dataptr[3] - dataptr[4];

    /* Even part per LL&M figure 1 --- note that published figure is faulty;
     * rotator "sqrt(2)*c1" should be "sqrt(2)*c6".
     */

    tmp10 = tmp0 + tmp3;
    tmp13 = tmp0 - tmp3;
    tmp11 = tmp1 + tmp2;
    tmp12 = tmp1 - tmp2;

    dataptr[0] = (DCTELEM) ((tmp10 + tmp11) << PASS1_BITS);
    dataptr[4] = (DCTELEM) ((tmp10 - tmp11) << PASS1_BITS);

    z1 = MULTIPLY(tmp12 + tmp13, FIX_0_541196100);
    dataptr[2] = (DCTELEM) DESCALE(z1 + MULTIPLY(tmp13, FIX_0_765366865),
				   CONST_BITS-PASS1_BITS);
    dataptr[6] = (DCTELEM) DESCALE(z1 + MULTIPLY(tmp12, - FIX_1_847759065),
				   CONST_BITS-PASS1_BITS);

    /* Odd part per figure 8 --- note paper omits factor of sqrt(2).
     * cK represents cos(K*pi/16).
     * i0..i3 in the paper are tmp4..tmp7 here.
     */

    z1 = tmp4 + tmp7;
    z2 = tmp5 + tmp6;
    z3 = tmp4 + tmp6;
    z4 = tmp5 + tmp7;
    z5 = MULTIPLY(z3 + z4, FIX_1_175875602); /* sqrt(2) * c3 */

    tmp4 = MULTIPLY(tmp4, FIX_0_298631336); /* sqrt(2) * (-c1+c3+c5-c7) */
    tmp5 = MULTIPLY(tmp5, FIX_2_053119869); /* sqrt(2) * ( c1+c3-c5+c7) */
    tmp6 = MULTIPLY(tmp6, FIX_3_072711026); /* sqrt(2) * ( c1+c3+c5-c7) */
    tmp7 = MULTIPLY(tmp7, FIX_1_501321110); /* sqrt(2) * ( c1+c3-c5-c7) */
    z1 = MULTIPLY(z1, - FIX_0_899976223); /* sqrt(2) * (c7-c3) */
    z2 = MULTIPLY(z2, - FIX_2_562915447); /* sqrt(2) * (-c1-c3) */
    z3 = MULTIPLY(z3, - FIX_1_961570560); /* sqrt(2) * (-c3-c5) */
    z4 = MULTIPLY(z4, - FIX_0_390180644); /* sqrt(2) * (c5-c3) */

    z3 += z5;
    z4 += z5;

    dataptr[7] = (DCTELEM) DESCALE(tmp4 + z1 + z3, CONST_BITS-PASS1_BITS);
    dataptr[5] = (DCTELEM) DESCALE(tmp5 + z2 + z4, CONST_BITS-PASS1_BITS);
    dataptr[3] = (DCTELEM) DESCALE(tmp6 + z2 + z3, CONST_BITS-PASS1_BITS);
    dataptr[1] = (DCTELEM) DESCALE(tmp7 + z1 + z4, CONST_BITS-PASS1_BITS);

    dataptr += DCTSIZE;		/* advance pointer to next row */
  }

  /* Pass 2: process columns.
   * We remove the PASS1_BITS scaling, but leave the results scaled up
   * by an overall factor of 8.
   */

  dataptr = data;
  for (ctr = DCTSIZE-1; ctr >= 0; ctr--) {
    tmp0 = dataptr[DCTSIZE*0] + dataptr[DCTSIZE*7];
    tmp7 = dataptr[DCTSIZE*0] - dataptr[DCTSIZE*7];
    tmp1 = dataptr[DCTSIZE*1] + dataptr[DCTSIZE*6];
    tmp6 = dataptr[DCTSIZE*1] - dataptr[DCTSIZE*6];
    tmp2 = dataptr[DCTSIZE*2] + dataptr[DCTSIZE*5];
    tmp5 = dataptr[DCTSIZE*2] - dataptr[DCTSIZE*5];
    tmp3 = dataptr[DCTSIZE*3] + dataptr[DCTSIZE*4];
    tmp4 = dataptr[DCTSIZE*3] - dataptr[DCTSIZE*4];

