.section #gm107_builtin_code // DIV U32 // // UNR recurrence (q = a / b): // look for z such that 2^32 - b <= b * z < 2^32 // then q - 1 <= (a * z) / 2^32 <= q // // INPUT: $r0: dividend, $r1: divisor // OUTPUT: $r0: result, $r1: modulus // CLOBBER: $r2 - $r3, $p0 - $p1 // SIZE: 22 / 14 * 8 bytes // gm107_div_u32: sched (st 0xd wr 0x0 wt 0x3f) (st 0x1 wt 0x1) (st 0x6) flo u32 $r2 $r1 lop xor 1 $r2 $r2 0x1f mov $r3 0x1 0xf sched (st 0x1) (st 0xf wr 0x0) (st 0x6 wr 0x0 wt 0x1) shl $r2 $r3 $r2 i2i u32 u32 $r1 neg $r1 imul u32 u32 $r3 $r1 $r2 sched (st 0x6 wr 0x0 wt 0x1) (st 0x6 wr 0x0 wt 0x1) (st 0x6 wr 0x0 wt 0x1) imad u32 u32 hi $r2 $r2 $r3 $r2 imul u32 u32 $r3 $r1 $r2 imad u32 u32 hi $r2 $r2 $r3 $r2 sched (st 0x6 wr 0x0 wt 0x1) (st 0x6 wr 0x0 wt 0x1) (st 0x6 wr 0x0 wt 0x1) imul u32 u32 $r3 $r1 $r2 imad u32 u32 hi $r2 $r2 $r3 $r2 imul u32 u32 $r3 $r1 $r2 sched (st 0x6 wr 0x0 wt 0x1) (st 0x6 wr 0x0 wt 0x1) (st 0x6 wr 0x0 rd 0x1 wt 0x1) imad u32 u32 hi $r2 $r2 $r3 $r2 imul u32 u32 $r3 $r1 $r2 imad u32 u32 hi $r2 $r2 $r3 $r2 sched (st 0x6 wt 0x2) (st 0x6 wr 0x0 rd 0x1 wt 0x1) (st 0xf wr 0x0 rd 0x1 wt 0x2) mov $r3 $r0 0xf imul u32 u32 hi $r0 $r0 $r2 i2i u32 u32 $r2 neg $r1 sched (st 0x6 wr 0x0 wt 0x3) (st 0xd wt 0x1) (st 0x1) imad u32 u32 $r1 $r1 $r0 $r3 isetp ge u32 and $p0 1 $r1 $r2 1 $p0 iadd $r1 $r1 neg $r2 sched (st 0x5) (st 0xd) (st 0x1) $p0 iadd $r0 $r0 0x1 $p0 isetp ge u32 and $p0 1 $r1 $r2 1 $p0 iadd $r1 $r1 neg $r2 sched (st 0x1) (st 0xf) (st 0xf) $p0 iadd $r0 $r0 0x1 ret nop 0 // DIV S32, like DIV U32 after taking ABS(inputs) // // INPUT: $r0: dividend, $r1: divisor // OUTPUT: $r0: result, $r1: modulus // CLOBBER: $r2 - $r3, $p0 - $p3 // gm107_div_s32: sched (st 0xd wt 0x3f) (st 0x1) (st 0x1 wr 0x0) isetp lt and $p2 0x1 $r0 0 1 isetp lt xor $p3 1 $r1 0 $p2 i2i s32 s32 $r0 abs $r0 sched (st 0xf wr 0x1) (st 0xd wr 0x1 wt 0x2) (st 0x1 wt 0x2) i2i s32 s32 $r1 abs $r1 flo u32 $r2 $r1 lop xor 1 $r2 $r2 0x1f sched (st 0x6) (st 0x1) (st 0xf wr 0x1) mov $r3 0x1 0xf shl $r2 $r3 $r2 i2i u32 u32 $r1 neg $r1 sched (st 0x6 wr 0x1 wt 0x2) (st 0x6 wr 0x1 wt 0x2) (st 0x6 wr 0x1 wt 0x2) imul u32 u32 $r3 $r1 $r2 imad u32 u32 hi $r2 $r2 $r3 $r2 imul u32 u32 $r3 $r1 $r2 sched (st 0x6 wr 0x1 wt 0x2) (st 0x6 wr 0x1 wt 0x2) (st 0x6 wr 0x1 wt 0x2) imad u32 u32 hi $r2 $r2 $r3 $r2 imul u32 u32 $r3 $r1 $r2 imad u32 u32 hi $r2 $r2 $r3 $r2 sched (st 0x6 wr 0x1 wt 0x2) (st 0x6 wr 0x1 wt 0x2) (st 0x6 wr 0x1 wt 0x2) imul u32 u32 $r3 $r1 $r2 imad u32 u32 hi $r2 $r2 $r3 $r2 imul u32 u32 $r3 $r1 $r2 sched (st 0x6 wr 0x1 rd 0x2 wt 0x2) (st 0x2 wt 0x5) (st 0x6 wr 0x0 rd 0x1 wt 0x2) imad u32 u32 hi $r2 $r2 $r3 $r2 mov $r3 $r0 0xf imul u32 u32 hi $r0 $r0 $r2 sched (st 0xf wr 0x1 rd 0x2 wt 0x2) (st 0x6 wr 0x0 wt 0x5) (st 0xd wt 0x3) i2i u32 u32 $r2 neg $r1 imad u32 u32 $r1 $r1 $r0 $r3 isetp ge u32 and $p0 1 $r1 $r2 1 sched (st 0x1) (st 0x5) (st 0xd) $p0 iadd $r1 $r1 neg $r2 $p0 iadd $r0 $r0 0x1 $p0 isetp ge u32 and $p0 1 $r1 $r2 1 sched (st 