ARPFloat

https://docs.rs/arpfloat
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ARPFloat is a Rust library providing arbitrary-precision floating-point data structures and...
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### BigInt usage example: multiply and power

Source: https://docs.rs/arpfloat/latest/src/arpfloat/bigint.rs.html

Shows how to create BigInt values from u64, multiply them, raise to a power, and print the decimal representation.

```rust
   use arpfloat::BigInt;

   let x = BigInt::from_u64(1995);
   let y = BigInt::from_u64(90210);

   let z = x * y;
   let z = z.powi(10);

   // Prints: 3564312949426686000....
   println!("{}", z.as_decimal());

```

--------------------------------

### Output of FP128 square root example

Source: https://docs.rs/arpfloat/latest/index.html

Expected output from the previous example, showing the square root of 5 in FP128 and FP64 precision.

```text
fp128: 2.2360679774997896964091736687312763
fp64:  2.23606797749979
```

--------------------------------

### Example: Create Float with Semantics

Source: https://docs.rs/arpfloat/latest/arpfloat/struct.Semantics.html

Shows how to create a new Semantics with 10 exponent bits, 100 precision bits, and Positive rounding mode, then create the number 1.0 and verify its double value.

```rust
    use arpfloat::{Float, RoundingMode, Semantics};

    // Create a new floating point semantics.
    let sem = Semantics::new(10, 100, RoundingMode::Positive);
    // Create the number 1.0 with the new semantics.
    let x = Float::one(sem, false);

    // Check that the value is correct when casting to `double`.
    assert_eq!(x.as_f64(), 1.0);
```

--------------------------------

### Output of continued fraction example

Source: https://docs.rs/arpfloat/latest/index.html

Expected output from the continued fraction example, showing the value of ln(2) and its successive fraction approximations.

```text
  ln(2) = .6931471805599453094172321214581765680755001343.....
  0/1
  1/1
  2/3
  7/10
  9/13
  61/88
  192/277
  253/365
  445/642
  1143/1649
  1588/2291
  2731/3940
  ....
```

--------------------------------

### Rounding mode API example

Source: https://docs.rs/arpfloat/latest/arpfloat

Shows how to use the rounding mode API with `mul_with_rm` to multiply two FP128 numbers using the `NearestTiesToEven` rounding mode.

```rust
    use arpfloat::FP128;
    use arpfloat::RoundingMode::NearestTiesToEven;
    use arpfloat::Float;

    let x = Float::from_u64(FP128, 1<<53);
    let y = Float::from_f64(1000.0).cast(FP128);

    let val = Float::mul_with_rm(&x, &y, NearestTiesToEven);
```

