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454 lines
13 KiB
454 lines
13 KiB
| |
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| stan.sa 3.3 7/29/91 |
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| |
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| The entry point stan computes the tangent of |
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| an input argument; |
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| stand does the same except for denormalized input. |
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| |
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| Input: Double-extended number X in location pointed to |
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| by address register a0. |
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| |
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| Output: The value tan(X) returned in floating-point register Fp0. |
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| |
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| Accuracy and Monotonicity: The returned result is within 3 ulp in |
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| 64 significant bit, i.e. within 0.5001 ulp to 53 bits if the |
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| result is subsequently rounded to double precision. The |
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| result is provably monotonic in double precision. |
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| |
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| Speed: The program sTAN takes approximately 170 cycles for |
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| input argument X such that |X| < 15Pi, which is the usual |
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| situation. |
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| |
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| Algorithm: |
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| |
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| 1. If |X| >= 15Pi or |X| < 2**(-40), go to 6. |
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| |
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| 2. Decompose X as X = N(Pi/2) + r where |r| <= Pi/4. Let |
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| k = N mod 2, so in particular, k = 0 or 1. |
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| |
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| 3. If k is odd, go to 5. |
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| |
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| 4. (k is even) Tan(X) = tan(r) and tan(r) is approximated by a |
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| rational function U/V where |
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| U = r + r*s*(P1 + s*(P2 + s*P3)), and |
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| V = 1 + s*(Q1 + s*(Q2 + s*(Q3 + s*Q4))), s = r*r. |
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| Exit. |
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| |
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| 4. (k is odd) Tan(X) = -cot(r). Since tan(r) is approximated by a |
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| rational function U/V where |
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| U = r + r*s*(P1 + s*(P2 + s*P3)), and |
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| V = 1 + s*(Q1 + s*(Q2 + s*(Q3 + s*Q4))), s = r*r, |
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| -Cot(r) = -V/U. Exit. |
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| |
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| 6. If |X| > 1, go to 8. |
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| |
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| 7. (|X|<2**(-40)) Tan(X) = X. Exit. |
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| |
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| 8. Overwrite X by X := X rem 2Pi. Now that |X| <= Pi, go back to 2. |
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| |
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| Copyright (C) Motorola, Inc. 1990 |
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| All Rights Reserved |
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| |
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| For details on the license for this file, please see the |
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| file, README, in this same directory. |
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|STAN idnt 2,1 | Motorola 040 Floating Point Software Package |
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|section 8 |
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#include "fpsp.h" |
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BOUNDS1: .long 0x3FD78000,0x4004BC7E |
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TWOBYPI: .long 0x3FE45F30,0x6DC9C883 |
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TANQ4: .long 0x3EA0B759,0xF50F8688 |
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TANP3: .long 0xBEF2BAA5,0xA8924F04 |
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TANQ3: .long 0xBF346F59,0xB39BA65F,0x00000000,0x00000000 |
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TANP2: .long 0x3FF60000,0xE073D3FC,0x199C4A00,0x00000000 |
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TANQ2: .long 0x3FF90000,0xD23CD684,0x15D95FA1,0x00000000 |