0x1) (st 0x2) (st 0xf wr 0x0) $p0 iadd $r1 $r1 neg $r2 $p0 iadd $r0 $r0 0x1 $p3 i2i s32 s32 $r0 neg $r0 sched (st 0xf wr 0x1) (st 0xf wt 0x3) (st 0xf) $p2 i2i s32 s32 $r1 neg $r1 ret nop 0 // RCP F64 // // INPUT: $r0d // OUTPUT: $r0d // CLOBBER: $r2 - $r9, $p0 // // The core of RCP and RSQ implementation is Newton-Raphson step, which is // used to find successively better approximation from an imprecise initial // value (single precision rcp in RCP and rsqrt64h in RSQ). // gm107_rcp_f64: // Step 1: classify input according to exponent and value, and calculate // result for 0/inf/nan. $r2 holds the exponent value, which starts at // bit 52 (bit 20 of the upper half) and is 11 bits in length sched (st 0x0) (st 0x0) (st 0x0) bfe u32 $r2 $r1 0xb14 iadd32i $r3 $r2 -1 ssy #rcp_rejoin // We want to check whether the exponent is 0 or 0x7ff (i.e. NaN, inf, // denorm, or 0). Do this by subtracting 1 from the exponent, which will // mean that it's > 0x7fd in those cases when doing unsigned comparison sched (st 0x0) (st 0x0) (st 0x0) isetp gt u32 and $p0 1 $r3 0x7fd 1 // $r3: 0 for norms, 0x36 for denorms, -1 for others mov $r3 0x0 0xf not $p0 sync // Process all special values: NaN, inf, denorm, 0 sched (st 0x0) (st 0x0) (st 0x0) mov32i $r3 0xffffffff 0xf // A number is NaN if its abs value is greater than or unordered with inf dsetp gtu and $p0 1 abs $r0 0x7ff0000000000000 1 not $p0 bra #rcp_inf_or_denorm_or_zero // NaN -> NaN, the next line sets the "quiet" bit of the result. This // behavior is both seen on the CPU and the blob sched (st 0x0) (st 0x0) (st 0x0) lop32i or $r1 $r1 0x80000 sync rcp_inf_or_denorm_or_zero: lop32i and $r4 $r1 0x7ff00000 sched (st 0x0) (st 0x0) (st 0x0) // Other values with nonzero in exponent field should be inf isetp eq and $p0 1 $r4 0x0 1 $p0 bra #rcp_denorm_or_zero // +/-Inf -> +/-0 lop32i xor $r1 $r1 0x7ff00000 sched (st 0x0) (st 0x0) (st 0x0) mov $r0 0x0 0xf sync rcp_denorm_or_zero: dsetp gtu and $p0 1 abs $r0 0x0 1 sched (st 0x0) (st 0x0) (st 0x0) $p0 bra #rcp_denorm // +/-0 -> +/-Inf lop32i or $r1 $r1 0x7ff00000 sync rcp_denorm: // non-0 denorms: multiply with 2^54 (the 0x36 in $r3), join with norms sched (st 0x0) (st 0x0) (st 0x0) dmul $r0 $r0 0x4350000000000000 mov $r3 0x36 0xf sync rcp_rejoin: // All numbers with -1 in $r3 have their result ready in $r0d, return them // others need further calculation sched (st 0x0) (st 0x0) (st 0x0) isetp lt and $p0 1 $r3 0x0 1 $p0 bra #rcp_end // Step 2: Before the real calculation goes on, renormalize the values to // range [1, 2) by setting exponent field to 0x3ff (the exponent of 1) // result in $r6d. The exponent will be recovered later. bfe u32 $r2 $r1 0xb14 sched (st 0x0) (st 0x0) (st 0x0) lop32i and $r7 $r1 0x800fffff iadd32i $r7 $r7 0x3ff00000 mov $r6 $r0 0xf // Step 3: Convert new value to float (no overflow will occur due to step // 2), calculate rcp and do newton-raphson step once sched (st 0x0) (st 0x0) (st 0x0) f2f ftz f64 f32 $r5 $r6 mufu rcp $r4 $r5 mov32i $r0 0xbf800000 0xf sched (st 0x0) (st 0x0) (st 0x0) ffma $r5 $r4 $r5 $r0 ffma $r0 $r5 neg $r4 $r4 // Step 4: convert result $r0 back to double, do newton-raphson steps f2f f32 f64 $r0 $r0 sched (st 0x0) (st 0x0) (st 0x0) f2f f64 f64 $r6 neg $r6 f2f f32 f64 $r8 0x3f800000 // 4 Newton-Raphson Steps, tmp in $r4d, result in $r0d // The formula used here (and above) is: // RCP_{n + 1} = 2 * RCP_{n} - x * RCP_{n} * RCP_{n} // The following code uses 2 FMAs for each step, and it will basically // looks like: // tmp = -src * RCP_{n} + 1 // RCP_{n + 1} = RCP_{n} * tmp + RCP_{n} dfma $r4 $r6 $r0 $r8 sched (st 0x0) (st 0x0) (st 0x0) dfma $r0 $r0 $r4 $r0 dfma $r4 $r6 $r0 $r8 dfma $r0 $r0 $r4 $r0 sched (st 0x0) (st 0x0) (st 0x0) dfma $r4 $r6 $r0 $r8 dfma $r0 $r0 $r4 $r0 dfma $r4 $r6 $r0 $r8 sched (st 0x0) (st 0x0) (st 0x0) dfma $r0 $r0 $r4 $r0 // Step 5: Exponent recovery and final processing // The exponent is recovered by adding what we added to the exponent. // Suppose we want to calculate rcp(x), but we have rcp(cx), then // rcp(x) = c * rcp(cx) // The delta in exponent comes from two sources: // 1) The renormalization in step 2. The delta is: // 0x3ff - $r2 // 2) (For the denorm input) The 2^54 we multiplied at rcp_denorm, stored // in $r3 // These 2 sources are calculated in the first two lines below, and then // added to the exponent extracted from the result above. // Note that after processing, the new exponent may >= 0x7ff (inf) // or <= 0 (denorm). Those cases will be handled respectively below iadd $r2 neg $r2 0x3ff iadd $r4 $r2 $r3 sched (st 0x0) (st 0x0) (st 0x0) bfe u32 $r3 $r1 0xb14 // New exponent in $r3 iadd $r3 $r3 $r4 iadd32i $r2 $r3 -1 // (exponent-1) < 0x7fe (unsigned) means the result is in norm range // (same logic as in step 1) sched (st 0x0) (st 0x0) (st 0x0) isetp lt u32 and $p0 1 $r2 0x7fe 1 not $p0 bra #rcp_result_inf_or_denorm // Norms: convert exponents back and return shl $r4 $r4 0x14 sched (st 0x0) (st 0x0) (st 0x0) iadd $r1 $r4 $r1 bra #rcp_end rcp_result_inf_or_denorm: // New exponent >= 0x7ff means that result is inf isetp ge and $p0 1 $r3 0x7ff 1 sched (st 0x0) (st 0x0) (st 0x0) not $p0 bra #rcp_result_denorm // Infinity lop32i and $r1 $r1 0x80000000 mov $r0 0x0 0xf sched (st 0x0) (st 0x0) (st 0x0) iadd32i $r1 $r1 0x7ff00000 bra #rcp_end rcp_result_denorm: // Denorm result comes from huge input. The greatest possible fp64, i.e. // 0x7fefffffffffffff's rcp is 0x0004000000000000, 1/4 of the smallest // normal value. Other rcp result should be greater than that. If we // set the exponent field to 1, we can recover the result by multiplying // it with 1/2 or 1/4. 