    /*
     * 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 long 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  ((long)  2446)	/* FIX(0.298631336) */
    #define FIX_0_390180644  ((long)  3196)	/* FIX(0.390180644) */
    #define FIX_0_541196100  ((long)  4433)	/* FIX(0.541196100) */
    #define FIX_0_765366865  ((long)  6270)	/* FIX(0.765366865) */
    #define FIX_0_899976223  ((long)  7373)	/* FIX(0.899976223) */
    #define FIX_1_175875602  ((long)  9633)	/* FIX(1.175875602) */
    #define FIX_1_501321110  ((long)  12299)	/* FIX(1.501321110) */
    #define FIX_1_847759065  ((long)  15137)	/* FIX(1.847759065) */
    #define FIX_1_961570560  ((long)  16069)	/* FIX(1.961570560) */
    #define FIX_2_053119869  ((long)  16819)	/* FIX(2.053119869) */
    #define FIX_2_562915447  ((long)  20995)	/* FIX(2.562915447) */
    #define FIX_3_072711026  ((long)  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 long variable by an long constant to yield an long 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)
    {
      long tmp0, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7;
      long tmp10, tmp11, tmp12, tmp13;
      long 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 */