ABS

The ABS cost is the latency difference between a probe chaining abs(tmp + x[i]) and one chaining only tmp + x[i] — probes f_add_abs and f_add. On ARM64 this compiles to a single fabs instruction, not a library call.

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

Inner-loop diff

--- f_add
+++ f_add_abs
  .L0:
  ldr  %d0, [%x0], #8
  fadd  %d1, %d1, %d0
+ fabs  %d1, %d1
  str  %d1, [%x1], #8
  subs  %x2, %x2, #1
  b.ne  .L0

Loop structure

• f_add -- 2 innermost loop(s): 30 instructions, 6 instructions
• f_add_abs -- 1 innermost loop(s): 7 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_abs

  cmp  x2, #1
  b.lt  LBB0_6
  cmp  x3, #1
  b.lt  LBB0_6
  ldr  x8, [sp, #56]
  ldr  x9, [sp]
  mov  x10, #22377
  movk  x10, #35604, lsl #16
  movk  x10, #48906, lsl #32
  movk  x10, #16389, lsl #48
  fmov  d0, x10
LBB0_3:
  mov  x10, x3
  mov  x11, x9
  mov  x12, x8
  mov.16b  v1, v0
LBB0_4:
  ldr  d2, [x11], #8
  fadd  d1, d1, d2
  fabs  d1, d1
  str  d1, [x12], #8
  subs  x10, x10, #1
  b.ne  LBB0_4
  subs  x2, x2, #1
  b.gt  LBB0_3
LBB0_6:
  str  xzr, [x0]
  mov  w0, #0
  ret

Discussion

The subtraction isolates exactly one fabs.

1. Intended instruction, and nothing else: the diff adds the single line + fabs … — loads, stores and loop control are otherwise identical.
2. In the dependency chain: fabs reads and writes the accumulator that feeds the next iteration's fadd, so each iteration waits for the full chain.
3. Loop-structure symmetry: same asymmetry as the SQRT page — f_add unrolls 8×, f_add_abs does not; the diff shows f_add's scalar remainder. As there, both sides stay latency-bound through the accumulator chain, so the subtraction holds.