1/2 is used if the "exponent" $r3 is 0, otherwise // 1/4 ($r3 should be -1 then). This is quite tricky but greatly simplifies // the logic here. isetp ne u32 and $p0 1 $r3 0x0 1 sched (st 0x0) (st 0x0) (st 0x0) lop32i and $r1 $r1 0x800fffff // 0x3e800000: 1/4 $p0 f2f f32 f64 $r6 0x3e800000 // 0x3f000000: 1/2 not $p0 f2f f32 f64 $r6 0x3f000000 sched (st 0x0) (st 0x0) (st 0x0) iadd32i $r1 $r1 0x00100000 dmul $r0 $r0 $r6 rcp_end: ret // RSQ F64 // // INPUT: $r0d // OUTPUT: $r0d // CLOBBER: $r2 - $r9, $p0 - $p1 // gm107_rsq_f64: // Before getting initial result rsqrt64h, two special cases should be // handled first. // 1. NaN: set the highest bit in mantissa so it'll be surely recognized // as NaN in rsqrt64h sched (st 0xd wr 0x0 wt 0x3f) (st 0xd wt 0x1) (st 0xd) dsetp gtu and $p0 1 abs $r0 0x7ff0000000000000 1 $p0 lop32i or $r1 $r1 0x00080000 lop32i and $r2 $r1 0x7fffffff // 2. denorms and small normal values: using their original value will // lose precision either at rsqrt64h or the first step in newton-raphson // steps below. Take 2 as a threshold in exponent field, and multiply // with 2^54 if the exponent is smaller or equal. (will multiply 2^27 // to recover in the end) sched (st 0xd) (st 0xd) (st 0xd) bfe u32 $r3 $r1 0xb14 isetp le u32 and $p1 1 $r3 0x2 1 lop or 1 $r2 $r0 $r2 sched (st 0xd wr 0x0) (st 0xd wr 0x0 wt 0x1) (st 0xd) $p1 dmul $r0 $r0 0x4350000000000000 mufu rsq64h $r5 $r1 // rsqrt64h will give correct result for 0/inf/nan, the following logic // checks whether the input is one of those (exponent is 0x7ff or all 0 // except for the sign bit) iset ne u32 and $r6 $r3 0x7ff 1 sched (st 0xd) (st 0xd) (st 0xd) lop and 1 $r2 $r2 $r6 isetp ne u32 and $p0 1 $r2 0x0 1 $p0 bra #rsq_norm // For 0/inf/nan, make sure the sign bit agrees with input and return sched (st 0xd) (st 0xd) (st 0xd wt 0x1) lop32i and $r1 $r1 0x80000000 mov $r0 0x0 0xf lop or 1 $r1 $r1 $r5 sched (st 0xd) (st 0xf) (st 0xf) ret nop 0 nop 0 rsq_norm: // For others, do 4 Newton-Raphson steps with the formula: // RSQ_{n + 1} = RSQ_{n} * (1.5 - 0.5 * x * RSQ_{n} * RSQ_{n}) // In the code below, each step is written as: // tmp1 = 0.5 * x * RSQ_{n} // tmp2 = -RSQ_{n} * tmp1 + 0.5 // RSQ_{n + 1} = RSQ_{n} * tmp2 + RSQ_{n} sched (st 0xd) (st 0xd wr 0x1) (st 0xd wr 0x1 rd 0x0 wt 0x3) mov $r4 0x0 0xf // 0x3f000000: 1/2 f2f f32 f64 $r8 0x3f000000 dmul $r2 $r0 $r8 sched (st 0xd wr 0x0 wt 0x3) (st 0xd wr 0x0 wt 0x1) (st 0xd wr 0x0 wt 0x1) dmul $r0 $r2 $r4 dfma $r6 $r0 neg $r4 $r8 dfma $r4 $r4 $r6 $r4 sched (st 0xd wr 0x0 wt 0x1) (st 0xd wr 0x0 wt 0x1) (st 0xd wr 0x0 wt 0x1) dmul $r0 $r2 $r4 dfma $r6 $r0 neg $r4 $r8 dfma $r4 $r4 $r6 $r4 sched (st 0xd wr 0x0 wt 0x1) (st 0xd wr 0x0 wt 0x1) (st 0xd wr 0x0 wt 0x1) dmul $r0 $r2 $r4 dfma $r6 $r0 neg $r4 $r8 dfma $r4 $r4 $r6 $r4 sched (st 0xd wr 0x0 wt 0x1) (st 0xd wr 0x0 wt 0x1) (st 0xd wr 0x0 wt 0x1) dmul $r0 $r2 $r4 dfma $r6 $r0 neg $r4 $r8 dfma $r4 $r4 $r6 $r4 // Multiply 2^27 to result for small inputs to recover sched (st 0xd wr 0x0 wt 0x1) (st 0xd wt 0x1) (st 0xd) $p1 dmul $r4 $r4 0x41a0000000000000 mov $r1 $r5 0xf mov $r0 $r4 0xf sched (st 0xd) (st 0xf) (st 0xf) ret nop 0 nop 0 .section #gm107_builtin_offsets .b64 #gm107_div_u32 .b64 #gm107_div_s32 .b64 #gm107_rcp_f64 .b64 #gm107_rsq_f64