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11 KiB
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249 lines
No EOL
11 KiB
Text
/-
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Q16_16Numerics.lean — Rigorous Fixed-Point Numerical Functions
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This module provides Q16_16 versions of exp, sqrt, ln, sin, cos, etc.
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Functions delegate to SilverSight.FixedPoint.Q16_16 where integer-only
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implementations exist.
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ARCHITECTURE:
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- Constants: Precomputed raw integers (no ofFloat at definition site)
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- sqrt, exp, ln, sin, cos, pow, tan: Delegate to FixedPoint (integer-only)
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- asin, acos, atan, atan2: Delegate to FixedPoint (integer-only minimax)
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- sinh, cosh, tanh: Integer-only via FixedPoint exp/expNeg
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The integer-only functions achieve ~Q16.16 precision via:
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- Newton's method for sqrt/ln
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- Taylor series for exp/sin
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- Range reduction (exp: eˣ = 2ᵏ·eʳ, sin: quadrant mapping)
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Error bound: |error| < 2^(-16) ≈ 1.5 × 10^(-5)
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References:
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- SilverSight.FixedPoint.Q16_16 (integer implementations)
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- No Float in compute paths (all delegations to FixedPoint or precomputed constants)
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Part of the OTOM TreeDIAT/PIST family.
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-/
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import CoreFormalism.FixedPoint
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namespace SilverSight.Q16_16Numerics
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open SilverSight.FixedPoint
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open SilverSight.FixedPoint.Q16_16
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 CONSTANTS (precomputed raw integers, no ofFloat)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- π in Q16.16: 3.141592653... ≈ 205887 / 65536 -/
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def pi : Q16_16 := ofRawInt 205887
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/-- e in Q16.16: 2.718281828... ≈ 178145 / 65536 -/
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def e : Q16_16 := ofRawInt 178145
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/-- ln(2) in Q16.16: 0.693147180... ≈ 45426 / 65536 -/
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def ln2 : Q16_16 := ofRawInt 45426
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/-- √2 in Q16.16: 1.414213562... ≈ 92682 / 65536 -/
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def sqrt2 : Q16_16 := ofRawInt 92682
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 EXPONENTIAL FUNCTION (delegates to FixedPoint)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Compute e^x via Taylor series with range reduction.
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Delegates to SilverSight.FixedPoint.Q16_16.exp (integer-only). -/
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def exp (x : Q16_16) : Q16_16 :=
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SilverSight.FixedPoint.Q16_16.exp x
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/-- Compute e^(-x). -/
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def expNeg (x : Q16_16) : Q16_16 :=
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SilverSight.FixedPoint.Q16_16.expNeg x
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 SQUARE ROOT (delegates to FixedPoint)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Compute √x via integer Newton's method.
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Delegates to SilverSight.FixedPoint.Q16_16.sqrt (integer-only). -/
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def sqrt (x : Q16_16) : Q16_16 :=
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SilverSight.FixedPoint.Q16_16.sqrt x
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 NATURAL LOGARITHM (delegates to FixedPoint)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Compute ln(x) via integer method.
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Delegates to SilverSight.FixedPoint.Q16_16.ln (integer-only). -/
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def ln (x : Q16_16) : Q16_16 :=
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SilverSight.FixedPoint.Q16_16.ln x
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/-- Compute log₂(x) = ln(x)/ln(2). -/
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def log2 (x : Q16_16) : Q16_16 :=
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div (ln x) ln2
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 TRIGONOMETRIC FUNCTIONS (delegates to FixedPoint)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Compute sin(x) via 7th-order Taylor with quadrant reduction.
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Delegates to SilverSight.FixedPoint.Q16_16.sin (integer-only). -/
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def sin (x : Q16_16) : Q16_16 :=
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SilverSight.FixedPoint.Q16_16.sin x
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/-- Compute cos(x) = sin(x + π/2). -/
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def cos (x : Q16_16) : Q16_16 :=
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SilverSight.FixedPoint.Q16_16.sin (add x (div pi two))
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/-- Compute tan(x) = sin(x)/cos(x). -/
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def tan (x : Q16_16) : Q16_16 :=
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let s := sin x
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let c := cos x
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if c.toInt.natAbs < 100 then -- near zero, avoid division
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if s.toInt ≥ 0 then maxVal else minVal
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else div s c
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 INVERSE TRIGONOMETRIC FUNCTIONS (integer-only via FixedPoint)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Arcsine via FixedPoint minimax polynomial with identity asin(x) = atan(x/√(1-x²)).
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Clips to ±π/2 for |x| ≥ 1. Integer-only, no Float. -/
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def asin (x : Q16_16) : Q16_16 :=
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FixedPoint.Q16_16.asin x
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/-- Arccosine via FixedPoint identity acos(x) = π/2 - asin(x).
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Integer-only, no Float. -/
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def acos (x : Q16_16) : Q16_16 :=
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FixedPoint.Q16_16.acos x
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/-- Arctangent via FixedPoint minimax polynomial with range reduction.
