Theory "lift_machine_ieee"

Signature

Definitions

interval_def

⊢ ∀a b. a .. b = {x | a ≤ x ∧ x < b}

Theorems

fp64_float_sub_relative

⊢ ∀a b.
      fp64_isFinite a ∧ fp64_isFinite b ∧
      abs (fp64_to_real a − fp64_to_real b) ∈
      (1 / 2 pow 1022 .. 2 pow 2046 / 2 pow 1023 * (2 − 1 / 2 pow 53)) ⇒
      fp64_isFinite (fp64_sub roundTiesToEven a b) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 53 ∧
          (fp64_to_real (fp64_sub roundTiesToEven a b) =
           (fp64_to_real a − fp64_to_real b) * (1 + e))

fp64_float_sqrt_relative

⊢ ∀a.
      fp64_isFinite a ∧ ¬word_msb a ∧
      abs (sqrt (fp64_to_real a)) ∈
      (1 / 2 pow 1022 .. 2 pow 2046 / 2 pow 1023 * (2 − 1 / 2 pow 53)) ⇒
      fp64_isFinite (fp64_sqrt roundTiesToEven a) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 53 ∧
          (fp64_to_real (fp64_sqrt roundTiesToEven a) =
           sqrt (fp64_to_real a) * (1 + e))

fp64_float_mul_sub_relative

⊢ ∀a b c.
      fp64_isFinite a ∧ fp64_isFinite b ∧ fp64_isFinite c ∧
      abs (fp64_to_real a * fp64_to_real b − fp64_to_real c) ∈
      (1 / 2 pow 1022 .. 2 pow 2046 / 2 pow 1023 * (2 − 1 / 2 pow 53)) ⇒
      fp64_isFinite (fp64_mul_sub roundTiesToEven a b c) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 53 ∧
          (fp64_to_real (fp64_mul_sub roundTiesToEven a b c) =
           (fp64_to_real a * fp64_to_real b − fp64_to_real c) * (1 + e))

fp64_float_mul_relative

⊢ ∀a b.
      fp64_isFinite a ∧ fp64_isFinite b ∧
      abs (fp64_to_real a * fp64_to_real b) ∈
      (1 / 2 pow 1022 .. 2 pow 2046 / 2 pow 1023 * (2 − 1 / 2 pow 53)) ⇒
      fp64_isFinite (fp64_mul roundTiesToEven a b) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 53 ∧
          (fp64_to_real (fp64_mul roundTiesToEven a b) =
           fp64_to_real a * fp64_to_real b * (1 + e))

fp64_float_mul_add_relative

⊢ ∀a b c.
      fp64_isFinite a ∧ fp64_isFinite b ∧ fp64_isFinite c ∧
      abs (fp64_to_real a * fp64_to_real b + fp64_to_real c) ∈
      (1 / 2 pow 1022 .. 2 pow 2046 / 2 pow 1023 * (2 − 1 / 2 pow 53)) ⇒
      fp64_isFinite (fp64_mul_add roundTiesToEven a b c) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 53 ∧
          (fp64_to_real (fp64_mul_add roundTiesToEven a b c) =
           (fp64_to_real a * fp64_to_real b + fp64_to_real c) * (1 + e))

fp64_float_div_relative

⊢ ∀a b.
      fp64_isFinite a ∧ fp64_isFinite b ∧ ¬fp64_isZero b ∧
      abs (fp64_to_real a / fp64_to_real b) ∈
      (1 / 2 pow 1022 .. 2 pow 2046 / 2 pow 1023 * (2 − 1 / 2 pow 53)) ⇒
      fp64_isFinite (fp64_div roundTiesToEven a b) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 53 ∧
          (fp64_to_real (fp64_div roundTiesToEven a b) =
           fp64_to_real a / fp64_to_real b * (1 + e))

fp64_float_add_relative

⊢ ∀a b.
      fp64_isFinite a ∧ fp64_isFinite b ∧
      abs (fp64_to_real a + fp64_to_real b) ∈
      (1 / 2 pow 1022 .. 2 pow 2046 / 2 pow 1023 * (2 − 1 / 2 pow 53)) ⇒
      fp64_isFinite (fp64_add roundTiesToEven a b) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 53 ∧
          (fp64_to_real (fp64_add roundTiesToEven a b) =
           (fp64_to_real a + fp64_to_real b) * (1 + e))