--------------------------------

### Module imports and Float impl start

Source: https://docs.rs/arpfloat/latest/src/arpfloat/arithmetic.rs.html

Imports the necessary modules and begins the implementation of arithmetic operations on the Float type. Includes the add_or_sub_normals function that handles addition and subtraction of normal numbers, following the APFloat implementation.

```rust
//! This module contains the implementation of the basic arithmetic operations:
//! Addition, Subtraction, Multiplication, Division.
extern crate alloc;
use crate::bigint::BigInt;

use super::bigint::LossFraction;
use super::float::{Category, Float, RoundingMode};
use core::cmp::Ordering;
use core::ops::{
    Add, AddAssign, Div, DivAssign, Mul, MulAssign, Sub, SubAssign,
};

impl Float {
    /// An inner function that performs the addition and subtraction of normal
    /// numbers (no NaN, Inf, Zeros).
    /// See Pg 247.  Chapter 8. Algorithms for the Five Basic Operations.
    /// This implementation follows the APFloat implementation, that does not
    /// swap the operands.
    fn add_or_sub_normals(
        a: &Self,
        b: &Self,
        subtract: bool,
    ) -> (Self, LossFraction) {
        debug_assert_eq!(a.get_semantics(), b.get_semantics());
        let sem = a.get_semantics();
        let loss;
        let mut a = a.clone();
        let mut b = b.clone();

        // Align the input numbers on the same exponent.
        let bits = a.get_exp() - b.get_exp();

        // Can transform (a-b) to (a + -b), either way, there are cases where
        // subtraction needs to happen.
        let subtract = subtract ^ (a.get_sign() ^ b.get_sign());
        if subtract {
            // Align the input numbers. We shift LHS one bit to the left to
            // allow carry/borrow in case of underflow as result of subtraction.
            match bits.cmp(&0) {
                Ordering::Equal => {
                    loss = LossFraction::ExactlyZero;
                }
                Ordering::Greater => {
                    loss = b.shift_significand_right((bits - 1) as u64);
                    a.shift_significand_left(1);
                }
                Ordering::Less => {
                    loss = a.shift_significand_right((-bits - 1) as u64);
                    b.shift_significand_left(1);
                }
            }

            let a_mantissa = a.get_mantissa();
            let b_mantissa = b.get_mantissa();
            let ab_mantissa;
            let mut sign = a.get_sign();

            // Figure out the carry from the shifting operations that dropped
            // bits.
            let c = !loss.is_exactly_zero() as u64;
            let c = BigInt::from_u64(c);

            // Figure out which mantissa is larger, to make sure that we don't
            // overflow the subtraction.
            if a_mantissa < b_mantissa {
                // A < B
                ab_mantissa = b_mantissa - a_mantissa - c;
                sign = !sign;
            } else {
                // A >= B
                ab_mantissa = a_mantissa - b_mantissa - c;
            }
            (
                Self::from_parts(sem, sign, a.get_exp(), ab_mantissa),
                loss.invert(),
            )
        } else {
            // Handle the easy case of Add:
            let mut b = b.clone();
            let mut a = a.clone();
            if bits > 0 {
                loss = b.shift_significand_right(bits as u64);
            } else {
                loss = a.shift_significand_right(-bits as u64);
            }
            debug_assert!(a.get_exp() == b.get_exp());
            let ab_mantissa = a.get_mantissa() + b.get_mantissa();
            (
                Self::from_parts(sem, a.get_sign(), a.get_exp(), ab_mantissa),
                loss,
            )
        }
    }

    /// Computes a+b using the rounding mode `rm`.
    pub fn add_with_rm(a: &Self, b: &Self, rm: RoundingMode) -> Self {
        Self::add_sub(a, b, false, rm)
    }
    /// Computes a-b using the rounding mode `rm`.
    pub fn sub_with_rm(a: &Self, b: &Self, rm: RoundingMode) -> Self {
        Self::add_sub(a, b, true, rm)
    }

    fn add_sub(a: &Self, b: &Self, subtract: bool, rm: RoundingMode) -> Self {
        let sem = a.get_semantics();
        // Table 8.2: Specification of addition for positive floating-point
        // data. Pg 247.
        match (a.get_category(), b.get_category()) {
            (Category::NaN, Category::Infinity)
            | (Category::NaN, Category::NaN)
            | (Category::NaN, Category::Normal)
            | (Category::NaN, Category::Zero)
            | (Category::Normal, Category::Zero)
            | (Category::Infinity, Category::Normal)
            | (Category::Infinity, Category::Zero) => a.clone(),

            (Category::Zero, Category::NaN)
            | (Category::Normal, Category::NaN)
            | (Category::Infinity, Category::NaN) => {

```

--------------------------------

### string

Source: https://docs.rs/arpfloat/latest/src/arpfloat/string.rs.html

string

```string
string
```

--------------------------------

### Test Pentium FDIV bug

Source: https://docs.rs/arpfloat/latest/src/arpfloat/arithmetic.rs.html

Tests that the division of two specific large numbers produces a result starting with '1.333820449136241002', referencing the Pentium FDIV bug. Uses FP128 precision.

```rust
#[cfg(feature = "std")]
#[test]
fn test_famous_pentium4_bug() {
    use crate::std::string::ToString;
    // https://en.wikipedia.org/wiki/Pentium_FDIV_bug
    use crate::FP128;

    let a = Float::from_u64(FP128, 4_195_835);
    let b = Float::from_u64(FP128, 3_145_727);
    let res = a / b;
    let result = res.to_string();
    assert!(result.starts_with("1.333820449136241002"));
}
```

--------------------------------

### string

Source: https://docs.rs/arpfloat/latest/src/arpfloat/string.rs.html

string

```rust
fn convert_to_string(&self) -> String {
    // In order to print decimal digits we need a minimum number of mantissa
    // bits for the conversion. Small floats (such as BF16) don't have
    // enough bits, so we cast to a larger number.
    if self.get_semantics().get_mantissa_len() < 16 {
        use crate::FP32;
        return self.cast(FP32).to_string();
    }

    let result = if self.get_sign() { "-" } else { "" };
    let mut result: String = result.to_string();

    let body: String = match self.get_category() {
        super::float::Category::Infinity => "Inf".to_string(),
        super::float::Category::NaN => "NaN".to_string(),
        super::float::Category::Normal => self.convert_normal_to_string(),
        super::float::Category::Zero => "0.0".to_string(),
    };

    result.push_str(&body);
    result
}
```

--------------------------------

### string

Source: https://docs.rs/arpfloat/latest/src/arpfloat/string.rs.html

string

```rust
impl Display for Float {
    fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {
        write!(f, "{}", self.convert_to_string())
    }
}
```

