ATAN2

The ATAN2 cost is the latency difference between a probe chaining atan2(tmp + x[i], x[i]) and one chaining only tmp + x[i] — probes f_add_atan2 and f_add. A libm call with two arguments: the chained value and the freshly loaded element. The probe's moderate input range keeps most magnitudes above 1, matching atan's general-case regime and the trig family's shared range; atan2's own cost is flat from there through huge arguments.

What Python code counts into ATAN2 is described in FLOP types.

Inner-loop diff

--- f_add
+++ f_add_atan2
  .L0:
- ldr  %d0, [%x0], #8
+ ldur  %d0, [%x0, #-8]
  fadd  %d1, %d1, %d0
- str  %d1, [%x1], #8
- subs  %x2, %x2, #1
+ blr  %x1
+ stur  %d1, [%x2, #-8]
+ add  %x3, %x3, #2
+ ldr  %d0, [%x0], #16
+ fadd  %d1, %d1, %d0
+ blr  %x1
+ str  %d1, [%x2], #16
+ cmp  %x4, %x3
  b.ne  .L0

Loop structure

• f_add -- 2 innermost loop(s): 30 instructions, 6 instructions
• f_add_atan2 -- 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_atan2

  stp  d9, d8, [sp, #-112]!
  stp  x28, x27, [sp, #16]
  stp  x26, x25, [sp, #32]
  stp  x24, x23, [sp, #48]
  stp  x22, x21, [sp, #64]
  stp  x20, x19, [sp, #80]
  stp  x29, x30, [sp, #96]
  mov  x19, x0
  cmp  x2, #1
  b.lt  LBB0_10
  mov  x20, x3
  cmp  x3, #1
  b.lt  LBB0_10
  mov  x21, x2
  ldr  x22, [sp, #168]
  ldr  x23, [sp, #112]
  and  x24, x20, #0x7ffffffffffffffe
  mov  x8, #22377
  movk  x8, #35604, lsl #16
  movk  x8, #48906, lsl #32
  movk  x8, #16389, lsl #48
  fmov  d8, x8
Lloh0:
  adrp  x25, _atan2@GOTPAGE
Lloh1:
  ldr  x25, [x25, _atan2@GOTPAGEOFF]
  b  LBB0_6
LBB0_3:
  mov  x26, #0
  mov.16b  v0, v8
LBB0_4:
  ldr  d1, [x23, x26, lsl #3]
  fadd  d0, d0, d1
  blr  x25
  str  d0, [x22, x26, lsl #3]
LBB0_5:
  subs  x21, x21, #1
  b.le  LBB0_10
LBB0_6:
  cmp  x20, #1
  b.eq  LBB0_3
  mov  x26, #0
  add  x27, x22, #8
  add  x28, x23, #8
  mov.16b  v0, v8
LBB0_8:
  ldur  d1, [x28, #-8]
  fadd  d0, d0, d1
  blr  x25
  stur  d0, [x27, #-8]
  add  x26, x26, #2
  ldr  d1, [x28], #16
  fadd  d0, d0, d1
  blr  x25
  str  d0, [x27], #16
  cmp  x24, x26
  b.ne  LBB0_8
  tbnz  w20, #0, LBB0_4
  b  LBB0_5
LBB0_10:
  str  xzr, [x19]
  mov  w0, #0
  ldp  x29, x30, [sp, #96]
  ldp  x20, x19, [sp, #80]
  ldp  x22, x21, [sp, #64]
  ldp  x24, x23, [sp, #48]
  ldp  x26, x25, [sp, #32]
  ldp  x28, x27, [sp, #16]
  ldp  d9, d8, [sp], #112
  ret
  .loh AdrpLdrGot  Lloh0, Lloh1

Discussion

The subtraction isolates exactly one call to libm's atan2 per element.

1. Intended call, and nothing else: the structural addition is one blr per element — the atan2 call through a preloaded call-target register. The remaining -/+ pairs are addressing differences between the two unrolling shapes (see point 3), not extra work.
2. In the dependency chain: fadd produces the first argument, the call's return value feeds the next iteration's fadd; the second argument is the freshly loaded element, off the chain.
3. Loop-structure symmetry: not symmetric — f_add unrolls 8×, and f_add_atan2 compiles 2×-unrolled (one call per element either way). As on the SQRT page, both sides stay latency-bound through the accumulator, so the subtraction holds.