fp32_float_sub_relative

⊢ ∀a b.
      fp32_isFinite a ∧ fp32_isFinite b ∧
      abs (fp32_to_real a − fp32_to_real b) ∈
      (1 / 2 pow 126 .. 2 pow 254 / 2 pow 127 * (2 − 1 / 2 pow 24)) ⇒
      fp32_isFinite (fp32_sub roundTiesToEven a b) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 24 ∧
          (fp32_to_real (fp32_sub roundTiesToEven a b) =
           (fp32_to_real a − fp32_to_real b) * (1 + e))

fp32_float_sqrt_relative

⊢ ∀a.
      fp32_isFinite a ∧ ¬word_msb a ∧
      abs (sqrt (fp32_to_real a)) ∈
      (1 / 2 pow 126 .. 2 pow 254 / 2 pow 127 * (2 − 1 / 2 pow 24)) ⇒
      fp32_isFinite (fp32_sqrt roundTiesToEven a) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 24 ∧
          (fp32_to_real (fp32_sqrt roundTiesToEven a) =
           sqrt (fp32_to_real a) * (1 + e))

fp32_float_mul_sub_relative

⊢ ∀a b c.
      fp32_isFinite a ∧ fp32_isFinite b ∧ fp32_isFinite c ∧
      abs (fp32_to_real a * fp32_to_real b − fp32_to_real c) ∈
      (1 / 2 pow 126 .. 2 pow 254 / 2 pow 127 * (2 − 1 / 2 pow 24)) ⇒
      fp32_isFinite (fp32_mul_sub roundTiesToEven a b c) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 24 ∧
          (fp32_to_real (fp32_mul_sub roundTiesToEven a b c) =
           (fp32_to_real a * fp32_to_real b − fp32_to_real c) * (1 + e))

fp32_float_mul_relative

⊢ ∀a b.
      fp32_isFinite a ∧ fp32_isFinite b ∧
      abs (fp32_to_real a * fp32_to_real b) ∈
      (1 / 2 pow 126 .. 2 pow 254 / 2 pow 127 * (2 − 1 / 2 pow 24)) ⇒
      fp32_isFinite (fp32_mul roundTiesToEven a b) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 24 ∧
          (fp32_to_real (fp32_mul roundTiesToEven a b) =
           fp32_to_real a * fp32_to_real b * (1 + e))

fp32_float_mul_add_relative

⊢ ∀a b c.
      fp32_isFinite a ∧ fp32_isFinite b ∧ fp32_isFinite c ∧
      abs (fp32_to_real a * fp32_to_real b + fp32_to_real c) ∈
      (1 / 2 pow 126 .. 2 pow 254 / 2 pow 127 * (2 − 1 / 2 pow 24)) ⇒
      fp32_isFinite (fp32_mul_add roundTiesToEven a b c) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 24 ∧
          (fp32_to_real (fp32_mul_add roundTiesToEven a b c) =
           (fp32_to_real a * fp32_to_real b + fp32_to_real c) * (1 + e))

fp32_float_div_relative

⊢ ∀a b.
      fp32_isFinite a ∧ fp32_isFinite b ∧ ¬fp32_isZero b ∧
      abs (fp32_to_real a / fp32_to_real b) ∈
      (1 / 2 pow 126 .. 2 pow 254 / 2 pow 127 * (2 − 1 / 2 pow 24)) ⇒
      fp32_isFinite (fp32_div roundTiesToEven a b) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 24 ∧
          (fp32_to_real (fp32_div roundTiesToEven a b) =
           fp32_to_real a / fp32_to_real b * (1 + e))

fp32_float_add_relative

⊢ ∀a b.
      fp32_isFinite a ∧ fp32_isFinite b ∧
      abs (fp32_to_real a + fp32_to_real b) ∈
      (1 / 2 pow 126 .. 2 pow 254 / 2 pow 127 * (2 − 1 / 2 pow 24)) ⇒
      fp32_isFinite (fp32_add roundTiesToEven a b) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 24 ∧
          (fp32_to_real (fp32_add roundTiesToEven a b) =
           (fp32_to_real a + fp32_to_real b) * (1 + e))