    /* Even part per LL&M figure 1 --- note that published figure is faulty;
     * rotator "sqrt(2)*c1" should be "sqrt(2)*c6".
     */

    tmp10 = tmp0 + tmp3;
    tmp13 = tmp0 - tmp3;
    tmp11 = tmp1 + tmp2;
    tmp12 = tmp1 - tmp2;

    dataptr[DCTSIZE*0] = (DCTELEM) DESCALE(tmp10 + tmp11, PASS1_BITS);
    dataptr[DCTSIZE*4] = (DCTELEM) DESCALE(tmp10 - tmp11, PASS1_BITS);

    z1 = MULTIPLY(tmp12 + tmp13, FIX_0_541196100);
    dataptr[DCTSIZE*2] = (DCTELEM) DESCALE(z1 + MULTIPLY(tmp13, FIX_0_765366865),
					   CONST_BITS+PASS1_BITS);
    dataptr[DCTSIZE*6] = (DCTELEM) DESCALE(z1 + MULTIPLY(tmp12, - FIX_1_847759065),
					   CONST_BITS+PASS1_BITS);

    /* Odd part per figure 8 --- note paper omits factor of sqrt(2).
     * cK represents cos(K*pi/16).
     * i0..i3 in the paper are tmp4..tmp7 here.
     */

    z1 = tmp4 + tmp7;
    z2 = tmp5 + tmp6;
    z3 = tmp4 + tmp6;
    z4 = tmp5 + tmp7;
    z5 = MULTIPLY(z3 + z4, FIX_1_175875602); /* sqrt(2) * c3 */

    tmp4 = MULTIPLY(tmp4, FIX_0_298631336); /* sqrt(2) * (-c1+c3+c5-c7) */
    tmp5 = MULTIPLY(tmp5, FIX_2_053119869); /* sqrt(2) * ( c1+c3-c5+c7) */
    tmp6 = MULTIPLY(tmp6, FIX_3_072711026); /* sqrt(2) * ( c1+c3+c5-c7) */
    tmp7 = MULTIPLY(tmp7, FIX_1_501321110); /* sqrt(2) * ( c1+c3-c5-c7) */
    z1 = MULTIPLY(z1, - FIX_0_899976223); /* sqrt(2) * (c7-c3) */
    z2 = MULTIPLY(z2, - FIX_2_562915447); /* sqrt(2) * (-c1-c3) */
    z3 = MULTIPLY(z3, - FIX_1_961570560); /* sqrt(2) * (-c3-c5) */
    z4 = MULTIPLY(z4, - FIX_0_390180644); /* sqrt(2) * (c5-c3) */

    z3 += z5;
    z4 += z5;

    dataptr[DCTSIZE*7] = (DCTELEM) DESCALE(tmp4 + z1 + z3,
					   CONST_BITS+PASS1_BITS);
    dataptr[DCTSIZE*5] = (DCTELEM) DESCALE(tmp5 + z2 + z4,
					   CONST_BITS+PASS1_BITS);
    dataptr[DCTSIZE*3] = (DCTELEM) DESCALE(tmp6 + z2 + z3,
					   CONST_BITS+PASS1_BITS);
    dataptr[DCTSIZE*1] = (DCTELEM) DESCALE(tmp7 + z1 + z4,
					   CONST_BITS+PASS1_BITS);