--------------------------------

### string

Source: https://docs.rs/arpfloat/latest/src/arpfloat/string.rs.html

string

```rust
impl RoundingMode {
    pub fn as_string(&self) -> &str {
        match self {
            RoundingMode::None => "None",
            RoundingMode::NearestTiesToEven => "NearestTiesToEven",
            RoundingMode::NearestTiesToAway => "NearestTiesToAway",
            RoundingMode::Zero => "Zero",
            RoundingMode::Positive => "Positive",
            RoundingMode::Negative => "Negative",
        }
    }
}
```

--------------------------------

### string

Source: https://docs.rs/arpfloat/latest/src/arpfloat/string.rs.html

string

```rust
impl Display for Semantics {
    fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {
        write!(
            f,
            "(exponent:{} precision:{} rm:{})",
            self.get_exponent_len(),
            self.get_precision(),
            self.get_rounding_mode().as_string()
        )
    }
}
```

--------------------------------

### string

Source: https://docs.rs/arpfloat/latest/src/arpfloat/string.rs.html

string

```rust
#[cfg(feature = "std")]
mod from {
    use core::fmt::{Debug, Display};
    use std::error::Error;

    use crate::{BigInt, Float, Semantics, FP64};

    impl Float {
        /// Try to construct a Float instance with semantics 'sem' from the
        /// string 'value'. Note that the operation of conversion might lose
        /// precision. If you care about precision you might want to use a
        /// higher precision float and downcast.
        pub fn try_from_str(
            value: &str,
            sem: Semantics,
        ) -> Result<Self, ParseError> {
            // Handle the empty case.
            if value.is_empty() {
                return Err(ParseError(ParseErrorKind::InputEmpty));
            }

            // Handle the plus or minus in front of the number.
            let chars = value.as_bytes();
            let (sign, skip) = if chars[0] == b'-' || chars[0] == b'+' {
                (chars[0] == b'-', 1)
            } else {
                (false, 0)
            };
            let value = &value[skip..];

            // Handle Nan.
            if value.eq_ignore_ascii_case("nan") {
                return Ok(Self::nan(sem, sign));
            }

            // Handle Inf.
            if value.eq_ignore_ascii_case("inf") {
                return Ok(Self::inf(sem, sign));
            }

            // Start handling the non trivial cases.

            let l_r = value.split_once('.');
            // Handle cases where we have no `.` and as such no mantissa.
            if l_r.is_none() {
                // No period. Just parse the integer.
                let ((num, _), exp_num) = parse_with_exp(value)?;
                let mut num = Float::from_bigint(sem, num);

                // Shift the number according to the exponent (in decimal).
                if let Some(exp) = exp_num {
                    if exp >= 0 {
                        num *= Float::from_bigint(
                            sem,
                            BigInt::from_u64(10).powi(exp as u64),
                        );
                    } else {
                        num /= Float::from_bigint(
                            sem,
                            BigInt::from_u64(10).powi((-exp) as u64),
                        );
                    }
                }
                num.set_sign(sign);
                return Ok(num);
            }
        }
    }
}
```

--------------------------------

### string

Source: https://docs.rs/arpfloat/latest/src/arpfloat/string.rs.html

string

```rust
impl Display for BigInt {
    fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {
        write!(f, "{}", self.as_binary())
    }
}
```

--------------------------------

### grow and shrink methods

Source: https://docs.rs/arpfloat/latest/src/arpfloat/bigint.rs.html

grow ensures at least 'size' words exist by pushing zeros. shrink removes leading zero words while keeping at least two words.

```rust
    /// Ensure that there are at least 'size' words in the bigint.
    pub fn grow(&mut self, size: usize) {
        for _ in self.len()..size {
            self.parts.push(0);
        }
    }

    /// Remove the leading zero words from the bigint.
    fn shrink(&mut self) {
        while self.len() > 2 && self.parts[self.len() - 1] == 0 {
            self.parts.pop();
        }
    }
```

--------------------------------

### Convert floating point numbers to Continued Fractions

Source: https://docs.rs/arpfloat/latest/index.html

Demonstrates converting the natural logarithm of 2 to a continued fraction approximation using the as_fraction method.

```rust
 use arpfloat::{Float, FP256, RoundingMode};

 let ln = Float::ln2(FP256);
 println!("ln(2) = {}", ln);
 for i in 1..20 {
   let (p,q) = ln.as_fraction(i);
   println!("{}/{}", p.as_decimal(), q.as_decimal());
 }
```

--------------------------------

### from_i64

Source: https://docs.rs/arpfloat/latest/arpfloat/struct.Float.html

Loads a signed 64-bit integer into a Float.

```APIDOC
## from_i64

### Description
Load the integer `val` into the float. Notice that the number may overflow or rounded.

### Method
N/A (Rust function)

### Parameters
- **sem** (`Semantics`) - Required - The semantics of the float.
- **val** (`i64`) - Required - The integer value to load.

### Returns
- **Self** - The resulting Float.
```

