    /*
     * jidctint.c
     *
     * Copyright (C) 1991-1998, 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
     * inverse DCT (Discrete Cosine Transform).  In the IJG code, this routine
     * must also perform dequantization of the input coefficients.
     *
     * A 2-D IDCT can be done by 1-D IDCT on each column followed by 1-D IDCT
     * on each row (or vice versa, but it's more convenient to emit a row at
     * a time).  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 IDCT step produces outputs which are a factor of sqrt(N)
     * larger than the true IDCT 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 IDCT,
     * because the y0 and y4 inputs need not be divided by sqrt(N).
     *
     * 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.  (To scale up 12-bit sample data further, an
     * intermediate long array would be needed.)
     *
     * 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

    /* Dequantize a coefficient by multiplying it by the multiplier-table
     * entry; produce an int result.  In this module, both inputs and result
     * are 16 bits or less, so either int or short multiply will work.
     */

    #define DEQUANTIZE(coef,quantval)  (((ISLOW_MULT_TYPE) (coef)) * (quantval))

    /*
     * Perform dequantization and inverse DCT on one block of coefficients.
     */

    GLOBAL(void)
    jpeg_idct_islow (j_decompress_ptr cinfo, jpeg_component_info * compptr,
    		 JCOEFPTR coef_block,
    		 JSAMPARRAY output_buf, JDIMENSION output_col)
    {
      long tmp0, tmp1, tmp2, tmp3;
      long tmp10, tmp11, tmp12, tmp13;
      long z1, z2, z3, z4, z5;
      JCOEFPTR inptr;
      ISLOW_MULT_TYPE * quantptr;
      int * wsptr;
      JSAMPROW outptr;
      JSAMPLE *range_limit = IDCT_range_limit(cinfo);
      int ctr;
      int workspace[DCTSIZE2];	/* buffers data between passes */
      SHIFT_TEMPS

      /* Pass 1: process columns from input, store into work array. */
      /* Note results are scaled up by sqrt(8) compared to a true IDCT; */
      /* furthermore, we scale the results by 2**PASS1_BITS. */

      inptr = coef_block;
      quantptr = (ISLOW_MULT_TYPE *) compptr->dct_table;
      wsptr = workspace;
      for (ctr = DCTSIZE; ctr > 0; ctr--) {
        /* Due to quantization, we will usually find that many of the input
         * coefficients are zero, especially the AC terms.  We can exploit this
         * by short-circuiting the IDCT calculation for any column in which all
         * the AC terms are zero.  In that case each output is equal to the
         * DC coefficient (with scale factor as needed).
         * With typical images and quantization tables, half or more of the
         * column DCT calculations can be simplified this way.
         */

        if (inptr[DCTSIZE*1] == 0 && inptr[DCTSIZE*2] == 0 &&
    	inptr[DCTSIZE*3] == 0 && inptr[DCTSIZE*4] == 0 &&
    	inptr[DCTSIZE*5] == 0 && inptr[DCTSIZE*6] == 0 &&
    	inptr[DCTSIZE*7] == 0) {
          /* AC terms all zero */
          int dcval = DEQUANTIZE(inptr[DCTSIZE*0], quantptr[DCTSIZE*0]) << PASS1_BITS;

          wsptr[DCTSIZE*0] = dcval;
          wsptr[DCTSIZE*1] = dcval;
          wsptr[DCTSIZE*2] = dcval;
          wsptr[DCTSIZE*3] = dcval;
          wsptr[DCTSIZE*4] = dcval;
          wsptr[DCTSIZE*5] = dcval;
          wsptr[DCTSIZE*6] = dcval;
          wsptr[DCTSIZE*7] = dcval;

          inptr++;			/* advance pointers to next column */
          quantptr++;
          wsptr++;
          continue;
        }

