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use crate::core::integer::*;
use crate::core::undefined::*;
use crate::{Circle, CircleConstants, Integer, Scalar, ScalarConstants};
use core::ops::*;
use i256::I256;
use num_traits::{AsPrimitive, WrappingAdd, WrappingMul, WrappingNeg, WrappingSub};
#[allow(private_bounds)]
impl<
F: Integer
+ FullInt
+ Shl<isize, Output = F>
+ Shr<isize, Output = F>
+ Shl<F, Output = F>
+ Shr<F, Output = F>
+ Shl<E, Output = F>
+ Shr<E, Output = F>
+ WrappingNeg
+ WrappingAdd
+ WrappingMul
+ WrappingSub,
E: Integer
+ FullInt
+ Shl<isize, Output = E>
+ Shr<isize, Output = E>
+ Shl<E, Output = E>
+ Shr<E, Output = E>
+ Shl<F, Output = E>
+ Shr<F, Output = E>
+ WrappingNeg
+ WrappingAdd
+ WrappingMul
+ WrappingSub,
> Circle<F, E>
where
Circle<F, E>: CircleConstants,
Scalar<F, E>: ScalarConstants,
u8: AsPrimitive<F>,
u16: AsPrimitive<F>,
u32: AsPrimitive<F>,
u64: AsPrimitive<F>,
u128: AsPrimitive<F>,
usize: AsPrimitive<F>,
i8: AsPrimitive<F>,
i16: AsPrimitive<F>,
i32: AsPrimitive<F>,
i64: AsPrimitive<F>,
i128: AsPrimitive<F>,
isize: AsPrimitive<F>,
I256: From<F>,
u8: AsPrimitive<E>,
u16: AsPrimitive<E>,
u32: AsPrimitive<E>,
u64: AsPrimitive<E>,
u128: AsPrimitive<E>,
usize: AsPrimitive<E>,
i8: AsPrimitive<E>,
i16: AsPrimitive<E>,
i32: AsPrimitive<E>,
i64: AsPrimitive<E>,
i128: AsPrimitive<E>,
isize: AsPrimitive<E>,
I256: From<E>,
{
/// Raises a complex number to a scalar power.
///
/// Implements z^s (complex raised to scalar power) using:
/// - Specialized algorithm for integer exponents
/// - General formula z^s = exp(s * ln(z)) for non-integer exponents
///
/// # Special Cases:
/// - Undefined or escaped values return appropriate undefined states
/// - 0^0 returns the ZERO_POWER_ZERO undefined state
/// - 0^s returns 0 if s is positive
/// - 0^s returns ZERO_NEGATIVE_POWER undefined state if s is negative
///
/// # Parameters:
/// - `self`: The complex base
/// - `exp`: The scalar exponent
///
/// # Returns:
/// - A complex number representing z^s
pub(crate) fn circle_power_scalar(&self, exp: &Scalar<F, E>) -> Self {
if !self.is_normal() || !exp.is_normal() {
if self.is_undefined() {
return *self;
}
if exp.is_undefined() {
return Self {
real: exp.fraction,
imaginary: exp.fraction,
exponent: exp.exponent,
};
}
// z^0 = 1 for any finite z (mathematical identity, 0^0 = 1 included) — matches the Scalar convention.
if exp.is_zero() {
return Self::ONE;
}
// Zero base: only the exponent's SIGN matters — 0^(+) = 0, 0^(−) = ∞ — for any positive/negative exponent, normal or escaped.
// NOTE: exp is an N0-convention Scalar, so use its is_negative(), never the raw fraction sign (stored MSB=1 reads POSITIVE, the opposite of a plain int).
if self.is_zero() {
return if exp.is_negative() {
Self::INFINITY
} else {
Self::ZERO
};
}
// Escaped base: the orientation θ is stored, so much more resolves than the Scalar case — see escaped_rotate_to_power for the non-integer rules.
if self.exploded() || self.vanished() {
let base_exploded = self.exploded();
if exp.is_normal() {
if exp.is_integer() {
// Multiply chain: class, parity, and orientation all survive — and z^1 ≡ z.
return self.integer_power(exp);
}
if exp.magnitude() > Scalar::<F, E>::ONE {
// Non-integer |p| > 1: magnitude class is determinate (dominance; p < 0 inverts thru the reciprocal) and the direction rotates to p·θ, computable from the stored orientation.
return self.escaped_rotate_to_power(exp, base_exploded);
}
// Non-integer 0 < |p| < 1: tiny^p / huge^p can re-enter normal range — class indeterminate.
let prefix: F = if base_exploded {
TRANSFINITE_POWER.prefix.sa()
} else {
VANISHED_POWER.prefix.sa()
};
return Self {
real: prefix,
imaginary: prefix,
exponent: Self::ambiguous_exponent(),
};
}
// Vanished exponent: p·lb|z| is a 0·∞ form → ℘^⬇. Anything transfinite (exploded/∞ exponent): the rotation p·θ mod 2π is unknowable → ℘^⬆.
let prefix: F = if exp.vanished() {
POWER_VANISHED.prefix.sa()
} else {
POWER_TRANSFINITE.prefix.sa()
};
return Self {
real: prefix,
imaginary: prefix,
exponent: Self::ambiguous_exponent(),
};
}
// Infinite base (the single unsigned point at infinity, no angle to lose): ∞^(+) = ∞, ∞^(−) = 0. (∞^0 = 1 returned above.)
if self.is_infinite() {
return if exp.is_negative() {
Self::ZERO
} else {
Self::INFINITY
};
}
// Normal base, non-normal exponent: transfinite exponents (exploded or ∞) can't commit a rotation → ℘^⬆; a vanished exponent is the 0·∞-adjacent form ℘^⬇.
let prefix: F = if exp.vanished() {
POWER_VANISHED.prefix.sa()
} else {
POWER_TRANSFINITE.prefix.sa()
};
return Self {
real: prefix,
imaginary: prefix,
exponent: Self::ambiguous_exponent(),
};
}
// Check if exponent is integer for exact computation
if exp.is_integer() {
return self.integer_power(exp);
}
let ln_z = self.ln();
let s_ln_z = ln_z * exp;
s_ln_z.exp()
}
/// Raise an escaped Circle to a non-integer real power with |p| > 1: the magnitude class is fixed by dominance (same class for p > 0, inverted thru the reciprocal for p < 0), and the direction rotates from the stored θ to p·θ.
