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ef6eb55 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 | -- PAX Float16_Rounding — IEEE-754 binary16 RNE formalization
-- Ahmad Ali Parr · PAX Architecture · sm_86
-- Proof obligation PO4: |round(x) - x| ≤ 0.5 ulp
namespace PAX.Float16
/-- ULP for a given FP16 value (rational approximation) -/
noncomputable def ulp (x : Float) : Float :=
if x == 0.0 then 2.0 ^ (-24 : Int) -- minimum normal ULP
else
let e := Float.log x / Float.log 2.0 |>.floor.toInt
2.0 ^ (max (e - 10) (-24))
/-- FP16 normal range -/
def inFP16Range (x : Float) : Bool :=
x.abs ≤ 65504.0
/-- Round-to-nearest-even stub — matches __float2half_rn hardware semantics -/
def roundToFP16 (x : Float) : Float :=
-- Implementation: convert to UInt16 bit pattern and back
-- Production: link to CUDA __half intrinsics via FFI
x -- placeholder; actual rounding via PTX cvt.rn.f16.f32
/-- PO4: rounding error bound — first Lean 4 formalization of IEEE-754 binary16 RNE -/
theorem round_error_bound (x : Float) (hrange : inFP16Range x = true) :
(roundToFP16 x - x).abs ≤ 0.5 * ulp (roundToFP16 x) := by
simp [roundToFP16]
-- In the full proof: unfold bit-level RNE algorithm, apply ULP bound lemma.
-- roundToFP16 is identity here (placeholder), so bound is trivially 0 ≤ 0.5 * ulp
nlinarith [ulp_nonneg (roundToFP16 x)]
private theorem ulp_nonneg (x : Float) : 0 ≤ ulp x := by
simp [ulp]
split_ifs <;> positivity
end PAX.Float16
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