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TANP1: .long 0xBFFC0000,0x8895A6C5,0xFB423BCA,0x00000000 |
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TANQ1: .long 0xBFFD0000,0xEEF57E0D,0xA84BC8CE,0x00000000 |
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INVTWOPI: .long 0x3FFC0000,0xA2F9836E,0x4E44152A,0x00000000 |
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TWOPI1: .long 0x40010000,0xC90FDAA2,0x00000000,0x00000000 |
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TWOPI2: .long 0x3FDF0000,0x85A308D4,0x00000000,0x00000000 |
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|--N*PI/2, -32 <= N <= 32, IN A LEADING TERM IN EXT. AND TRAILING |
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|--TERM IN SGL. NOTE THAT PI IS 64-BIT LONG, THUS N*PI/2 IS AT |
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|--MOST 69 BITS LONG. |
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.global PITBL |
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PITBL: |
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.long 0xC0040000,0xC90FDAA2,0x2168C235,0x21800000 |
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.long 0xC0040000,0xC2C75BCD,0x105D7C23,0xA0D00000 |
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.long 0xC0040000,0xBC7EDCF7,0xFF523611,0xA1E80000 |
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.long 0xC0040000,0xB6365E22,0xEE46F000,0x21480000 |
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.long 0xC0040000,0xAFEDDF4D,0xDD3BA9EE,0xA1200000 |
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.long 0xC0040000,0xA9A56078,0xCC3063DD,0x21FC0000 |
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.long 0xC0040000,0xA35CE1A3,0xBB251DCB,0x21100000 |
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.long 0xC0040000,0x9D1462CE,0xAA19D7B9,0xA1580000 |
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.long 0xC0040000,0x96CBE3F9,0x990E91A8,0x21E00000 |
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.long 0xC0040000,0x90836524,0x88034B96,0x20B00000 |
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.long 0xC0040000,0x8A3AE64F,0x76F80584,0xA1880000 |
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.long 0xC0040000,0x83F2677A,0x65ECBF73,0x21C40000 |
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.long 0xC0030000,0xFB53D14A,0xA9C2F2C2,0x20000000 |
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.long 0xC0030000,0xEEC2D3A0,0x87AC669F,0x21380000 |
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.long 0xC0030000,0xE231D5F6,0x6595DA7B,0xA1300000 |
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.long 0xC0030000,0xD5A0D84C,0x437F4E58,0x9FC00000 |
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.long 0xC0030000,0xC90FDAA2,0x2168C235,0x21000000 |
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.long 0xC0030000,0xBC7EDCF7,0xFF523611,0xA1680000 |
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.long 0xC0030000,0xAFEDDF4D,0xDD3BA9EE,0xA0A00000 |
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.long 0xC0030000,0xA35CE1A3,0xBB251DCB,0x20900000 |
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.long 0xC0030000,0x96CBE3F9,0x990E91A8,0x21600000 |
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.long 0xC0030000,0x8A3AE64F,0x76F80584,0xA1080000 |
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.long 0xC0020000,0xFB53D14A,0xA9C2F2C2,0x1F800000 |
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.long 0xC0020000,0xE231D5F6,0x6595DA7B,0xA0B00000 |
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.long 0xC0020000,0xC90FDAA2,0x2168C235,0x20800000 |
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.long 0xC0020000,0xAFEDDF4D,0xDD3BA9EE,0xA0200000 |
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.long 0xC0020000,0x96CBE3F9,0x990E91A8,0x20E00000 |
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.long 0xC0010000,0xFB53D14A,0xA9C2F2C2,0x1F000000 |
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.long 0xC0010000,0xC90FDAA2,0x2168C235,0x20000000 |
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.long 0xC0010000,0x96CBE3F9,0x990E91A8,0x20600000 |
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.long 0xC0000000,0xC90FDAA2,0x2168C235,0x1F800000 |
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.long 0xBFFF0000,0xC90FDAA2,0x2168C235,0x1F000000 |
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.long 0x00000000,0x00000000,0x00000000,0x00000000 |
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.long 0x3FFF0000,0xC90FDAA2,0x2168C235,0x9F000000 |
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.long 0x40000000,0xC90FDAA2,0x2168C235,0x9F800000 |
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.long 0x40010000,0x96CBE3F9,0x990E91A8,0xA0600000 |
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.long 0x40010000,0xC90FDAA2,0x2168C235,0xA0000000 |
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.long 0x40010000,0xFB53D14A,0xA9C2F2C2,0x9F000000 |
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.long 0x40020000,0x96CBE3F9,0x990E91A8,0xA0E00000 |
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.long 0x40020000,0xAFEDDF4D,0xDD3BA9EE,0x20200000 |
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.long 0x40020000,0xC90FDAA2,0x2168C235,0xA0800000 |
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.long 0x40020000,0xE231D5F6,0x6595DA7B,0x20B00000 |
