g_math 0.7.0

Deterministic fixed-point arithmetic, every format a type: correctly rounded transcendentals, decimal and symbolic domains, linear algebra, Lie groups, manifolds, ODE solvers, certified intervals. Pure Rust, zero-float
Documentation 
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
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195

//! The exact evaluation of a canonical tree: every leaf at its exact value,
//! every node on a backend of `crate::fixed_point::exact` (the compute tier
//! with tracked bounds first, then `Mpf` at growing precision), and one
//! rounding to storage when the bound decides it
//! (`docs/design/CORRECT_ROUNDING.md`).

use super::{FascFormat, LazyExpr, StackEvaluator, StackValue};
use crate::fixed_point::domains::binary_fixed::transcendental::mpf::{Mpf, MpfErr, C17, C2, C3, C33, C5, C65, C9};
use crate::fixed_point::domains::symbolic::rational::rational_number::{RationalNumber, RationalParts};
use crate::fixed_point::exact::{fast::Fast, Num};
use crate::fixed_point::universal::fasc::lazy_expr::ConstantId;
use crate::fixed_point::universal::tier_types::{CompactShadow, ShadowConstantId};
use crate::fixed_point::width::WideInt;

type R<X> = Result<X, MpfErr>;

/// `10^k` exactly (or with the backend's bound).
fn pow10<T: FascFormat, B: Num<T>>(k: u32) -> R<B> {
    let mut v = B::int(1)?;
    let ten = B::int(10)?;
    for _ in 0..k { v = v.mul(ten)?; }
    Ok(v)
}

/// An unsigned 128-bit denominator exactly.
fn from_u128<T: FascFormat, B: Num<T>>(d: u128) -> R<B> {
    B::int((d >> 1) as i128)?.mul(B::int(2)?)?.add(B::int((d & 1) as i128)?)
}

fn rational<T: FascFormat, B: Num<T>>(r: &RationalNumber) -> R<B> {
    match r.extract_native() {
        RationalParts::Tiny(n, d) => B::int(n as i128)?.div(B::int(d as i128)?),
        RationalParts::Small(n, d) => B::int(n as i128)?.div(B::int(d as i128)?),
        RationalParts::Medium(n, d) => B::int(n as i128)?.div(B::int(d as i128)?),
        RationalParts::Large(n, d) => B::int(n as i128)?.div(B::int(d as i128)?),
        RationalParts::Huge(n, d) => B::int(n)?.div(from_u128::<T, B>(d)?),
        RationalParts::Massive(n, d) => B::from_word(n, 0)?.div(B::from_word(d, 0)?),
        RationalParts::Ultra(n, d) => B::from_word(n, 0)?.div(B::from_word(d, 0)?),
        #[cfg(feature = "infinite-precision")]
        RationalParts::Infinite(..) => Err(MpfErr::Unsupported),
    }
}

fn constant<T: FascFormat, B: Num<T>>(id: ConstantId) -> R<B> {
    match id {
        ConstantId::Pi => B::pi(),
        ConstantId::E => B::int(1)?.exp(),
        ConstantId::Sqrt2 => B::int(2)?.sqrt(),
        ConstantId::Phi => Ok(B::int(1)?.add(B::int(5)?.sqrt()?)?.half()),
        ConstantId::EulerGamma => B::euler_gamma(),
    }
}

fn shadow_value<T: FascFormat, B: Num<T>>(s: &CompactShadow) -> Option<R<B>> {
    if let CompactShadow::ConstantRef(id) = s {
        return Some(match id {
            ShadowConstantId::Pi => constant::<T, B>(ConstantId::Pi),
            ShadowConstantId::E => constant::<T, B>(ConstantId::E),
            ShadowConstantId::Sqrt2 => constant::<T, B>(ConstantId::Sqrt2),
            ShadowConstantId::Phi => constant::<T, B>(ConstantId::Phi),
            ShadowConstantId::Ln2 => B::int(2).and_then(|v| v.ln()),
            ShadowConstantId::Ln10 => B::int(10).and_then(|v| v.ln()),
            ShadowConstantId::EulerGamma => B::euler_gamma(),
        });
    }
    let (n, d) = s.as_rational()?;
    Some(B::int(n).and_then(|n| n.div(from_u128::<T, B>(d)?)))
}