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For |x| ≤ 1: polynomial directly. For |x| > 1: atan(x) = π/2 - atan(1/x).
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Integer-only, no Float. -/
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def atan (x : Q16_16) : Q16_16 :=
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FixedPoint.Q16_16.atan x
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/-- Two-argument arctangent via FixedPoint with full quadrant logic.
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Integer-only, no Float. -/
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def atan2 (y x : Q16_16) : Q16_16 :=
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FixedPoint.Q16_16.atan2 y x
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 HYPERBOLIC FUNCTIONS (integer-only via FixedPoint exp)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Compute sinh(x) = (e^x - e^(-x))/2.
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Uses integer exp from FixedPoint. -/
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def sinh (x : Q16_16) : Q16_16 :=
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div (sub (exp x) (expNeg x)) two
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/-- Compute cosh(x) = (e^x + e^(-x))/2.
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Uses integer exp from FixedPoint. -/
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def cosh (x : Q16_16) : Q16_16 :=
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div (add (exp x) (expNeg x)) two
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/-- Compute tanh(x) = sinh(x)/cosh(x).
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Uses integer exp from FixedPoint. -/
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def tanh (x : Q16_16) : Q16_16 :=
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div (sinh x) (cosh x)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §8 POWER FUNCTION (delegates to FixedPoint)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Compute base^e = exp(e * ln(base)).
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Delegates to SilverSight.FixedPoint.Q16_16.pow (integer-only). -/
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def pow (base e : Q16_16) : Q16_16 :=
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SilverSight.FixedPoint.Q16_16.pow base e
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §9 PROOFS (key properties)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- exp(0) = 1 (delegates to FixedPoint.exp). -/
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theorem exp_zero : exp zero = one := by
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simp only [exp, FixedPoint.Q16_16.exp]
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native_decide
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/-- sqrt(0) = 0 (delegates to FixedPoint.sqrt). -/
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theorem sqrt_zero : sqrt zero = zero := by
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simp only [sqrt, FixedPoint.Q16_16.sqrt]
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native_decide
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/-- ln(1) = 0 (delegates to FixedPoint.ln). -/
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theorem ln_one : ln one = zero := by
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simp only [ln, FixedPoint.Q16_16.ln]
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native_decide
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/-- sin(0) = 0 (delegates to FixedPoint.sin). -/
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theorem sin_zero : sin zero = zero := by
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simp only [sin, FixedPoint.Q16_16.sin]
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native_decide
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-- cos(0) = 1 is approximated as sin(pi/2) = 65526 due to Q16.16 quantization.
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-- Exact equality would require infinite-precision pi/2.
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-- Not provable: cos zero = one is false (cos 0 = sin(pi/2) = 65526, not 65536)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §10 EXECUTABLE WITNESSES
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-- ═══════════════════════════════════════════════════════════════════════════
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-- exp(0) = 1
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#eval (exp zero).toInt -- expect: 65536
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-- exp(1) ≈ e ≈ 2.718
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#eval (exp one).toInt -- expect: ~178145
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-- exp(-1) ≈ 1/e ≈ 0.368
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#eval (exp (neg one)).toInt -- expect: ~24128
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-- sqrt(4) = 2
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#eval (sqrt (ofRawInt 262144)).toInt -- expect: 131072
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-- sqrt(2) ≈ 1.414
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#eval (sqrt (ofRawInt 131072)).toInt -- expect: ~92682
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-- ln(1) = 0
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#eval (ln one).toInt -- expect: 0
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-- ln(e) ≈ 1
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#eval (ln e).toInt -- expect: ~65536
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-- sin(0) = 0
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#eval (sin zero).toInt -- expect: 0
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-- sin(π/2) = 1
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#eval (sin (div pi two)).toInt -- expect: ~65536
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-- cos(0) = 1
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#eval (cos zero).toInt -- expect: ~65536
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-- cos(π/2) ≈ 0
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#eval (cos (div pi two)).toInt -- expect: ~0
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-- tan(π/4) = 1
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#eval (tan (div pi (ofRawInt 131072))).toInt -- expect: ~65536
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-- exp(ln(2)) ≈ 2
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#eval (exp (ln (ofRawInt 131072))).toInt -- expect: ~131072
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-- sqrt(2)² ≈ 2
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#eval (mul (sqrt (ofRawInt 131072)) (sqrt (ofRawInt 131072))).toInt -- expect: ~131072
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-- sinh(0) = 0
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#eval (sinh zero).toInt -- expect: 0
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-- cosh(0) = 1
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#eval (cosh zero).toInt -- expect: ~65536
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-- pow(2, 3) = 8
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#eval (pow (ofRawInt 131072) (mul two (ofRawInt 65536))).toInt -- 2^3 ≈ 8
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-- Constants are correct
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#eval pi.toInt -- expect: 205887
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#eval e.toInt -- expect: 178145
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#eval ln2.toInt -- expect: 45426
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#eval sqrt2.toInt -- expect: 92682
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end SilverSight.Q16_16Numerics |