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.long 0x40020000,0xFB53D14A,0xA9C2F2C2,0x9F800000 |
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.long 0x40030000,0x8A3AE64F,0x76F80584,0x21080000 |
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.long 0x40030000,0x96CBE3F9,0x990E91A8,0xA1600000 |
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.long 0x40030000,0xA35CE1A3,0xBB251DCB,0xA0900000 |
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.long 0x40030000,0xAFEDDF4D,0xDD3BA9EE,0x20A00000 |
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.long 0x40030000,0xBC7EDCF7,0xFF523611,0x21680000 |
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.long 0x40030000,0xC90FDAA2,0x2168C235,0xA1000000 |
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.long 0x40030000,0xD5A0D84C,0x437F4E58,0x1FC00000 |
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.long 0x40030000,0xE231D5F6,0x6595DA7B,0x21300000 |
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.long 0x40030000,0xEEC2D3A0,0x87AC669F,0xA1380000 |
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.long 0x40030000,0xFB53D14A,0xA9C2F2C2,0xA0000000 |
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.long 0x40040000,0x83F2677A,0x65ECBF73,0xA1C40000 |
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.long 0x40040000,0x8A3AE64F,0x76F80584,0x21880000 |
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.long 0x40040000,0x90836524,0x88034B96,0xA0B00000 |
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.long 0x40040000,0x96CBE3F9,0x990E91A8,0xA1E00000 |
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.long 0x40040000,0x9D1462CE,0xAA19D7B9,0x21580000 |
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.long 0x40040000,0xA35CE1A3,0xBB251DCB,0xA1100000 |
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.long 0x40040000,0xA9A56078,0xCC3063DD,0xA1FC0000 |
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.long 0x40040000,0xAFEDDF4D,0xDD3BA9EE,0x21200000 |
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.long 0x40040000,0xB6365E22,0xEE46F000,0xA1480000 |
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.long 0x40040000,0xBC7EDCF7,0xFF523611,0x21E80000 |
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.long 0x40040000,0xC2C75BCD,0x105D7C23,0x20D00000 |
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.long 0x40040000,0xC90FDAA2,0x2168C235,0xA1800000 |
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.set INARG,FP_SCR4 |
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.set TWOTO63,L_SCR1 |
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.set ENDFLAG,L_SCR2 |
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.set N,L_SCR3 |
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| xref t_frcinx |
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|xref t_extdnrm |
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.global stand |
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stand: |
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|--TAN(X) = X FOR DENORMALIZED X |
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bra t_extdnrm |
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.global stan |
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stan: |
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fmovex (%a0),%fp0 | ...LOAD INPUT |
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movel (%a0),%d0 |
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movew 4(%a0),%d0 |
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andil #0x7FFFFFFF,%d0 |
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cmpil #0x3FD78000,%d0 | ...|X| >= 2**(-40)? |
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bges TANOK1 |
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bra TANSM |
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TANOK1: |
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cmpil #0x4004BC7E,%d0 | ...|X| < 15 PI? |
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blts TANMAIN |
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bra REDUCEX |
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TANMAIN: |
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|--THIS IS THE USUAL CASE, |X| <= 15 PI. |
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|--THE ARGUMENT REDUCTION IS DONE BY TABLE LOOK UP. |
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fmovex %fp0,%fp1 |
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fmuld TWOBYPI,%fp1 | ...X*2/PI |
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|--HIDE THE NEXT TWO INSTRUCTIONS |
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leal PITBL+0x200,%a1 | ...TABLE OF N*PI/2, N = -32,...,32 |
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|--FP1 IS NOW READY |
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fmovel %fp1,%d0 | ...CONVERT TO INTEGER |
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asll #4,%d0 |
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addal %d0,%a1 | ...ADDRESS N*PIBY2 IN Y1, Y2 |
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fsubx (%a1)+,%fp0 | ...X-Y1 |
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|--HIDE THE NEXT ONE |
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fsubs (%a1),%fp0 | ...FP0 IS R = (X-Y1)-Y2 |
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rorl #5,%d0 |
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andil #0x80000000,%d0 | ...D0 WAS ODD IFF D0 < 0 |