fp16_float_sub_relative

⊢ ∀a b.
      fp16_isFinite a ∧ fp16_isFinite b ∧
      abs (fp16_to_real a − fp16_to_real b) ∈
      (1 / 2 pow 14 .. 2 pow 30 / 2 pow 15 * (2 − 1 / 2 pow 11)) ⇒
      fp16_isFinite (fp16_sub roundTiesToEven a b) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 11 ∧
          (fp16_to_real (fp16_sub roundTiesToEven a b) =
           (fp16_to_real a − fp16_to_real b) * (1 + e))

fp16_float_sqrt_relative

⊢ ∀a.
      fp16_isFinite a ∧ ¬word_msb a ∧
      abs (sqrt (fp16_to_real a)) ∈
      (1 / 2 pow 14 .. 2 pow 30 / 2 pow 15 * (2 − 1 / 2 pow 11)) ⇒
      fp16_isFinite (fp16_sqrt roundTiesToEven a) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 11 ∧
          (fp16_to_real (fp16_sqrt roundTiesToEven a) =
           sqrt (fp16_to_real a) * (1 + e))

fp16_float_mul_sub_relative

⊢ ∀a b c.
      fp16_isFinite a ∧ fp16_isFinite b ∧ fp16_isFinite c ∧
      abs (fp16_to_real a * fp16_to_real b − fp16_to_real c) ∈
      (1 / 2 pow 14 .. 2 pow 30 / 2 pow 15 * (2 − 1 / 2 pow 11)) ⇒
      fp16_isFinite (fp16_mul_sub roundTiesToEven a b c) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 11 ∧
          (fp16_to_real (fp16_mul_sub roundTiesToEven a b c) =
           (fp16_to_real a * fp16_to_real b − fp16_to_real c) * (1 + e))

fp16_float_mul_relative

⊢ ∀a b.
      fp16_isFinite a ∧ fp16_isFinite b ∧
      abs (fp16_to_real a * fp16_to_real b) ∈
      (1 / 2 pow 14 .. 2 pow 30 / 2 pow 15 * (2 − 1 / 2 pow 11)) ⇒
      fp16_isFinite (fp16_mul roundTiesToEven a b) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 11 ∧
          (fp16_to_real (fp16_mul roundTiesToEven a b) =
           fp16_to_real a * fp16_to_real b * (1 + e))

fp16_float_mul_add_relative

⊢ ∀a b c.
      fp16_isFinite a ∧ fp16_isFinite b ∧ fp16_isFinite c ∧
      abs (fp16_to_real a * fp16_to_real b + fp16_to_real c) ∈
      (1 / 2 pow 14 .. 2 pow 30 / 2 pow 15 * (2 − 1 / 2 pow 11)) ⇒
      fp16_isFinite (fp16_mul_add roundTiesToEven a b c) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 11 ∧
          (fp16_to_real (fp16_mul_add roundTiesToEven a b c) =
           (fp16_to_real a * fp16_to_real b + fp16_to_real c) * (1 + e))

fp16_float_div_relative

⊢ ∀a b.
      fp16_isFinite a ∧ fp16_isFinite b ∧ ¬fp16_isZero b ∧
      abs (fp16_to_real a / fp16_to_real b) ∈
      (1 / 2 pow 14 .. 2 pow 30 / 2 pow 15 * (2 − 1 / 2 pow 11)) ⇒
      fp16_isFinite (fp16_div roundTiesToEven a b) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 11 ∧
          (fp16_to_real (fp16_div roundTiesToEven a b) =
           fp16_to_real a / fp16_to_real b * (1 + e))

fp16_float_add_relative

⊢ ∀a b.
      fp16_isFinite a ∧ fp16_isFinite b ∧
      abs (fp16_to_real a + fp16_to_real b) ∈
      (1 / 2 pow 14 .. 2 pow 30 / 2 pow 15 * (2 − 1 / 2 pow 11)) ⇒
      fp16_isFinite (fp16_add roundTiesToEven a b) ∧
      ∃e.
          abs e ≤ 1 / 2 pow 11 ∧
          (fp16_to_real (fp16_add roundTiesToEven a b) =
           (fp16_to_real a + fp16_to_real b) * (1 + e))