    dataptr++;			/* advance pointer to next column */
  }
}

#endif /* DCT_ISLOW_SUPPORTED */
Commit	Line	Data
	1	/*
	2	* jfdctint.c
	3	*
	4	* Copyright (C) 1991-1996, Thomas G. Lane.
	5	* This file is part of the Independent JPEG Group's software.
	6	* For conditions of distribution and use, see the accompanying README file.
	7	*
	8	* This file contains a slow-but-accurate integer implementation of the
	9	* forward DCT (Discrete Cosine Transform).
	10	*
	11	* A 2-D DCT can be done by 1-D DCT on each row followed by 1-D DCT
	12	* on each column. Direct algorithms are also available, but they are
	13	* much more complex and seem not to be any faster when reduced to code.
	14	*
	15	* This implementation is based on an algorithm described in
	16	* C. Loeffler, A. Ligtenberg and G. Moschytz, "Practical Fast 1-D DCT
	17	* Algorithms with 11 Multiplications", Proc. Int'l. Conf. on Acoustics,
	18	* Speech, and Signal Processing 1989 (ICASSP '89), pp. 988-991.
	19	* The primary algorithm described there uses 11 multiplies and 29 adds.
	20	* We use their alternate method with 12 multiplies and 32 adds.
	21	* The advantage of this method is that no data path contains more than one
	22	* multiplication; this allows a very simple and accurate implementation in
	23	* scaled fixed-point arithmetic, with a minimal number of shifts.
	24	*/
	25
	26	#define JPEG_INTERNALS
	27	#include "jinclude.h"
	28	#include "jpeglib.h"
	29	#include "jdct.h" /* Private declarations for DCT subsystem */
	30
	31	#ifdef DCT_ISLOW_SUPPORTED
	32
	33
	34	/*
	35	* This module is specialized to the case DCTSIZE = 8.
	36	*/
	37
	38	#if DCTSIZE != 8
	39	Sorry, this code only copes with 8x8 DCTs. /* deliberate syntax err */
	40	#endif
	41
	42
	43	/*
	44	* The poop on this scaling stuff is as follows:
	45	*
	46	* Each 1-D DCT step produces outputs which are a factor of sqrt(N)
	47	* larger than the true DCT outputs. The final outputs are therefore
	48	* a factor of N larger than desired; since N=8 this can be cured by
	49	* a simple right shift at the end of the algorithm. The advantage of
	50	* this arrangement is that we save two multiplications per 1-D DCT,
	51	* because the y0 and y4 outputs need not be divided by sqrt(N).
	52	* In the IJG code, this factor of 8 is removed by the quantization step
	53	* (in jcdctmgr.c), NOT in this module.
	54	*
	55	* We have to do addition and subtraction of the integer inputs, which
	56	* is no problem, and multiplication by fractional constants, which is
	57	* a problem to do in integer arithmetic. We multiply all the constants
	58	* by CONST_SCALE and convert them to integer constants (thus retaining
	59	* CONST_BITS bits of precision in the constants). After doing a
	60	* multiplication we have to divide the product by CONST_SCALE, with proper
	61	* rounding, to produce the correct output. This division can be done
	62	* cheaply as a right shift of CONST_BITS bits. We postpone shifting