--------------------------------

### as_fraction method implementation

Source: https://docs.rs/arpfloat/latest/src/arpfloat/operations/frac.rs.html

Implements Float::as_fraction, which converts a Float to a continued fraction (a0 + 1/(a1 + ...)) using n iterations. Returns (0, 0) for Inf and NaN, and (0, 1) for zero. The algorithm is based on 'Elementary Functions: Algorithms and Implementation' section 9.3.1.

```rust
//! This module contains the implementation of methods that compute continued
//! fraction.

use crate::{bigint::BigInt, Float};

impl Float {
    /// Convert the number to a Continued Fraction of two integers.
    /// The fraction is computed using 'n' iterations of the form:
    /// a0 + 1/(a1 + 1/(a2 + 1/( ... ))).
    /// This method discards the sign, and returns (0, 0) for Inf and NaN.
    pub fn as_fraction(&self, n: usize) -> (BigInt, BigInt) {
        if self.is_zero() {
            return (BigInt::zero(), BigInt::one()); // Zero.
        } else if self.is_inf() || self.is_nan() {
            return (BigInt::zero(), BigInt::zero()); // Invalid.
        }

        // Algorithm from:
        // Elementary Functions: Algorithms and Implementation
        // 9.3.1 A few basic notions on continued fractions - Page 180.
        extern crate alloc;
        use alloc::vec::Vec;
        let sem = self.get_semantics();
        let rm = sem.get_rounding_mode();

        let one = Self::one(sem, false);
        let mut real = self.clone();
        let mut a: Vec<BigInt> = Vec::new();

        for _ in 0..n.max(2) {
            let int = real.trunc();
            a.push(int.convert_normal_to_integer(rm));
            real = &one / (real - int);
        }

        let one = BigInt::one();
        let mut p = (&one + &(&a[0] * &a[1]), a[0].clone());
        let mut q = (a[1].clone(), one);

        if n < 2 {
            return (p.1, q.1);
        }

        for elem in a.iter().skip(2) {
            p = (&p.1 + &(elem * &p.0), p.0);
            q = (&q.1 + &(elem * &q.0), q.0);
        }

        (p.0, q.0)
    }
}
```

--------------------------------

### Log and exponential functions implementation

Source: https://docs.rs/arpfloat/latest/src/arpfloat/operations/exp.rs.html

This code defines the log and exp functions for the Float type, including Taylor series and range reduction methods. It handles special cases and uses a loop for convergence.

```rust
//! This module contains the implementation of log- and exp-related methods.
//!

use crate::RoundingMode;

use crate::float::Float;

impl Float {
    /// Computes the taylor series, centered around 1, and valid in [0..2].
    /// z = (x - 1)/(x + 1)
    /// log(x) = 2 (z + z^3/3 + z^5/5 + z^7/7 ... )
    fn log_taylor(x: &Self) -> Self {
        use RoundingMode::None as rm;
        let sem = x.get_semantics();
        let one = Self::one(sem, false);
        let up = Float::sub_with_rm(x, &one, rm);
        let down = Float::add_with_rm(x, &one, rm);
        let z = Float::div_with_rm(&up, &down, rm);
        let z2 = z.sqr();

        let mut top = z;
        let mut sum = Self::zero(sem, false);
        let mut prev = Self::one(sem, true);
        for i in 0..50 {
            if prev == sum {
                break; // Stop if we are not making progress.
            }
            prev = sum.clone();

            let bottom = &Self::from_u64(sem, i * 2 + 1);
            let elem = Float::div_with_rm(&top, bottom, rm);
            sum = Float::add_with_rm(&sum, &elem, rm);

            // Prepare the next iteration.
            top = Float::mul_with_rm(&top, &z2, rm);
        }

        sum.scale(1, RoundingMode::Zero)
    }

    /// Reduce the range of 'x' with the identity:
    /// ln(x) = ln(sqrt(x)^2) = 2 * ln(sqrt(x)) and
    /// ln(x) = -ln(1/x)
    fn log_range_reduce(x: &Self) -> Self {
        use RoundingMode::NearestTiesToEven as even;
        let sem = x.get_semantics();
        let up = Self::from_f64(1.001).cast(sem);
        let one = Self::from_u64(sem, 1);

        if x > &up {
            let sx = x.sqrt();
            return Self::log_range_reduce(&sx).scale(1, even);
        }

        if x < &one {
            let re = Float::div_with_rm(&one, x, RoundingMode::None);
            return Self::log_range_reduce(&re).neg();
        }

        Self::log_taylor(x)
    }

    /// Computes logarithm of 'x'.
    pub fn log(&self) -> Self {
        use RoundingMode::None as rm;
        let sem = self.get_semantics();

        //Fast Logarithm function for Arbitrary Precision number,
        // by Henrik Vestermark.

        // Handle all of the special cases:
        if !self.is_normal() || self.is_negative() {
            return Self::nan(sem, self.get_sign());
        }

        let orig_sem = self.get_semantics();
        let sem = orig_sem.grow_log(10).increase_exponent(10);

        let x = &self.cast_with_rm(sem, rm);
        Self::log_range_reduce(x).cast_with_rm(orig_sem, rm)
    }
}

#[test]
fn test_log() {
    use crate::FP128;
    let x = Float::from_f64(0.1).cast(FP128).log();
    assert_eq!(x.as_f64(), -2.3025850929940455);

    for x in [
        0.1, 0.5, 2.3, 4.5, 9.8, 11.2, 15.2, 91.2, 102.2, 192.4, 1024.2,
        90210.2,
    ] {
        let lhs = Float::from_f64(x).cast(FP128).log().as_f64();
        let rhs = x.ln();
        assert_eq!(lhs, rhs);
    }
}

impl Float {
    /// Computes the taylor series:
    /// exp(x) = 1 + x/1! + x^2/2! + x^3/3! ...
    fn exp_taylor(x: &Self) -> Self {
        let sem = x.get_semantics();
        use crate::bigint::BigInt;
        let mut top = Self::one(sem, false);
        let mut bottom = BigInt::one();

        let mut sum = Self::zero(sem, false);
        let mut prev = Self::one(sem, true);
        for k in 1..50 {
            if prev == sum {
                break; // Stop if we are not making progress.
            }
            prev = sum.clone();

            let elem = &top / &Self::from_bigint(sem, bottom.clone());
            sum += elem;

            // Prepare the next iteration.
            bottom *= BigInt::from_u64(k);
            top = &top * x;
        }

        sum
    }

    /// Reduce the range of 'x' with the identity:
    /// e^x = (e^(x/2))^2
    fn exp_range_reduce(x: &Self) -> Self {
        let sem = x.get_semantics();

        let one = Self::from_u64(sem, 1);

        if x > &one {
            let sx = x.scale(-3, RoundingMode::Zero);
            let esx = Self::exp_range_reduce(&sx);
            return esx.sqr().sqr().sqr();
        }

        Self::exp_taylor(x)
    }

    /// Computes exponential function `e^self`.
    pub fn exp(&self) -> Self {
        let sem = self.get_semantics();

        // Handle all of the special cases:
        if self.is_zero() {
            return Self::one(sem, false);
        } else if !self.is_normal() {
            return Self::nan(sem, self.get_sign());
        }