/// The stored escaped fractions ARE the unit direction (N1 shape for exploded, N2 for vanished), so rebuild them as a normal Circle, extract θ via atan2, rotate, and stamp the result class shape.
/// If p·θ's period position exceeds precision, sin/cos return their imprecision undefineds — propagated honestly.
fn escaped_rotate_to_power(&self, exp: &Scalar<F, E>, base_exploded: bool) -> Self {
let n_shift: isize = if base_exploded { 0 } else { 1 };
let dir = Self {
real: self.real << n_shift,
imaginary: self.imaginary << n_shift,
exponent: Self::ONE.exponent,
};
let theta = dir.i().atan2(dir.r());
let phi = theta * *exp;
let unit = Self::from((phi.cos(), phi.sin()));
if !unit.is_normal() {
return unit;
}
let result_exploded = if exp.is_positive() {
base_exploded
} else {
!base_exploded
};
if result_exploded {
Self {
real: unit.real,
imaginary: unit.imaginary,
exponent: Self::ambiguous_exponent(),
}
} else {
Self {
real: unit.real >> 1isize,
imaginary: unit.imaginary >> 1isize,
exponent: Self::ambiguous_exponent(),
}
}
}
pub(crate) fn integer_power(&self, n: &Scalar<F, E>) -> Self {
if n.is_zero() {
return Self::ONE;
}
let mut result = Self::ONE;
let mut base = if n.is_negative() {
Self::ONE / *self
} else {
*self
};
let mut exp = n.magnitude();
// Two's-complement boundary: n = -2^MAX_EXP escapes under magnitude() and would hang the squaring loop (see the Scalar twin). For a Circle the result DIRECTION is n·θ with |n| = 2^MAX_EXP — a period position far beyond any stored precision — so the honest result is undefined, transfinite-power tagged.
if !exp.is_normal() {
if base.is_undefined() {
return base;
}
let prefix: F = POWER_TRANSFINITE.prefix.sa();
return Self {
real: prefix,
imaginary: prefix,
exponent: Self::ambiguous_exponent(),
};
}
while !exp.is_zero() {
if (exp & Scalar::<F, E>::ONE) == 1 {
result *= base;
}
base = base.square();
exp = (exp >> 1u8).floor();
}
result
}
/// Computes the logarithm of a complex number with a scalar base.
///
/// Implements c log s (logarithm of complex c with scalar base s) using: c log s = ln(c) / ln(s)
///
/// # Special Cases:
/// - Undefined or escaped values return appropriate undefined states
/// - Exploded or vanished values return appropriate undefined states
/// - 0 log s returns ZERO_LOG undefined state
/// - c log 0 returns LOG_ZERO undefined state
/// - c log -# returns LOG_NEGATIVE undefined state
///
/// # Parameters:
/// - `self`: The complex number to take the logarithm of
/// - `base`: The scalar base of the logarithm
///
/// # Returns:
/// - A complex number representing c log s
pub(crate) fn circle_logarithm_scalar(&self, base: &Scalar<F, E>) -> Self {
if !self.is_normal() || !base.is_normal() {
if self.is_undefined() {
return *self;
}
if base.is_undefined() {
return Self {
real: base.fraction,
imaginary: 0.as_(),
exponent: base.exponent,
};
}
if self.is_zero() {
let prefix: F = NEGLIGIBLE_LOG.prefix.sa();
return Self {
real: prefix,
imaginary: prefix,
exponent: Self::ambiguous_exponent(),
};
}
if base.is_zero() {
let prefix: F = LOG_NEGLIGIBLE.prefix.sa();
return Self {
real: prefix,
imaginary: prefix,
exponent: Self::ambiguous_exponent(),
};
}
if self.exploded() {
let prefix: F = TRANSFINITE_LOG.prefix.sa();
return Self {
real: prefix,
imaginary: prefix,
exponent: Self::ambiguous_exponent(),
};
}
if self.vanished() {
let prefix: F = NEGLIGIBLE_LOG.prefix.sa();
return Self {
real: prefix,
imaginary: prefix,
exponent: Self::ambiguous_exponent(),
};
}
if base.exploded() {
let prefix: F = LOG_TRANSFINITE.prefix.sa();
return Self {
real: prefix,
imaginary: prefix,
exponent: Self::ambiguous_exponent(),
};
}
if base.vanished() {
let prefix: F = LOG_NEGLIGIBLE.prefix.sa();
return Self {
real: prefix,
imaginary: prefix,
exponent: Self::ambiguous_exponent(),
};
}
}
if base.fraction.is_negative() {
let prefix: F = LOG_NEGATIVE.prefix.sa();
return Self {
real: prefix,
imaginary: prefix,
exponent: Self::ambiguous_exponent(),
};
}
let ln_z = self.ln();
let ln_base = base.ln();
ln_z / ln_base
}
}