	63	* as long as possible so that partial sums can be added together with
	64	* full fractional precision.
	65	*
	66	* The outputs of the first pass are scaled up by PASS1_BITS bits so that
	67	* they are represented to better-than-integral precision. These outputs
	68	* require BITS_IN_JSAMPLE + PASS1_BITS + 3 bits; this fits in a 16-bit word
	69	* with the recommended scaling. (For 12-bit sample data, the intermediate
	70	* array is INT32 anyway.)
	71	*
	72	* To avoid overflow of the 32-bit intermediate results in pass 2, we must
	73	* have BITS_IN_JSAMPLE + CONST_BITS + PASS1_BITS <= 26. Error analysis
	74	* shows that the values given below are the most effective.
	75	*/
	76
	77	#if BITS_IN_JSAMPLE == 8
	78	#define CONST_BITS 13
	79	#define PASS1_BITS 2
	80	#else
	81	#define CONST_BITS 13
	82	#define PASS1_BITS 1 /* lose a little precision to avoid overflow */
	83	#endif
	84
	85	/* Some C compilers fail to reduce "FIX(constant)" at compile time, thus
	86	* causing a lot of useless floating-point operations at run time.
	87	* To get around this we use the following pre-calculated constants.
	88	* If you change CONST_BITS you may want to add appropriate values.
	89	* (With a reasonable C compiler, you can just rely on the FIX() macro...)
	90	*/
	91
	92	#if CONST_BITS == 13
	93	#define FIX_0_298631336 ((JPEG_INT32) 2446) /* FIX(0.298631336) */
	94	#define FIX_0_390180644 ((JPEG_INT32) 3196) /* FIX(0.390180644) */
	95	#define FIX_0_541196100 ((JPEG_INT32) 4433) /* FIX(0.541196100) */
	96	#define FIX_0_765366865 ((JPEG_INT32) 6270) /* FIX(0.765366865) */
	97	#define FIX_0_899976223 ((JPEG_INT32) 7373) /* FIX(0.899976223) */
	98	#define FIX_1_175875602 ((JPEG_INT32) 9633) /* FIX(1.175875602) */
	99	#define FIX_1_501321110 ((JPEG_INT32) 12299) /* FIX(1.501321110) */
	100	#define FIX_1_847759065 ((JPEG_INT32) 15137) /* FIX(1.847759065) */
	101	#define FIX_1_961570560 ((JPEG_INT32) 16069) /* FIX(1.961570560) */
	102	#define FIX_2_053119869 ((JPEG_INT32) 16819) /* FIX(2.053119869) */
	103	#define FIX_2_562915447 ((JPEG_INT32) 20995) /* FIX(2.562915447) */
	104	#define FIX_3_072711026 ((JPEG_INT32) 25172) /* FIX(3.072711026) */
	105	#else
	106	#define FIX_0_298631336 FIX(0.298631336)
	107	#define FIX_0_390180644 FIX(0.390180644)
	108	#define FIX_0_541196100 FIX(0.541196100)
	109	#define FIX_0_765366865 FIX(0.765366865)
	110	#define FIX_0_899976223 FIX(0.899976223)
	111	#define FIX_1_175875602 FIX(1.175875602)
	112	#define FIX_1_501321110 FIX(1.501321110)
	113	#define FIX_1_847759065 FIX(1.847759065)
	114	#define FIX_1_961570560 FIX(1.961570560)
	115	#define FIX_2_053119869 FIX(2.053119869)
	116	#define FIX_2_562915447 FIX(2.562915447)
	117	#define FIX_3_072711026 FIX(3.072711026)
	118	#endif
	119
	120
	121	/* Multiply an INT32 variable by an INT32 constant to yield an INT32 result.
	122	* For 8-bit samples with the recommended scaling, all the variable
	123	* and constant values involved are no more than 16 bits wide, so a