```

--------------------------------

### arpfloat utils.rs: mask, Lfsr, and mul_part helpers

Source: https://docs.rs/arpfloat/latest/src/arpfloat/utils.rs.html

The complete source of arpfloat's utils.rs module, containing helper functions and test helpers. It defines mask(), get_special_test_values(), the Lfsr struct with its Iterator implementation, and the mul_part() function, along with tests for each. The Lfsr is used as a random number generator for tests, and mul_part() splits a 64-bit multiplication into low and high parts.

```rust
//! This file contains simple helper functions and test helpers.

/// Returns a mask full of 1s, of `b` bits.
pub fn mask(b: usize) -> usize {
    (1 << (b)) - 1
}

#[test]
fn test_masking() {
    assert_eq!(mask(0), 0x0);
    assert_eq!(mask(1), 0x1);
    assert_eq!(mask(8), 255);
}

#[cfg(feature = "std")]
#[allow(dead_code)]
/// Returns list of interesting values that various tests use to catch edge cases.
pub fn get_special_test_values() -> [f64; 22] {
    [
        -f64::NAN,
        f64::NAN,
        f64::INFINITY,
        f64::NEG_INFINITY,
        f64::EPSILON,
        -f64::EPSILON,
        0.000000000000000000000000000000000000001,
        f64::MIN,
        f64::MAX,
        std::f64::consts::PI,
        std::f64::consts::LN_2,
        std::f64::consts::SQRT_2,
        std::f64::consts::E,
        0.0,
        -0.0,
        10.,
        -10.,
        -0.00001,
        0.1,
        355. / 113.,
        -1.0,
        -1.1,
    ]
}

// Linear-feedback shift register. We use this as a random number generator for
// tests.
pub struct Lfsr {
    state: u32,
}

impl Default for Lfsr {
    fn default() -> Self {
        Self::new()
    }
}

impl Lfsr {
    /// Generate a new LFSR number generator.
    pub fn new() -> Lfsr {
        Lfsr { state: 0x13371337 }
    }

    /// Generate a new LFSR number generator that starts with a specific state.
    pub fn new_with_seed(seed: u32) -> Lfsr {
        Lfsr {
            state: 0x13371337 ^ seed,
        }
    }

    pub fn next(&mut self) {
        let a = (self.state >> 24) & 1;
        let b = (self.state >> 23) & 1;
        let c = (self.state >> 22) & 1;
        let d = (self.state >> 17) & 1;
        let n = a ^ b ^ c ^ d ^ 1;
        self.state <<= 1;
        self.state |= n;
    }

    fn get(&mut self) -> u32 {
        let mut res: u32 = 0;
        for _ in 0..32 {
            self.next();
            res <<= 1;
            res ^= self.state & 0x1;
        }
        res
    }

    pub fn get64(&mut self) -> u64 {
        ((self.get() as u64) << 32) | self.get() as u64
    }
}

// Implement `Iterator` for `Lfsr`.
impl Iterator for Lfsr {
    type Item = u64;
    fn next(&mut self) -> Option<Self::Item> {
        Some(self.get64())
    }
}

#[test]
fn test_lfsr_balance() {
    let mut lfsr = Lfsr::new();

    // Count the number of items, and the number of 1s.
    let mut items = 0;
    let mut ones = 0;

    for _ in 0..10000 {
        let mut u = lfsr.get();
        for _ in 0..32 {
            items += 1;
            ones += u & 1;
            u >>= 1;
        }
    }
    // Make sure that we have around 50% 1s and 50% zeros.
    assert!((ones as f64) < (0.55 * items as f64));
    assert!((ones as f64) > (0.45 * items as f64));
}
#[test]
fn test_repetition() {
    let mut lfsr = Lfsr::new();
    let first = lfsr.get();
    let second = lfsr.get();

    // Make sure that the items don't repeat themselves too frequently.
    for _ in 0..30000 {
        assert_ne!(first, lfsr.get());
        assert_ne!(second, lfsr.get());
    }
}

// Multiply a and b, and return the (low, high) parts.
#[allow(dead_code)]
fn mul_part(a: u64, b: u64) -> (u64, u64) {
    let half_bits = u64::BITS / 2;
    let half_mask = (1 << half_bits) - 1;

    let a_lo = a & half_mask;
    let a_hi = a >> half_bits;
    let b_lo = b & half_mask;
    let b_hi = b >> half_bits;

    let ab_hi = a_hi * b_hi;
    let ab_mid = a_hi * b_lo;
    let ba_mid = b_hi * a_lo;
    let ab_low = a_lo * b_lo;

    let carry =
        ((ab_mid & half_mask) + (ba_mid & half_mask) + (ab_low >> half_bits))
            >> half_bits;
    let low = (ab_mid << half_bits)
        .overflowing_add(ba_mid << half_bits)
        .0
        .overflowing_add(ab_low)
        .0;

    let high = (ab_hi + (ab_mid >> half_bits) + (ba_mid >> half_bits)) + carry;
    (low, high)
}

#[test]
fn test_mul_parts() {
    use super::utils::Lfsr;

    let mut lfsr = Lfsr::new();

    for _ in 0..500 {
        let v0 = lfsr.get64();
        let v1 = lfsr.get64();
        let res = mul_part(v0, v1);
        let full = v0 as u128 * v1 as u128;
        assert_eq!(full as u64, res.0);
        assert_eq!((full >> 64) as u64, res.1);
    }
}

```

--------------------------------

### Implement arithmetic operations on Float (powi, sqr, sqrt, abs, max, min)

Source: https://docs.rs/arpfloat/latest/src/arpfloat/operations/functions.rs.html

