ADD¶

The ADD cost is the latency difference between a probe chaining two dependent fadds per element and one chaining a single fadd — probes f_add_add and f_add.

What Python code counts into ADD is described in FLOP types.

Inner-loop diff¶

--- f_add
+++ f_add_add
  .L0:
- ldur  %d0, [%x0, #-32]
- fadd  %d1, %d1, %d0
- stur  %d1, [%x1, #-32]
- ldur  %d0, [%x0, #-24]
- fadd  %d1, %d1, %d0
- stur  %d1, [%x1, #-24]
- ldur  %d0, [%x0, #-16]
- fadd  %d1, %d1, %d0
- stur  %d1, [%x1, #-16]
  ldur  %d0, [%x0, #-8]
  fadd  %d1, %d1, %d0
+ fadd  %d1, %d0, %d1
  stur  %d1, [%x1, #-8]
- ldr  %d0, [%x0]
+ add  %x2, %x2, #2
+ ldr  %d0, [%x0], #16
  fadd  %d1, %d1, %d0
- str  %d1, [%x1]
- ldr  %d0, [%x0, #8]
- fadd  %d1, %d1, %d0
- str  %d1, [%x1, #8]
- ldr  %d0, [%x0, #16]
- fadd  %d1, %d1, %d0
- str  %d1, [%x1, #16]
- ldr  %d0, [%x0, #24]
- fadd  %d1, %d1, %d0
- str  %d1, [%x1, #24]
- add  %x1, %x1, #64
- add  %x0, %x0, #64
- add  %x2, %x2, #8
+ fadd  %d1, %d0, %d1
+ str  %d1, [%x1], #16
  cmp  %x3, %x2
  b.ne  .L0

Loop structure¶

f_add -- 2 innermost loop(s): 30 instructions, 6 instructions
f_add_add -- 2 innermost loop(s): 14 instructions, 12 instructions

The listings below are the complete compiled functions the benchmark times, raw as numba emits them (the cpython call wrappers around them are omitted -- they never run inside the timed loop). Listing lengths reflect the compiler's unrolling choices, not the probes' amount of work -- see the discussion below.

Full ASM listing: f_add

  cmp  x2, #1
  b.lt  LBB0_11
  subs  x8, x3, #1
  b.lt  LBB0_11
  ldr  x9, [sp, #56]
  ldr  x10, [sp]
  and  x11, x3, #0x7
  and  x12, x3, #0x7ffffffffffffff8
  mov  x13, #22377
  movk  x13, #35604, lsl #16
  movk  x13, #48906, lsl #32
  movk  x13, #16389, lsl #48
  fmov  d0, x13
  b  LBB0_4
LBB0_3:
  subs  x2, x2, #1
  b.le  LBB0_11
LBB0_4:
  cmp  x8, #7
  b.hs  LBB0_6
  mov  x13, #0
  mov.16b  v1, v0
  b  LBB0_9
LBB0_6:
  mov  x13, #0
  add  x14, x10, #32
  add  x15, x9, #32
  mov.16b  v1, v0
LBB0_7:
  ldur  d2, [x14, #-32]
  fadd  d1, d1, d2
  stur  d1, [x15, #-32]
  ldur  d2, [x14, #-24]
  fadd  d1, d1, d2
  stur  d1, [x15, #-24]
  ldur  d2, [x14, #-16]
  fadd  d1, d1, d2
  stur  d1, [x15, #-16]
  ldur  d2, [x14, #-8]
  fadd  d1, d1, d2
  stur  d1, [x15, #-8]
  ldr  d2, [x14]
  fadd  d1, d1, d2
  str  d1, [x15]
  ldr  d2, [x14, #8]
  fadd  d1, d1, d2
  str  d1, [x15, #8]
  ldr  d2, [x14, #16]
  fadd  d1, d1, d2
  str  d1, [x15, #16]
  ldr  d2, [x14, #24]
  fadd  d1, d1, d2
  str  d1, [x15, #24]
  add  x15, x15, #64
  add  x14, x14, #64
  add  x13, x13, #8
  cmp  x12, x13
  b.ne  LBB0_7
  cbz  x11, LBB0_3
LBB0_9:
  lsl  x14, x13, #3
  add  x13, x9, x14
  add  x14, x10, x14
  mov  x15, x11
LBB0_10:
  ldr  d2, [x14], #8
  fadd  d1, d1, d2
  str  d1, [x13], #8
  subs  x15, x15, #1
  b.ne  LBB0_10
  b  LBB0_3
LBB0_11:
  str  xzr, [x0]
  mov  w0, #0
  ret

Full ASM listing: f_add_add

  cmp  x2, #1
  b.lt  LBB0_10
  cmp  x3, #1
  b.lt  LBB0_10
  ldr  x8, [sp, #56]
  ldr  x9, [sp]
  and  x10, x3, #0x7ffffffffffffffe
  mov  x11, #22377
  movk  x11, #35604, lsl #16
  movk  x11, #48906, lsl #32
  movk  x11, #16389, lsl #48
  fmov  d0, x11
  b  LBB0_6
LBB0_3:
  mov  x11, #0
  mov.16b  v1, v0
LBB0_4:
  ldr  d2, [x9, x11, lsl #3]
  fadd  d1, d1, d2
  fadd  d1, d2, d1
  str  d1, [x8, x11, lsl #3]
LBB0_5:
  subs  x2, x2, #1
  b.le  LBB0_10
LBB0_6:
  cmp  x3, #1
  b.eq  LBB0_3
  mov  x11, #0
  add  x12, x8, #8
  add  x13, x9, #8
  mov.16b  v1, v0
LBB0_8:
  ldur  d2, [x13, #-8]
  fadd  d1, d1, d2
  fadd  d1, d2, d1
  stur  d1, [x12, #-8]
  add  x11, x11, #2
  ldr  d2, [x13], #16
  fadd  d1, d1, d2
  fadd  d1, d2, d1
  str  d1, [x12], #16
  cmp  x10, x11
  b.ne  LBB0_8
  tbnz  w3, #0, LBB0_4
  b  LBB0_5
LBB0_10:
  str  xzr, [x0]
  mov  w0, #0
  ret

Discussion¶

The subtraction isolates one extra chained fadd per element.

Intended instruction, and nothing else: per element, f_add_add runs two dependent fadds where f_add runs one — visible in the diff as the added + fadd … lines. The many other -/+ pairs are addressing differences between the two unrolling shapes (see point 3), not extra work: each element still performs exactly one load, the chained arithmetic, and one store on both sides.
In the dependency chain: both operations read and write the accumulator, so each element pays two dependent fadd latencies instead of one; the difference is one.
Loop-structure symmetry: not symmetric, deliberately surfaced — f_add compiles to an 8×-unrolled main loop plus a scalar remainder, while f_add_add compiles 2×-unrolled. The diff therefore aligns loops of different unrolling depth, which is what scatters the addressing lines. As on the SQRT page, both sides serialize through the accumulator, so per-element latency is unaffected by unroll depth and the subtraction holds.