// The functions are complex with many branches, and explicit
// `return`s makes it clear where function exit points are
#![allow(clippy::needless_return)]
use float::Float;
use int::{CastInto, DInt, HInt, Int};
fn div32<F: Float>(a: F, b: F) -> F
where
u32: CastInto<F::Int>,
F::Int: CastInto<u32>,
i32: CastInto<F::Int>,
F::Int: CastInto<i32>,
F::Int: HInt,
{
let one = F::Int::ONE;
let zero = F::Int::ZERO;
// let bits = F::BITS;
let significand_bits = F::SIGNIFICAND_BITS;
let max_exponent = F::EXPONENT_MAX;
let exponent_bias = F::EXPONENT_BIAS;
let implicit_bit = F::IMPLICIT_BIT;
let significand_mask = F::SIGNIFICAND_MASK;
let sign_bit = F::SIGN_MASK as F::Int;
let abs_mask = sign_bit - one;
let exponent_mask = F::EXPONENT_MASK;
let inf_rep = exponent_mask;
let quiet_bit = implicit_bit >> 1;
let qnan_rep = exponent_mask | quiet_bit;
#[inline(always)]
fn negate_u32(a: u32) -> u32 {
(<i32>::wrapping_neg(a as i32)) as u32
}
let a_rep = a.repr();
let b_rep = b.repr();
let a_exponent = (a_rep >> significand_bits) & max_exponent.cast();
let b_exponent = (b_rep >> significand_bits) & max_exponent.cast();
let quotient_sign = (a_rep ^ b_rep) & sign_bit;
let mut a_significand = a_rep & significand_mask;
let mut b_significand = b_rep & significand_mask;
let mut scale = 0;
// Detect if a or b is zero, denormal, infinity, or NaN.
if a_exponent.wrapping_sub(one) >= (max_exponent - 1).cast()
|| b_exponent.wrapping_sub(one) >= (max_exponent - 1).cast()
{
let a_abs = a_rep & abs_mask;
let b_abs = b_rep & abs_mask;
// NaN / anything = qNaN
if a_abs > inf_rep {
return F::from_repr(a_rep | quiet_bit);
}
// anything / NaN = qNaN
if b_abs > inf_rep {
return F::from_repr(b_rep | quiet_bit);
}
if a_abs == inf_rep {
if b_abs == inf_rep {
// infinity / infinity = NaN
return F::from_repr(qnan_rep);
} else {
// infinity / anything else = +/- infinity
return F::from_repr(a_abs | quotient_sign);
}
}
// anything else / infinity = +/- 0
if b_abs == inf_rep {
return F::from_repr(quotient_sign);
}
if a_abs == zero {
if b_abs == zero {
// zero / zero = NaN
return F::from_repr(qnan_rep);
} else {
// zero / anything else = +/- zero
return F::from_repr(quotient_sign);
}
}
// anything else / zero = +/- infinity
if b_abs == zero {
return F::from_repr(inf_rep | quotient_sign);
}
// one or both of a or b is denormal, the other (if applicable) is a
// normal number. Renormalize one or both of a and b, and set scale to
// include the necessary exponent adjustment.
if a_abs < implicit_bit {
let (exponent, significand) = F::normalize(a_significand);
scale += exponent;
a_significand = significand;
}
if b_abs < implicit_bit {
let (exponent, significand) = F::normalize(b_significand);
scale -= exponent;
b_significand = significand;
}
}
// Or in the implicit significand bit. (If we fell through from the
// denormal path it was already set by normalize( ), but setting it twice
// won't hurt anything.)
a_significand |= implicit_bit;
b_significand |= implicit_bit;
let mut quotient_exponent: i32 = CastInto::<i32>::cast(a_exponent)
.wrapping_sub(CastInto::<i32>::cast(b_exponent))
.wrapping_add(scale);
// Align the significand of b as a Q31 fixed-point number in the range
// [1, 2.0) and get a Q32 approximate reciprocal using a small minimax
// polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2. This
// is accurate to about 3.5 binary digits.