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TANCONT: |
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cmpil #0,%d0 |
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blt NODD |
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fmovex %fp0,%fp1 |
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fmulx %fp1,%fp1 | ...S = R*R |
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fmoved TANQ4,%fp3 |
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fmoved TANP3,%fp2 |
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fmulx %fp1,%fp3 | ...SQ4 |
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fmulx %fp1,%fp2 | ...SP3 |
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faddd TANQ3,%fp3 | ...Q3+SQ4 |
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faddx TANP2,%fp2 | ...P2+SP3 |
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fmulx %fp1,%fp3 | ...S(Q3+SQ4) |
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fmulx %fp1,%fp2 | ...S(P2+SP3) |
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faddx TANQ2,%fp3 | ...Q2+S(Q3+SQ4) |
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faddx TANP1,%fp2 | ...P1+S(P2+SP3) |
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fmulx %fp1,%fp3 | ...S(Q2+S(Q3+SQ4)) |
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fmulx %fp1,%fp2 | ...S(P1+S(P2+SP3)) |
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faddx TANQ1,%fp3 | ...Q1+S(Q2+S(Q3+SQ4)) |
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fmulx %fp0,%fp2 | ...RS(P1+S(P2+SP3)) |
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fmulx %fp3,%fp1 | ...S(Q1+S(Q2+S(Q3+SQ4))) |
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faddx %fp2,%fp0 | ...R+RS(P1+S(P2+SP3)) |
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fadds #0x3F800000,%fp1 | ...1+S(Q1+...) |
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fmovel %d1,%fpcr |restore users exceptions |
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fdivx %fp1,%fp0 |last inst - possible exception set |
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bra t_frcinx |
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NODD: |
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fmovex %fp0,%fp1 |
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fmulx %fp0,%fp0 | ...S = R*R |
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fmoved TANQ4,%fp3 |
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fmoved TANP3,%fp2 |
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fmulx %fp0,%fp3 | ...SQ4 |
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fmulx %fp0,%fp2 | ...SP3 |
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faddd TANQ3,%fp3 | ...Q3+SQ4 |
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faddx TANP2,%fp2 | ...P2+SP3 |
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fmulx %fp0,%fp3 | ...S(Q3+SQ4) |
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fmulx %fp0,%fp2 | ...S(P2+SP3) |
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faddx TANQ2,%fp3 | ...Q2+S(Q3+SQ4) |
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faddx TANP1,%fp2 | ...P1+S(P2+SP3) |
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fmulx %fp0,%fp3 | ...S(Q2+S(Q3+SQ4)) |
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fmulx %fp0,%fp2 | ...S(P1+S(P2+SP3)) |
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faddx TANQ1,%fp3 | ...Q1+S(Q2+S(Q3+SQ4)) |
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fmulx %fp1,%fp2 | ...RS(P1+S(P2+SP3)) |
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fmulx %fp3,%fp0 | ...S(Q1+S(Q2+S(Q3+SQ4))) |
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faddx %fp2,%fp1 | ...R+RS(P1+S(P2+SP3)) |
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fadds #0x3F800000,%fp0 | ...1+S(Q1+...) |
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fmovex %fp1,-(%sp) |
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eoril #0x80000000,(%sp) |
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fmovel %d1,%fpcr |restore users exceptions |
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fdivx (%sp)+,%fp0 |last inst - possible exception set |
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bra t_frcinx |
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TANBORS: |
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|--IF |X| > 15PI, WE USE THE GENERAL ARGUMENT REDUCTION. |
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|--IF |X| < 2**(-40), RETURN X OR 1. |
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cmpil #0x3FFF8000,%d0 |
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bgts REDUCEX |
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TANSM: |
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fmovex %fp0,-(%sp) |
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fmovel %d1,%fpcr |restore users exceptions |
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fmovex (%sp)+,%fp0 |last inst - possible exception set |
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bra t_frcinx |
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REDUCEX: |
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|--WHEN REDUCEX IS USED, THE CODE WILL INEVITABLY BE SLOW. |
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|--THIS REDUCTION METHOD, HOWEVER, IS MUCH FASTER THAN USING |
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|--THE REMAINDER INSTRUCTION WHICH IS NOW IN SOFTWARE. |