/// Fraction bits of a binary raw at a UGOD tier (as `StackValue::to_rational`).
pub(super) fn binary_tier_frac<T: FascFormat>(tier: u8) -> u32 {
    match tier { 1 => T::TIER1_RATIONAL_FRAC_BITS, 2 => 32, 3 => 64, 4 => 128, 5 => 256, 6 => 512, _ => 64 }
}

/// The exact value of an evaluated leaf.
fn leaf<T: FascFormat, B: Num<T>>(v: &StackValue<T>) -> R<B> {
    let shadow = match v {
        StackValue::<T>::Binary(_, _, s) | StackValue::<T>::BinaryCompute(_, _, s) | StackValue::<T>::Decimal(_, _, s)
        | StackValue::<T>::DecimalCompute(_, _, s) | StackValue::<T>::Ternary(_, _, s) => Some(s),
        _ => None,
    };
    if let Some(r) = shadow.and_then(shadow_value::<T, B>) { return r; }
    match v {
        StackValue::<T>::Binary(tier, raw, _) => B::from_word(*raw, binary_tier_frac::<T>(*tier)),
        StackValue::<T>::BinaryCompute(_, raw, _) => B::from_word(*raw, 2 * T::FRAC_BITS),
        StackValue::<T>::Decimal(dp, scaled, _) => B::from_word(*scaled, 0)?.div(pow10::<T, B>(*dp as u32)?),
        StackValue::<T>::DecimalCompute(_, raw, _) => B::from_word(*raw, 0)?.div(pow10::<T, B>(T::DECIMAL_COMPUTE_DP as u32)?),
        StackValue::<T>::Ternary(..) | StackValue::<T>::Symbolic(_) => rational::<T, B>(&v.to_rational().map_err(|_| MpfErr::Unsupported)?),
        StackValue::<T>::Error(_) => Err(MpfErr::Unsupported),
    }
}

/// The tree on a backend.
pub(super) fn tree<T: FascFormat, B: Num<T>>(ev: &mut StackEvaluator<T>, e: &LazyExpr<T>) -> R<B> {
    use LazyExpr as L;
    match e {
        L::Literal(_) | L::Value(_) => {
            let v = ev.evaluate(e).map_err(|_| MpfErr::Unsupported)?;
            leaf::<T, B>(&v)
        }
        L::Constant(id) => constant::<T, B>(*id),
        L::Variable(_) => Err(MpfErr::Unsupported),
        L::Negate(a) => Ok(tree::<T, B>(ev, a)?.neg()),
        L::Add(a, b) => tree::<T, B>(ev, a)?.add(tree::<T, B>(ev, b)?),
        L::Sub(a, b) => tree::<T, B>(ev, a)?.sub(tree::<T, B>(ev, b)?),
        L::Mul(a, b) => tree::<T, B>(ev, a)?.mul(tree::<T, B>(ev, b)?),
        L::Div(a, b) => tree::<T, B>(ev, a)?.div(tree::<T, B>(ev, b)?),
        L::Pow(a, b) => tree::<T, B>(ev, a)?.pow(tree::<T, B>(ev, b)?),
        L::Atan2(a, b) => tree::<T, B>(ev, a)?.atan2(tree::<T, B>(ev, b)?),
        L::Exp(a) => tree::<T, B>(ev, a)?.exp(),
        L::Ln(a) => tree::<T, B>(ev, a)?.ln(),
        L::Sqrt(a) => tree::<T, B>(ev, a)?.sqrt(),
        L::Sin(a) => tree::<T, B>(ev, a)?.sin(),
        L::Cos(a) => tree::<T, B>(ev, a)?.cos(),
        L::Tan(a) => tree::<T, B>(ev, a)?.tan(),
        L::Atan(a) => tree::<T, B>(ev, a)?.atan(),
        L::Asin(a) => tree::<T, B>(ev, a)?.asin(),
        L::Acos(a) => tree::<T, B>(ev, a)?.acos(),
        L::Sinh(a) => tree::<T, B>(ev, a)?.sinh(),
        L::Cosh(a) => tree::<T, B>(ev, a)?.cosh(),
        L::Tanh(a) => tree::<T, B>(ev, a)?.tanh(),
        L::Asinh(a) => tree::<T, B>(ev, a)?.asinh(),
        L::Acosh(a) => tree::<T, B>(ev, a)?.acosh(),
        L::Atanh(a) => tree::<T, B>(ev, a)?.atanh(),
    }
}