let q31b = CastInto::<u32>::cast(b_significand << 8.cast());
let mut reciprocal = (0x7504f333u32).wrapping_sub(q31b);
// Now refine the reciprocal estimate using a Newton-Raphson iteration:
//
// x1 = x0 * (2 - x0 * b)
//
// This doubles the number of correct binary digits in the approximation
// with each iteration, so after three iterations, we have about 28 binary
// digits of accuracy.
let mut correction: u32 =
negate_u32(((reciprocal as u64).wrapping_mul(q31b as u64) >> 32) as u32);
reciprocal = ((reciprocal as u64).wrapping_mul(correction as u64) >> 31) as u32;
correction = negate_u32(((reciprocal as u64).wrapping_mul(q31b as u64) >> 32) as u32);
reciprocal = ((reciprocal as u64).wrapping_mul(correction as u64) >> 31) as u32;
correction = negate_u32(((reciprocal as u64).wrapping_mul(q31b as u64) >> 32) as u32);
reciprocal = ((reciprocal as u64).wrapping_mul(correction as u64) >> 31) as u32;
// Exhaustive testing shows that the error in reciprocal after three steps
// is in the interval [-0x1.f58108p-31, 0x1.d0e48cp-29], in line with our
// expectations. We bump the reciprocal by a tiny value to force the error
// to be strictly positive (in the range [0x1.4fdfp-37,0x1.287246p-29], to
// be specific). This also causes 1/1 to give a sensible approximation
// instead of zero (due to overflow).
reciprocal = reciprocal.wrapping_sub(2);
// The numerical reciprocal is accurate to within 2^-28, lies in the
// interval [0x1.000000eep-1, 0x1.fffffffcp-1], and is strictly smaller
// than the true reciprocal of b. Multiplying a by this reciprocal thus
// gives a numerical q = a/b in Q24 with the following properties:
//
// 1. q < a/b
// 2. q is in the interval [0x1.000000eep-1, 0x1.fffffffcp0)
// 3. the error in q is at most 2^-24 + 2^-27 -- the 2^24 term comes
// from the fact that we truncate the product, and the 2^27 term
// is the error in the reciprocal of b scaled by the maximum
// possible value of a. As a consequence of this error bound,
// either q or nextafter(q) is the correctly rounded
let mut quotient = (a_significand << 1).widen_mul(reciprocal.cast()).hi();
// Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0).
// In either case, we are going to compute a residual of the form
//
// r = a - q*b
//
// We know from the construction of q that r satisfies:
//
// 0 <= r < ulp(q)*b
//
// if r is greater than 1/2 ulp(q)*b, then q rounds up. Otherwise, we
// already have the correct result. The exact halfway case cannot occur.
// We also take this time to right shift quotient if it falls in the [1,2)
// range and adjust the exponent accordingly.
let residual = if quotient < (implicit_bit << 1) {
quotient_exponent = quotient_exponent.wrapping_sub(1);
(a_significand << (significand_bits + 1)).wrapping_sub(quotient.wrapping_mul(b_significand))
} else {
quotient >>= 1;
(a_significand << significand_bits).wrapping_sub(quotient.wrapping_mul(b_significand))
};
let written_exponent = quotient_exponent.wrapping_add(exponent_bias as i32);
if written_exponent >= max_exponent as i32 {
// If we have overflowed the exponent, return infinity.
return F::from_repr(inf_rep | quotient_sign);
} else if written_exponent < 1 {
// Flush denormals to zero. In the future, it would be nice to add
// code to round them correctly.