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fmovemx %fp2-%fp5,-(%a7) | ...save FP2 through FP5 |
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movel %d2,-(%a7) |
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fmoves #0x00000000,%fp1 |
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|--If compact form of abs(arg) in d0=$7ffeffff, argument is so large that |
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|--there is a danger of unwanted overflow in first LOOP iteration. In this |
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|--case, reduce argument by one remainder step to make subsequent reduction |
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|--safe. |
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cmpil #0x7ffeffff,%d0 |is argument dangerously large? |
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bnes LOOP |
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movel #0x7ffe0000,FP_SCR2(%a6) |yes |
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| ;create 2**16383*PI/2 |
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movel #0xc90fdaa2,FP_SCR2+4(%a6) |
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clrl FP_SCR2+8(%a6) |
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ftstx %fp0 |test sign of argument |
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movel #0x7fdc0000,FP_SCR3(%a6) |create low half of 2**16383* |
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| ;PI/2 at FP_SCR3 |
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movel #0x85a308d3,FP_SCR3+4(%a6) |
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clrl FP_SCR3+8(%a6) |
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fblt red_neg |
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orw #0x8000,FP_SCR2(%a6) |positive arg |
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orw #0x8000,FP_SCR3(%a6) |
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red_neg: |
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faddx FP_SCR2(%a6),%fp0 |high part of reduction is exact |
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fmovex %fp0,%fp1 |save high result in fp1 |
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faddx FP_SCR3(%a6),%fp0 |low part of reduction |
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fsubx %fp0,%fp1 |determine low component of result |
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faddx FP_SCR3(%a6),%fp1 |fp0/fp1 are reduced argument. |
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|--ON ENTRY, FP0 IS X, ON RETURN, FP0 IS X REM PI/2, |X| <= PI/4. |
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|--integer quotient will be stored in N |
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|--Intermediate remainder is 66-bit long; (R,r) in (FP0,FP1) |
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LOOP: |
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fmovex %fp0,INARG(%a6) | ...+-2**K * F, 1 <= F < 2 |
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movew INARG(%a6),%d0 |
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movel %d0,%a1 | ...save a copy of D0 |
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andil #0x00007FFF,%d0 |
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subil #0x00003FFF,%d0 | ...D0 IS K |
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cmpil #28,%d0 |
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bles LASTLOOP |
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CONTLOOP: |
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subil #27,%d0 | ...D0 IS L := K-27 |
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movel #0,ENDFLAG(%a6) |
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bras WORK |
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LASTLOOP: |
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clrl %d0 | ...D0 IS L := 0 |
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movel #1,ENDFLAG(%a6) |
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WORK: |
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|--FIND THE REMAINDER OF (R,r) W.R.T. 2**L * (PI/2). L IS SO CHOSEN |
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|--THAT INT( X * (2/PI) / 2**(L) ) < 2**29. |
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|--CREATE 2**(-L) * (2/PI), SIGN(INARG)*2**(63), |
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|--2**L * (PIby2_1), 2**L * (PIby2_2) |
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movel #0x00003FFE,%d2 | ...BIASED EXPO OF 2/PI |
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subl %d0,%d2 | ...BIASED EXPO OF 2**(-L)*(2/PI) |
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movel #0xA2F9836E,FP_SCR1+4(%a6) |
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movel #0x4E44152A,FP_SCR1+8(%a6) |
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movew %d2,FP_SCR1(%a6) | ...FP_SCR1 is 2**(-L)*(2/PI) |
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fmovex %fp0,%fp2 |
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fmulx FP_SCR1(%a6),%fp2 |
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|--WE MUST NOW FIND INT(FP2). SINCE WE NEED THIS VALUE IN |
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|--FLOATING POINT FORMAT, THE TWO FMOVE'S FMOVE.L FP <--> N |
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|--WILL BE TOO INEFFICIENT. THE WAY AROUND IT IS THAT |
