CBRT¶

The CBRT cost is the latency difference between a probe chaining cbrt(tmp + x[i]) and one chaining only tmp + x[i] — probes f_add_cbrt and f_add. A libm call with one wrinkle: numba's own route (np.cbrt) wraps the libm call in a NaN check and negative-argument handling that CPython's math.cbrt never executes, so the probe calls libm's cbrt through a ctypes binding instead — the bare call CPython executes, compiling to the same indirect-call loop shape as the REMAINDER probe.

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

Inner-loop diff¶

--- f_add
+++ f_add_cbrt
  .L0:
- ldr  %d0, [%x0], #8
+ ldr  %x0, [%x1]
+ ldr  %d0, [%x2], #8
  fadd  %d1, %d1, %d0
- str  %d1, [%x1], #8
- subs  %x2, %x2, #1
+ blr  %x0
+ str  %d1, [%x3], #8
+ subs  %x4, %x4, #1
  b.ne  .L0

Loop structure¶

f_add -- 2 innermost loop(s): 30 instructions, 6 instructions
f_add_cbrt -- 1 innermost loop(s): 8 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_cbrt

  .cfi_startproc
  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]
  .cfi_def_cfa_offset 112
  .cfi_offset w30, -8
  .cfi_offset w29, -16
  .cfi_offset w19, -24
  .cfi_offset w20, -32
  .cfi_offset w21, -40
  .cfi_offset w22, -48
  .cfi_offset w23, -56
  .cfi_offset w24, -64
  .cfi_offset w25, -72
  .cfi_offset w26, -80
  .cfi_offset w27, -88
  .cfi_offset w28, -96
  .cfi_offset b8, -104
  .cfi_offset b9, -112
  mov  x19, x0
  cmp  x2, #1
  b.lt  LBB0_6
  mov  x20, x3
  cmp  x3, #1
  b.lt  LBB0_6
  mov  x21, x2
  ldr  x22, [sp, #168]
  ldr  x23, [sp, #112]
  mov  x8, #22377
  movk  x8, #35604, lsl #16
  movk  x8, #48906, lsl #32
  movk  x8, #16389, lsl #48
  fmov  d8, x8
Lloh0:
  adrp  x24, _numba.dynamic.globals.<addr>@GOTPAGE
Lloh1:
  ldr  x24, [x24, _numba.dynamic.globals.<addr>@GOTPAGEOFF]
LBB0_3:
  mov  x25, x20
  mov  x26, x23
  mov  x27, x22
  mov.16b  v0, v8
LBB0_4:
  ldr  x8, [x24]
  ldr  d1, [x26], #8
  fadd  d0, d0, d1
  blr  x8
  str  d0, [x27], #8
  subs  x25, x25, #1
  b.ne  LBB0_4
  subs  x21, x21, #1
  b.gt  LBB0_3
LBB0_6:
  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
  .cfi_endproc

Discussion¶

The subtraction isolates exactly one call to libm's cbrt.

Intended call, and nothing else: the structural additions are the blr — the cbrt call — plus one integer load per iteration re-fetching the call-target pointer from the ctypes closure (the libm probes numba compiles directly keep their target in a register; the ctypes route reloads it). That load is a stride-0 L1 hit on the integer side. The remaining -/+ pairs are the canonical-index shift described on the index page. No NaN check and no sign branches: numba's np.cbrt would add those around the call, which is exactly why the probe binds libm directly — CPython's math.cbrt executes the bare call.
In the dependency chain: the accumulator flows through the call — fadd produces the argument, the call returns the result the next iteration's fadd consumes. The pointer reload is integer-side work that runs off the chain, overlapping earlier latency rather than extending it.
Loop-structure symmetry: same asymmetry as the SQRT page — f_add unrolls 8×, f_add_cbrt does not; the diff shows f_add's scalar remainder loop. As there, both sides stay latency-bound through the accumulator chain, so the subtraction holds.