This snippet contains the full implementation of arithmetic operations for the Float type, including powi, sqr, sqrt, abs, max, and min. It uses the Newton-Raphson method for sqrt and handles special values like zero, NaN, infinity, and negative numbers. The powi method uses a binary exponentiation algorithm.

```rust
//! This module contains the implementation of several arithmetic operations.

use crate::RoundingMode;

use crate::float::Float;

impl Float {
    /// Return this number raised to the power of 'n'.
    pub fn powi(&self, mut n: u64) -> Self {
        let sem = self.get_semantics().increase_precision(2);
        let mut elem = Self::one(sem, false);
        // This algorithm is similar to binary conversion. Each bit in 'n'
        // represents a power-of-two number, like 1,2,4,8 ... We know how to
        // generate numbers to the power of an even number by squaring the
        // number log2 times. So, we just multiply all of the numbers together
        // to get the result. This is like converting a binary number to integer
        // except that instead of adding we multiply the values.
        let mut val = self.cast(sem);
        while n > 0 {
            if n & 1 == 1 {
                elem *= &val;
            }
            val *= &val.clone();
            n >>= 1;
        }
        elem.cast(self.get_semantics())
    }

    /// Calculates the power of two.
    pub fn sqr(&self) -> Self {
        self.powi(2)
    }
    /// Calculates the square root of the number.
    pub fn sqrt(&self) -> Self {
        let sem = self.get_semantics();
        if self.is_zero() {
            return self.clone(); // (+/-) zero
        } else if self.is_nan() || self.is_negative() {
            return Self::nan(sem, self.get_sign()); // (-/+)Nan, -Number.
        } else if self.is_inf() {
            return self.clone(); // Inf+.
        }

        let target = self.clone();
        let two = Self::from_u64(sem, 2);

        // Start the search at max(2, x).
        let mut x = if target < two { two } else { target.clone() };
        let mut prev = x.clone();

        // Use the Newton Raphson method.
        loop {
            x += &target / &x;
            x = x.scale(-1, RoundingMode::NearestTiesToEven);
            // Stop when value did not change or regressed.
            if prev < x || x == prev {
                return x;
            }
            prev = x.clone();
        }
    }

    /// Returns the absolute value of this float.
    pub fn abs(&self) -> Self {
        let mut x = self.clone();
        x.set_sign(false);
        x
    }

    /// Returns the greater of self and `other`.
    pub fn max(&self, other: &Self) -> Self {
        if self.is_nan() {
            return other.clone();
        } else if other.is_nan() {
            return self.clone();
        } else if self.get_sign() != other.get_sign() {
            return if self.get_sign() {
                other.clone()
            } else {
                self.clone()
            }; // Handle (+-)0.
        }
        if self > other {
            self.clone()
        } else {
            other.clone()
        }
    }

    /// Returns the smaller of self and `other`.
    pub fn min(&self, other: &Self) -> Self {
        if self.is_nan() {
            return other.clone();
        } else if other.is_nan() {
            return self.clone();
        } else if self.get_sign() != other.get_sign() {
            return if self.get_sign() {
                self.clone()
            } else {
                other.clone()
            }; // Handle (+-)0.
        }
        if self > other {
            other.clone()
        } else {
            self.clone()
        }
    }
}

```

--------------------------------

### from_u64

Source: https://docs.rs/arpfloat/latest/arpfloat/struct.Float.html

Loads an unsigned 64-bit integer into a Float.

```APIDOC
