$p0 sub b32 $r1 $r1 $r2
$p0 add b32 $r0 $r0 0x1
$p3 cvt s32 $r0 neg s32 $r0
- sched 0x04 0x2e 0x04 0x28 0x04 0x20 0x2c
+ sched 0x04 0x2e 0x28 0x04 0x28 0x28 0x28
$p2 cvt s32 $r1 neg s32 $r1
ret
+// 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).
+//
gk110_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
+ ext u32 $r2 $r1 0xb14
+ add b32 $r3 $r2 0xffffffff
+ joinat #rcp_rejoin
+ // We want to check whether the exponent is 0 or 0x7ff (i.e. NaN, inf,
+ // denorm, or 0). Do this by substracting 1 from the exponent, which will
+ // mean that it's > 0x7fd in those cases when doing unsigned comparison
+ set b32 $p0 0x1 gt u32 $r3 0x7fd
+ // $r3: 0 for norms, 0x36 for denorms, -1 for others
+ mov b32 $r3 0x0
+ sched 0x2f 0x04 0x2d 0x2b 0x2f 0x28 0x28
+ join (not $p0) nop
+ // Process all special values: NaN, inf, denorm, 0
+ mov b32 $r3 0xffffffff
+ // A number is NaN if its abs value is greater than or unordered with inf
+ set $p0 0x1 gtu f64 abs $r0d 0x7ff0000000000000
+ (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
+ join or b32 $r1 $r1 0x80000
+rcp_inf_or_denorm_or_zero:
+ and b32 $r4 $r1 0x7ff00000
+ // Other values with nonzero in exponent field should be inf
+ set b32 $p0 0x1 eq s32 $r4 0x0
+ sched 0x2b 0x04 0x2f 0x2d 0x2b 0x2f 0x20
+ $p0 bra #rcp_denorm_or_zero
+ // +/-Inf -> +/-0
+ xor b32 $r1 $r1 0x7ff00000
+ join mov b32 $r0 0x0
+rcp_denorm_or_zero:
+ set $p0 0x1 gtu f64 abs $r0d 0x0
+ $p0 bra #rcp_denorm
+ // +/-0 -> +/-Inf
+ join or b32 $r1 $r1 0x7ff00000
+rcp_denorm:
+ // non-0 denorms: multiply with 2^54 (the 0x36 in $r3), join with norms
+ mul rn f64 $r0d $r0d 0x4350000000000000
+ sched 0x2f 0x28 0x2b 0x28 0x28 0x04 0x28
+ join mov b32 $r3 0x36
+rcp_rejoin:
+ // All numbers with -1 in $r3 have their result ready in $r0d, return them
+ // others need further calculation
+ set b32 $p0 0x1 lt s32 $r3 0x0
+ $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.
+ ext u32 $r2 $r1 0xb14
+ and b32 $r7 $r1 0x800fffff
+ add b32 $r7 $r7 0x3ff00000
+ mov b32 $r6 $r0
+ sched 0x2b 0x04 0x28 0x28 0x2a 0x2b 0x2e
+ // Step 3: Convert new value to float (no overflow will occur due to step
+ // 2), calculate rcp and do newton-raphson step once
+ cvt rz f32 $r5 f64 $r6d
+ rcp f32 $r4 $r5
+ mov b32 $r0 0xbf800000
+ fma rn f32 $r5 $r4 $r5 $r0
+ fma rn f32 $r0 neg $r4 $r5 $r4
+ // Step 4: convert result $r0 back to double, do newton-raphson steps
+ cvt f64 $r0d f32 $r0
+ cvt f64 $r6d f64 neg $r6d
+ sched 0x2e 0x29 0x29 0x29 0x29 0x29 0x29
+ cvt f64 $r8d f32 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}
+ fma rn f64 $r4d $r6d $r0d $r8d
+ fma rn f64 $r0d $r0d $r4d $r0d
+ fma rn f64 $r4d $r6d $r0d $r8d
+ fma rn f64 $r0d $r0d $r4d $r0d
+ fma rn f64 $r4d $r6d $r0d $r8d
+ fma rn f64 $r0d $r0d $r4d $r0d
+ sched 0x29 0x20 0x28 0x28 0x28 0x28 0x28
+ fma rn f64 $r4d $r6d $r0d $r8d
+ fma rn f64 $r0d $r0d $r4d $r0d
+ // 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
+ subr b32 $r2 $r2 0x3ff
+ add b32 $r4 $r2 $r3
+ ext u32 $r3 $r1 0xb14
+ // New exponent in $r3
+ add b32 $r3 $r3 $r4
+ add b32 $r2 $r3 0xffffffff
+ sched 0x28 0x2b 0x28 0x2b 0x28 0x28 0x2b
+ // (exponent-1) < 0x7fe (unsigned) means the result is in norm range
+ // (same logic as in step 1)
+ set b32 $p0 0x1 lt u32 $r2 0x7fe
+ (not $p0) bra #rcp_result_inf_or_denorm
+ // Norms: convert exponents back and return
+ shl b32 $r4 $r4 clamp 0x14
+ add b32 $r1 $r4 $r1
+ bra #rcp_end
+rcp_result_inf_or_denorm:
+ // New exponent >= 0x7ff means that result is inf
+ set b32 $p0 0x1 ge s32 $r3 0x7ff
+ (not $p0) bra #rcp_result_denorm
+ sched 0x20 0x25 0x28 0x2b 0x23 0x25 0x2f
+ // Infinity
+ and b32 $r1 $r1 0x80000000
+ mov b32 $r0 0x0
+ add b32 $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.
+ set b32 $p0 0x1 ne u32 $r3 0x0
+ and b32 $r1 $r1 0x800fffff
+ // 0x3e800000: 1/4
+ $p0 cvt f64 $r6d f32 0x3e800000
+ sched 0x2f 0x28 0x2c 0x2e 0x2e 0x00 0x00
+ // 0x3f000000: 1/2
+ (not $p0) cvt f64 $r6d f32 0x3f000000
+ add b32 $r1 $r1 0x00100000
+ mul rn f64 $r0d $r0d $r6d
+rcp_end:
+ ret
+
gk110_rsq_f64:
ret
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