/// The storage raw of the tree, rounded once (`Err` carries why not): the
/// evaluation runs at the narrowest capacity that holds the format's word
/// with room, and again at the next ones while the rounding is undecided.
pub(super) fn rounded<T: FascFormat>(ev: &mut StackEvaluator<T>, e: &LazyExpr<T>) -> Result<T::Storage, MpfErr> {
    let bits = <T::Storage as WideInt>::BITS;
    // the compute tier first: most trees are decided there
    if let Ok(r) = tree::<T, Fast<T>>(ev, e).and_then(|v: Fast<T>| Num::<T>::round(&v, false)) { return Ok(r); }
    macro_rules! attempt {
        ($C:ty, $tie:expr) => {
            match tree::<T, Mpf<$C>>(ev, e).and_then(|v| Num::<T>::round(&v, $tie)) {
                Ok(r) => return Ok(r),
                Err(MpfErr::Undecided) | Err(MpfErr::NearTie) => {}
                Err(x) => return Err(x),
            }
        };
    }
    if bits <= 64 { attempt!(C2, false); }
    if bits <= 128 { attempt!(C3, false); }
    if bits <= 256 { attempt!(C5, false); }
    attempt!(C9, false);
    attempt!(C17, false);
    attempt!(C33, false);
    attempt!(C65, true);
    Err(MpfErr::Undecided)
}



/// The decimal reading of `v`: the nearest integer to `v 10^dp`, ties to
/// even, at the most places from `smax` down whose value fits the storage
/// word (as decimal compute values are read).
fn decimal_at<T: FascFormat, B: Num<T>>(v: B, smax: u8, tie: bool) -> R<(u8, T::Storage)> {
    let mut dp = smax as i32;
    while dp >= 0 {
        let y = v.mul(pow10::<T, B>(dp as u32)?)?;
        match y.round_int_even(tie) {
            Ok(r) => return Ok((dp as u8, r)),
            Err(MpfErr::Overflow) => dp -= 1,
            Err(e) => return Err(e),
        }
    }
    Err(MpfErr::Overflow)
}

/// The tree's decimal reading `(places, scaled)`, rounded once from the
/// exact value: the compute tier first, then `Mpf` while undecided.
pub(super) fn rounded_decimal<T: FascFormat>(ev: &mut StackEvaluator<T>, e: &LazyExpr<T>) -> R<(u8, T::Storage)> {
    let smax = T::DECIMAL_STORAGE_MAX_DP;
    if let Ok(r) = tree::<T, Fast<T>>(ev, e).and_then(|v| decimal_at::<T, Fast<T>>(v, smax, false)) { return Ok(r); }
    let bits = <T::Storage as WideInt>::BITS;
    macro_rules! attempt {
        ($C:ty, $tie:expr) => {
            match tree::<T, Mpf<$C>>(ev, e).and_then(|v| decimal_at::<T, Mpf<$C>>(v, smax, $tie)) {
                Ok(r) => return Ok(r),
                Err(MpfErr::Undecided) | Err(MpfErr::NearTie) => {}
                Err(x) => return Err(x),
            }
        };
    }
    if bits <= 64 { attempt!(C2, false); }
    if bits <= 128 { attempt!(C3, false); }
    if bits <= 256 { attempt!(C5, false); }
    attempt!(C9, false);
    attempt!(C17, false);
    attempt!(C33, false);
    attempt!(C65, true);
    Err(MpfErr::Undecided)
}