## from_u64

### Description
Load the integer `val` into the float. Notice that the number may overflow, or rounded to the nearest even integer.

### Method
N/A (Rust function)

### Parameters
- **sem** (`Semantics`) - Required - The semantics of the float.
- **val** (`u64`) - Required - The integer value to load.

### Returns
- **Self** - The resulting Float.
```

--------------------------------

### Float::pi, Float::e, and Float::ln2 implementations with tests

Source: https://docs.rs/arpfloat/latest/src/arpfloat/operations/constants.rs.html

The source code for the constants module, including the pi, e, and ln2 methods and their unit tests. The pi method uses Brent's algorithm, e uses Euler's continued fraction, and ln2 uses a series. Tests are gated behind the 'std' feature and compare results with standard library constants.

```rust
//! This module contains the implementation of methods that compute mathematical
//! constants.
//!
use crate::RoundingMode;
use crate::{Float, Semantics};

impl Float {
    /// Computes pi.
    pub fn pi(sem: Semantics) -> Self {
        // Algorithm description in Pg 246:
        // Fast Multiple-Precision Evaluation of Elementary Functions
        // by Richard P. Brent.

        // Increase the precision, because the arithmetic operations below
        // require rounding, so if we want to get the accurate results we need
        // to operate with increased precision.
        let orig_sem = sem;
        let sem = sem.grow_log(4);

        use RoundingMode::NearestTiesToEven as rm;

        let one = Self::from_i64(sem, 1);
        let two = Self::from_i64(sem, 2);
        let four = Self::from_i64(sem, 4);

        let mut a = one.clone();
        let mut b = one.clone() / two.sqrt();
        let mut t = one.clone() / four;
        let mut x = one;

        while a != b {
            let y = a.clone();
            a = (&a + &b).scale(-1, rm);
            b = (&b * &y).sqrt();
            t -= &x * (&a - &y).sqr();
            x = x.scale(1, rm);
        }
        (a.sqr() / t).cast(orig_sem)
    }

    /// Computes e.
    pub fn e(sem: Semantics) -> Self {
        let orig_sem = sem;
        let sem = sem.increase_precision(1);

        let one = Self::one(sem, false);
        let mut term = one.clone();

        // Use Euler's continued fraction, which is a simple series.
        let iterations: i64 = (sem.get_exponent_len() * 2) as i64;
        for i in (1..iterations).rev() {
            let v = Self::from_i64(sem, i);
            term = &v + &v / &term;
        }

        (one / term + 2).cast(orig_sem)
    }

    /// Compute log(2).
    pub fn ln2(sem: Semantics) -> Self {
        use RoundingMode::None as rm;
        let sem2 = sem.increase_precision(8);

        // Represent log(2) using the sum 1/k*2^k
        let one = Self::one(sem2, false);
        let mut sum = Self::zero(sem2, false);
        let mut prev = Self::inf(sem2, true);
        for k in 1..500 {
            let k2 = Self::from_u64(sem2, 1).scale(k, rm);
            let k = Self::from_u64(sem2, k as u64);
            let kk2 = &Float::mul_with_rm(&k, &k2, rm);
            let term = Float::div_with_rm(&one, kk2, rm);
            sum = Float::add_with_rm(&sum, &term, rm);
            if prev == sum {
                break;
            }
            prev = sum.clone();
        }
        sum.cast(sem)
    }
}

#[cfg(feature = "std")]
#[test]
fn test_pi() {
    use crate::FP32;
    use crate::FP64;
    assert_eq!(Float::pi(FP64).as_f64(), std::f64::consts::PI);
    assert_eq!(Float::pi(FP32).as_f32(), std::f32::consts::PI);
}

#[cfg(feature = "std")]
#[test]
fn test_e() {
    use crate::FP32;
    use crate::FP64;
    assert_eq!(Float::e(FP64).as_f64(), std::f64::consts::E);
    assert_eq!(Float::e(FP32).as_f32(), std::f32::consts::E);
}

#[cfg(feature = "std")]
#[test]
fn test_ln2() {
    use crate::FP64;
    assert_eq!(Float::ln2(FP64).as_f64(), std::f64::consts::LN_2);
}