	124	* 16x16->32 bit multiply can be used instead of a full 32x32 multiply.
	125	* For 12-bit samples, a full 32-bit multiplication will be needed.
	126	*/
	127
	128	#if BITS_IN_JSAMPLE == 8
	129	#define MULTIPLY(var,const) MULTIPLY16C16(var,const)
	130	#else
	131	#define MULTIPLY(var,const) ((var) * (const))
	132	#endif
	133
	134
	135	/*
	136	* Perform the forward DCT on one block of samples.
	137	*/
	138
	139	GLOBAL(void)
	140	jpeg_fdct_islow (DCTELEM * data)
	141	{
	142	JPEG_INT32 tmp0, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7;
	143	JPEG_INT32 tmp10, tmp11, tmp12, tmp13;
	144	JPEG_INT32 z1, z2, z3, z4, z5;
	145	DCTELEM *dataptr;
	146	int ctr;
	147	SHIFT_TEMPS
	148
	149	/* Pass 1: process rows. */
	150	/* Note results are scaled up by sqrt(8) compared to a true DCT; */
	151	/* furthermore, we scale the results by 2*PASS1_BITS. /
	152
	153	dataptr = data;
	154	for (ctr = DCTSIZE-1; ctr >= 0; ctr--) {
	155	tmp0 = dataptr[0] + dataptr[7];
	156	tmp7 = dataptr[0] - dataptr[7];
	157	tmp1 = dataptr[1] + dataptr[6];
	158	tmp6 = dataptr[1] - dataptr[6];
	159	tmp2 = dataptr[2] + dataptr[5];
	160	tmp5 = dataptr[2] - dataptr[5];
	161	tmp3 = dataptr[3] + dataptr[4];
	162	tmp4 = dataptr[3] - dataptr[4];
	163
	164	/* Even part per LL&M figure 1 --- note that published figure is faulty;
	165	* rotator "sqrt(2)c1" should be "sqrt(2)c6".
	166	*/
	167
	168	tmp10 = tmp0 + tmp3;
	169	tmp13 = tmp0 - tmp3;
	170	tmp11 = tmp1 + tmp2;
	171	tmp12 = tmp1 - tmp2;
	172
	173	dataptr[0] = (DCTELEM) ((tmp10 + tmp11) << PASS1_BITS);
	174	dataptr[4] = (DCTELEM) ((tmp10 - tmp11) << PASS1_BITS);
	175
	176	z1 = MULTIPLY(tmp12 + tmp13, FIX_0_541196100);
	177	dataptr[2] = (DCTELEM) DESCALE(z1 + MULTIPLY(tmp13, FIX_0_765366865),
	178	CONST_BITS-PASS1_BITS);
	179	dataptr[6] = (DCTELEM) DESCALE(z1 + MULTIPLY(tmp12, - FIX_1_847759065),
	180	CONST_BITS-PASS1_BITS);
	181
	182	/* Odd part per figure 8 --- note paper omits factor of sqrt(2).
	183	* cK represents cos(K*pi/16).
	184	* i0..i3 in the paper are tmp4..tmp7 here.
	185	*/
	186
	187	z1 = tmp4 + tmp7;
	188	z2 = tmp5 + tmp6;
	189	z3 = tmp4 + tmp6;
	190	z4 = tmp5 + tmp7;
	191	z5 = MULTIPLY(z3 + z4, FIX_1_175875602); /* sqrt(2) * c3 */
	192
	193	tmp4 = MULTIPLY(tmp4, FIX_0_298631336); /* sqrt(2) * (-c1+c3+c5-c7) */
	194	tmp5 = MULTIPLY(tmp5, FIX_2_053119869); /* sqrt(2) * ( c1+c3-c5+c7) */
	195	tmp6 = MULTIPLY(tmp6, FIX_3_072711026); /* sqrt(2) * ( c1+c3+c5-c7) */
	196	tmp7 = MULTIPLY(tmp7, FIX_1_501321110); /* sqrt(2) * ( c1+c3-c5-c7) */
	197	z1 = MULTIPLY(z1, - FIX_0_899976223); /* sqrt(2) * (c7-c3) */
	198	z2 = MULTIPLY(z2, - FIX_2_562915447); /* sqrt(2) * (-c1-c3) */
	199	z3 = MULTIPLY(z3, - FIX_1_961570560); /* sqrt(2) * (-c3-c5) */
	200	z4 = MULTIPLY(z4, - FIX_0_390180644); /* sqrt(2) * (c5-c3) */
	201
	202	z3 += z5;
	203	z4 += z5;
	204
	205	dataptr[7] = (DCTELEM) DESCALE(tmp4 + z1 + z3, CONST_BITS-PASS1_BITS);
	206	dataptr[5] = (DCTELEM) DESCALE(tmp5 + z2 + z4, CONST_BITS-PASS1_BITS);