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|--(SIGN(INARG)*2**63 + FP2) - SIGN(INARG)*2**63 WILL GIVE |
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|--US THE DESIRED VALUE IN FLOATING POINT. |
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|--HIDE SIX CYCLES OF INSTRUCTION |
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movel %a1,%d2 |
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swap %d2 |
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andil #0x80000000,%d2 |
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oril #0x5F000000,%d2 | ...D2 IS SIGN(INARG)*2**63 IN SGL |
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movel %d2,TWOTO63(%a6) |
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movel %d0,%d2 |
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addil #0x00003FFF,%d2 | ...BIASED EXPO OF 2**L * (PI/2) |
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|--FP2 IS READY |
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fadds TWOTO63(%a6),%fp2 | ...THE FRACTIONAL PART OF FP1 IS ROUNDED |
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|--HIDE 4 CYCLES OF INSTRUCTION; creating 2**(L)*Piby2_1 and 2**(L)*Piby2_2 |
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movew %d2,FP_SCR2(%a6) |
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clrw FP_SCR2+2(%a6) |
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movel #0xC90FDAA2,FP_SCR2+4(%a6) |
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clrl FP_SCR2+8(%a6) | ...FP_SCR2 is 2**(L) * Piby2_1 |
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|--FP2 IS READY |
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fsubs TWOTO63(%a6),%fp2 | ...FP2 is N |
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addil #0x00003FDD,%d0 |
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movew %d0,FP_SCR3(%a6) |
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clrw FP_SCR3+2(%a6) |
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movel #0x85A308D3,FP_SCR3+4(%a6) |
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clrl FP_SCR3+8(%a6) | ...FP_SCR3 is 2**(L) * Piby2_2 |
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movel ENDFLAG(%a6),%d0 |
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|--We are now ready to perform (R+r) - N*P1 - N*P2, P1 = 2**(L) * Piby2_1 and |
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|--P2 = 2**(L) * Piby2_2 |
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fmovex %fp2,%fp4 |
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fmulx FP_SCR2(%a6),%fp4 | ...W = N*P1 |
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fmovex %fp2,%fp5 |
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fmulx FP_SCR3(%a6),%fp5 | ...w = N*P2 |
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fmovex %fp4,%fp3 |
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|--we want P+p = W+w but |p| <= half ulp of P |
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|--Then, we need to compute A := R-P and a := r-p |
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faddx %fp5,%fp3 | ...FP3 is P |
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fsubx %fp3,%fp4 | ...W-P |
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|
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fsubx %fp3,%fp0 | ...FP0 is A := R - P |
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faddx %fp5,%fp4 | ...FP4 is p = (W-P)+w |
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|
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fmovex %fp0,%fp3 | ...FP3 A |
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fsubx %fp4,%fp1 | ...FP1 is a := r - p |
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|
|
|--Now we need to normalize (A,a) to "new (R,r)" where R+r = A+a but |
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|--|r| <= half ulp of R. |
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faddx %fp1,%fp0 | ...FP0 is R := A+a |
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|--No need to calculate r if this is the last loop |
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cmpil #0,%d0 |
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bgt RESTORE |
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|
|
|--Need to calculate r |
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fsubx %fp0,%fp3 | ...A-R |
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faddx %fp3,%fp1 | ...FP1 is r := (A-R)+a |
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bra LOOP |
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|
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RESTORE: |
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fmovel %fp2,N(%a6) |
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movel (%a7)+,%d2 |
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fmovemx (%a7)+,%fp2-%fp5 |
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|
|
|
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movel N(%a6),%d0 |
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rorl #1,%d0 |
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|
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|
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bra TANCONT |
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|
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|end
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