        /* Even part: reverse the even part of the forward DCT. */
        /* The rotator is sqrt(2)*c(-6). */

        z2 = DEQUANTIZE(inptr[DCTSIZE*2], quantptr[DCTSIZE*2]);
        z3 = DEQUANTIZE(inptr[DCTSIZE*6], quantptr[DCTSIZE*6]);

        z1 = MULTIPLY(z2 + z3, FIX_0_541196100);
        tmp2 = z1 + MULTIPLY(z3, - FIX_1_847759065);
        tmp3 = z1 + MULTIPLY(z2, FIX_0_765366865);

        z2 = DEQUANTIZE(inptr[DCTSIZE*0], quantptr[DCTSIZE*0]);
        z3 = DEQUANTIZE(inptr[DCTSIZE*4], quantptr[DCTSIZE*4]);

        tmp0 = (z2 + z3) << CONST_BITS;
        tmp1 = (z2 - z3) << CONST_BITS;

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

        /* Odd part per figure 8; the matrix is unitary and hence its
         * transpose is its inverse.  i0..i3 are y7,y5,y3,y1 respectively.
         */

        tmp0 = DEQUANTIZE(inptr[DCTSIZE*7], quantptr[DCTSIZE*7]);
        tmp1 = DEQUANTIZE(inptr[DCTSIZE*5], quantptr[DCTSIZE*5]);
        tmp2 = DEQUANTIZE(inptr[DCTSIZE*3], quantptr[DCTSIZE*3]);
        tmp3 = DEQUANTIZE(inptr[DCTSIZE*1], quantptr[DCTSIZE*1]);

        z1 = tmp0 + tmp3;
        z2 = tmp1 + tmp2;
        z3 = tmp0 + tmp2;
        z4 = tmp1 + tmp3;
        z5 = MULTIPLY(z3 + z4, FIX_1_175875602); /* sqrt(2) * c3 */

        tmp0 = MULTIPLY(tmp0, FIX_0_298631336); /* sqrt(2) * (-c1+c3+c5-c7) */
        tmp1 = MULTIPLY(tmp1, FIX_2_053119869); /* sqrt(2) * ( c1+c3-c5+c7) */
        tmp2 = MULTIPLY(tmp2, FIX_3_072711026); /* sqrt(2) * ( c1+c3+c5-c7) */
        tmp3 = MULTIPLY(tmp3, 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;

        tmp0 += z1 + z3;
        tmp1 += z2 + z4;
        tmp2 += z2 + z3;
        tmp3 += z1 + z4;

        /* Final output stage: inputs are tmp10..tmp13, tmp0..tmp3 */

        wsptr[DCTSIZE*0] = (int) DESCALE(tmp10 + tmp3, CONST_BITS-PASS1_BITS);
        wsptr[DCTSIZE*7] = (int) DESCALE(tmp10 - tmp3, CONST_BITS-PASS1_BITS);
        wsptr[DCTSIZE*1] = (int) DESCALE(tmp11 + tmp2, CONST_BITS-PASS1_BITS);
        wsptr[DCTSIZE*6] = (int) DESCALE(tmp11 - tmp2, CONST_BITS-PASS1_BITS);
        wsptr[DCTSIZE*2] = (int) DESCALE(tmp12 + tmp1, CONST_BITS-PASS1_BITS);
        wsptr[DCTSIZE*5] = (int) DESCALE(tmp12 - tmp1, CONST_BITS-PASS1_BITS);
        wsptr[DCTSIZE*3] = (int) DESCALE(tmp13 + tmp0, CONST_BITS-PASS1_BITS);
        wsptr[DCTSIZE*4] = (int) DESCALE(tmp13 - tmp0, CONST_BITS-PASS1_BITS);

        inptr++;			/* advance pointers to next column */
        quantptr++;
        wsptr++;
      }

      /* Pass 2: process rows from work array, store into output array. */
      /* Note that we must descale the results by a factor of 8 == 2**3, */
      /* and also undo the PASS1_BITS scaling. */

      wsptr = workspace;
      for (ctr = 0; ctr < DCTSIZE; ctr++) {
        outptr = output_buf[ctr] + output_col;
        /* Rows of zeroes can be exploited in the same way as we did with columns.
         * However, the column calculation has created many nonzero AC terms, so
         * the simplification applies less often (typically 5% to 10% of the time).
         * On machines with very fast multiplication, it's possible that the
         * test takes more time than it's worth.  In that case this section
         * may be commented out.
         */