	207	dataptr[3] = (DCTELEM) DESCALE(tmp6 + z2 + z3, CONST_BITS-PASS1_BITS);
	208	dataptr[1] = (DCTELEM) DESCALE(tmp7 + z1 + z4, CONST_BITS-PASS1_BITS);
	209
	210	dataptr += DCTSIZE; /* advance pointer to next row */
	211	}
	212
	213	/* Pass 2: process columns.
	214	* We remove the PASS1_BITS scaling, but leave the results scaled up
	215	* by an overall factor of 8.
	216	*/
	217
	218	dataptr = data;
	219	for (ctr = DCTSIZE-1; ctr >= 0; ctr--) {
	220	tmp0 = dataptr[DCTSIZE0] + dataptr[DCTSIZE7];
	221	tmp7 = dataptr[DCTSIZE0] - dataptr[DCTSIZE7];
	222	tmp1 = dataptr[DCTSIZE1] + dataptr[DCTSIZE6];
	223	tmp6 = dataptr[DCTSIZE1] - dataptr[DCTSIZE6];
	224	tmp2 = dataptr[DCTSIZE2] + dataptr[DCTSIZE5];
	225	tmp5 = dataptr[DCTSIZE2] - dataptr[DCTSIZE5];
	226	tmp3 = dataptr[DCTSIZE3] + dataptr[DCTSIZE4];
	227	tmp4 = dataptr[DCTSIZE3] - dataptr[DCTSIZE4];
	228
	229	/* Even part per LL&M figure 1 --- note that published figure is faulty;
	230	* rotator "sqrt(2)c1" should be "sqrt(2)c6".
	231	*/
	232
	233	tmp10 = tmp0 + tmp3;
	234	tmp13 = tmp0 - tmp3;
	235	tmp11 = tmp1 + tmp2;
	236	tmp12 = tmp1 - tmp2;
	237
	238	dataptr[DCTSIZE*0] = (DCTELEM) DESCALE(tmp10 + tmp11, PASS1_BITS);
	239	dataptr[DCTSIZE*4] = (DCTELEM) DESCALE(tmp10 - tmp11, PASS1_BITS);
	240
	241	z1 = MULTIPLY(tmp12 + tmp13, FIX_0_541196100);
	242	dataptr[DCTSIZE*2] = (DCTELEM) DESCALE(z1 + MULTIPLY(tmp13, FIX_0_765366865),
	243	CONST_BITS+PASS1_BITS);
	244	dataptr[DCTSIZE*6] = (DCTELEM) DESCALE(z1 + MULTIPLY(tmp12, - FIX_1_847759065),
	245	CONST_BITS+PASS1_BITS);
	246
	247	/* Odd part per figure 8 --- note paper omits factor of sqrt(2).
	248	* cK represents cos(K*pi/16).
	249	* i0..i3 in the paper are tmp4..tmp7 here.
	250	*/
	251
	252	z1 = tmp4 + tmp7;
	253	z2 = tmp5 + tmp6;
	254	z3 = tmp4 + tmp6;
	255	z4 = tmp5 + tmp7;
	256	z5 = MULTIPLY(z3 + z4, FIX_1_175875602); /* sqrt(2) * c3 */
	257
	258	tmp4 = MULTIPLY(tmp4, FIX_0_298631336); /* sqrt(2) * (-c1+c3+c5-c7) */
	259	tmp5 = MULTIPLY(tmp5, FIX_2_053119869); /* sqrt(2) * ( c1+c3-c5+c7) */
	260	tmp6 = MULTIPLY(tmp6, FIX_3_072711026); /* sqrt(2) * ( c1+c3+c5-c7) */
	261	tmp7 = MULTIPLY(tmp7, FIX_1_501321110); /* sqrt(2) * ( c1+c3-c5-c7) */
	262	z1 = MULTIPLY(z1, - FIX_0_899976223); /* sqrt(2) * (c7-c3) */
	263	z2 = MULTIPLY(z2, - FIX_2_562915447); /* sqrt(2) * (-c1-c3) */
	264	z3 = MULTIPLY(z3, - FIX_1_961570560); /* sqrt(2) * (-c3-c5) */
	265	z4 = MULTIPLY(z4, - FIX_0_390180644); /* sqrt(2) * (c5-c3) */
	266
	267	z3 += z5;
	268	z4 += z5;
	269
	270	dataptr[DCTSIZE*7] = (DCTELEM) DESCALE(tmp4 + z1 + z3,
	271	CONST_BITS+PASS1_BITS);
	272	dataptr[DCTSIZE*5] = (DCTELEM) DESCALE(tmp5 + z2 + z4,
	273	CONST_BITS+PASS1_BITS);
	274	dataptr[DCTSIZE*3] = (DCTELEM) DESCALE(tmp6 + z2 + z3,
	275	CONST_BITS+PASS1_BITS);
	276	dataptr[DCTSIZE*1] = (DCTELEM) DESCALE(tmp7 + z1 + z4,
	277	CONST_BITS+PASS1_BITS);
	278
	279	dataptr++; /* advance pointer to next column */
	280	}
	281	}
	282
	283	#endif /* DCT_ISLOW_SUPPORTED */