```

--------------------------------

### Scale method implementation

Source: https://docs.rs/arpfloat/latest/src/arpfloat/operations/functions.rs.html

Implementation of the scale method on Float, which adds or subtracts to the exponent, scaling by 2^exp. It handles non-normal numbers by returning a clone and uses the specified rounding mode.

```rust
impl Float {
    /// Similar to 'scalbln'. Adds or subtracts to the exponent of the number,
    /// and scaling it by 2^exp.
    pub fn scale(&self, scale: i64, rm: RoundingMode) -> Self {
        use crate::bigint::LossFraction;
        if !self.is_normal() {
            return self.clone();
        }

        let mut r = Self::from_parts(
            self.get_semantics(),
            self.get_sign(),
            self.get_exp() + scale,
            self.get_mantissa(),
        );
        r.normalize(rm, LossFraction::ExactlyZero);
        r
    }
```

--------------------------------

### Test loading and storing all f32 values

Source: https://docs.rs/arpfloat/latest/src/arpfloat/cast.rs.html

Iterates over all possible f32 bit patterns (except NaN) and verifies that loading and storing preserves the value.

```rust
// Try to load and store normals and denormals.
for i in 0..(1u64 << 16) {
    let in_f = f32::from_bits((i << 10) as u32);
    let fp_f = Float::from_f32(in_f);
    let out_f = fp_f.as_f32();
    assert_eq!(in_f.is_nan(), out_f.is_nan());
    assert_eq!(in_f.is_infinite(), out_f.is_infinite());
    assert!(in_f.is_nan() || (in_f.to_bits() == out_f.to_bits()));
}
```

--------------------------------

### Test casting from i64 to f64 for FP16

Source: https://docs.rs/arpfloat/latest/src/arpfloat/cast.rs.html

Verifies that casting various i64 values to FP16 and then to f64 produces expected results, including saturation to infinity for large values.

```rust
assert_eq!(Float::from_i64(FP16, 0).as_f64(), 0.);
assert_eq!(Float::from_i64(FP16, 65500).as_f64(), 65504.0);
assert_eq!(Float::from_i64(FP16, 65504).as_f64(), 65504.0);
assert_eq!(Float::from_i64(FP16, 65519).as_f64(), 65504.0);
assert_eq!(Float::from_i64(FP16, 65520).as_f64(), f64::INFINITY);
assert_eq!(Float::from_i64(FP16, 65536).as_f64(), f64::INFINITY);
```

--------------------------------

### from_f32

Source: https://docs.rs/arpfloat/latest/arpfloat/struct.Float.html

Loads a native f32 value into a Float.

```APIDOC
## from_f32

### Description
Loads and converts a native fp32 value. Notice that the number may overflow or rounded (see cast and cast_with_rm).

### Method
N/A (Rust function)

### Parameters
- **float** (`f32`) - Required - The f32 value to load.

### Returns
- **Self** - The resulting Float.
```

--------------------------------

### Test for Simple Division (test_div_simple)

Source: https://docs.rs/arpfloat/latest/src/arpfloat/arithmetic.rs.html

Tests division of two f64 values by converting them to Float, performing division with rounding to nearest ties to even, and asserting the result matches the f64 division.

```rust
#[test]
fn test_div_simple() {
    let a: f64 = 1.0;
    let b: f64 = 7.0;

    let af = Float::from_f64(a);
    let bf = Float::from_f64(b);
    let cf = Float::div_with_rm(&af, &bf, RoundingMode::NearestTiesToEven);

    let r0 = cf.as_f64();
    let r1: f64 = a / b;
    assert_eq!(r0, r1);
}
```
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