    #ifndef NO_ZERO_ROW_TEST
        if (wsptr[1] == 0 && wsptr[2] == 0 && wsptr[3] == 0 && wsptr[4] == 0 &&
    	wsptr[5] == 0 && wsptr[6] == 0 && wsptr[7] == 0) {
          /* AC terms all zero */
          JSAMPLE dcval = range_limit[(int) DESCALE((long) wsptr[0], PASS1_BITS+3)
    				  & RANGE_MASK];

          outptr[0] = dcval;
          outptr[1] = dcval;
          outptr[2] = dcval;
          outptr[3] = dcval;
          outptr[4] = dcval;
          outptr[5] = dcval;
          outptr[6] = dcval;
          outptr[7] = dcval;

          wsptr += DCTSIZE;		/* advance pointer to next row */
          continue;
        }
    #endif

        /* Even part: reverse the even part of the forward DCT. */
        /* The rotator is sqrt(2)*c(-6). */

        z2 = (long) wsptr[2];
        z3 = (long) wsptr[6];

        z1 = MULTIPLY(z2 + z3, FIX_0_541196100);
        tmp2 = z1 + MULTIPLY(z3, - FIX_1_847759065);
        tmp3 = z1 + MULTIPLY(z2, FIX_0_765366865);

        tmp0 = ((long) wsptr[0] + (long) wsptr[4]) << CONST_BITS;
        tmp1 = ((long) wsptr[0] - (long) wsptr[4]) << CONST_BITS;

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

        /* Odd part per figure 8; the matrix is unitary and hence its
         * transpose is its inverse.  i0..i3 are y7,y5,y3,y1 respectively.
         */

        tmp0 = (long) wsptr[7];
        tmp1 = (long) wsptr[5];
        tmp2 = (long) wsptr[3];
        tmp3 = (long) wsptr[1];

        z1 = tmp0 + tmp3;
        z2 = tmp1 + tmp2;
        z3 = tmp0 + tmp2;
        z4 = tmp1 + tmp3;
        z5 = MULTIPLY(z3 + z4, FIX_1_175875602); /* sqrt(2) * c3 */

        tmp0 = MULTIPLY(tmp0, FIX_0_298631336); /* sqrt(2) * (-c1+c3+c5-c7) */
        tmp1 = MULTIPLY(tmp1, FIX_2_053119869); /* sqrt(2) * ( c1+c3-c5+c7) */
        tmp2 = MULTIPLY(tmp2, FIX_3_072711026); /* sqrt(2) * ( c1+c3+c5-c7) */
        tmp3 = MULTIPLY(tmp3, 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;

        tmp0 += z1 + z3;
        tmp1 += z2 + z4;
        tmp2 += z2 + z3;
        tmp3 += z1 + z4;

        /* Final output stage: inputs are tmp10..tmp13, tmp0..tmp3 */

        outptr[0] = range_limit[(int) DESCALE(tmp10 + tmp3,
    					  CONST_BITS+PASS1_BITS+3)
    			    & RANGE_MASK];
        outptr[7] = range_limit[(int) DESCALE(tmp10 - tmp3,
    					  CONST_BITS+PASS1_BITS+3)
    			    & RANGE_MASK];
        outptr[1] = range_limit[(int) DESCALE(tmp11 + tmp2,
    					  CONST_BITS+PASS1_BITS+3)
    			    & RANGE_MASK];
        outptr[6] = range_limit[(int) DESCALE(tmp11 - tmp2,
    					  CONST_BITS+PASS1_BITS+3)
    			    & RANGE_MASK];
        outptr[2] = range_limit[(int) DESCALE(tmp12 + tmp1,
    					  CONST_BITS+PASS1_BITS+3)
    			    & RANGE_MASK];
        outptr[5] = range_limit[(int) DESCALE(tmp12 - tmp1,
    					  CONST_BITS+PASS1_BITS+3)
    			    & RANGE_MASK];
        outptr[3] = range_limit[(int) DESCALE(tmp13 + tmp0,
    					  CONST_BITS+PASS1_BITS+3)
    			    & RANGE_MASK];
        outptr[4] = range_limit[(int) DESCALE(tmp13 - tmp0,
    					  CONST_BITS+PASS1_BITS+3)
    			    & RANGE_MASK];

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

    #endif /* DCT_ISLOW_SUPPORTED */