return F::from_repr(quotient_sign);
} else {
let round = ((residual << 1) > b_significand) as u32;
// Clear the implicit bits
let mut abs_result = quotient & significand_mask;
// Insert the exponent
abs_result |= written_exponent.cast() << significand_bits;
// Round
abs_result = abs_result.wrapping_add(round.cast());
// Insert the sign and return
return F::from_repr(abs_result | quotient_sign);
}
}
fn div64<F: Float>(a: F, b: F) -> F
where
u32: CastInto<F::Int>,
F::Int: CastInto<u32>,
i32: CastInto<F::Int>,
F::Int: CastInto<i32>,
u64: CastInto<F::Int>,
F::Int: CastInto<u64>,
i64: CastInto<F::Int>,
F::Int: CastInto<i64>,
F::Int: HInt,
{
let one = F::Int::ONE;
let zero = F::Int::ZERO;
// let bits = F::BITS;
let significand_bits = F::SIGNIFICAND_BITS;
let max_exponent = F::EXPONENT_MAX;
let exponent_bias = F::EXPONENT_BIAS;
let implicit_bit = F::IMPLICIT_BIT;
let significand_mask = F::SIGNIFICAND_MASK;
let sign_bit = F::SIGN_MASK as F::Int;
let abs_mask = sign_bit - one;
let exponent_mask = F::EXPONENT_MASK;
let inf_rep = exponent_mask;
let quiet_bit = implicit_bit >> 1;
let qnan_rep = exponent_mask | quiet_bit;
// let exponent_bits = F::EXPONENT_BITS;
#[inline(always)]
fn negate_u32(a: u32) -> u32 {
(<i32>::wrapping_neg(a as i32)) as u32
}
#[inline(always)]
fn negate_u64(a: u64) -> u64 {
(<i64>::wrapping_neg(a as i64)) as u64
}
let a_rep = a.repr();
let b_rep = b.repr();
let a_exponent = (a_rep >> significand_bits) & max_exponent.cast();
let b_exponent = (b_rep >> significand_bits) & max_exponent.cast();
let quotient_sign = (a_rep ^ b_rep) & sign_bit;
let mut a_significand = a_rep & significand_mask;
let mut b_significand = b_rep & significand_mask;
let mut scale = 0;
// Detect if a or b is zero, denormal, infinity, or NaN.
if a_exponent.wrapping_sub(one) >= (max_exponent - 1).cast()
|| b_exponent.wrapping_sub(one) >= (max_exponent - 1).cast()
{
let a_abs = a_rep & abs_mask;
let b_abs = b_rep & abs_mask;
// NaN / anything = qNaN
if a_abs > inf_rep {
return F::from_repr(a_rep | quiet_bit);
}
// anything / NaN = qNaN
if b_abs > inf_rep {
return F::from_repr(b_rep | quiet_bit);
}
if a_abs == inf_rep {
if b_abs == inf_rep {
// infinity / infinity = NaN
return F::from_repr(qnan_rep);
} else {
// infinity / anything else = +/- infinity
return F::from_repr(a_abs | quotient_sign);
}
}
// anything else / infinity = +/- 0
if b_abs == inf_rep {
return F::from_repr(quotient_sign);
}
if a_abs == zero {
if b_abs == zero {
// zero / zero = NaN
return F::from_repr(qnan_rep);
} else {
// zero / anything else = +/- zero
return F::from_repr(quotient_sign);
}
}
// anything else / zero = +/- infinity
if b_abs == zero {
return F::from_repr(inf_rep | quotient_sign);
}
// one or both of a or b is denormal, the other (if applicable) is a
// normal number. Renormalize one or both of a and b, and set scale to
// include the necessary exponent adjustment.
if a_abs < implicit_bit {
let (exponent, significand) = F::normalize(a_significand);
scale += exponent;
a_significand = significand;
}
if b_abs < implicit_bit {
let (exponent, significand) = F::normalize(b_significand);
scale -= exponent;
b_significand = significand;
}
}
// Or in the implicit significand bit. (If we fell through from the
// denormal path it was already set by normalize( ), but setting it twice
// won't hurt anything.)
a_significand |= implicit_bit;
b_significand |= implicit_bit;
let mut quotient_exponent: i32 = CastInto::<i32>::cast(a_exponent)
.wrapping_sub(CastInto::<i32>::cast(b_exponent))
.wrapping_add(scale);
// Align the significand of b as a Q31 fixed-point number in the range
// [1, 2.0) and get a Q32 approximate reciprocal using a small minimax
// polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2. This
// is accurate to about 3.5 binary digits.
let q31b = CastInto::<u32>::cast(b_significand >> 21.cast());
let mut recip32 = (0x7504f333u32).wrapping_sub(q31b);
// Now refine the reciprocal estimate using a Newton-Raphson iteration:
//
// x1 = x0 * (2 - x0 * b)
//
// This doubles the number of correct binary digits in the approximation
// with each iteration, so after three iterations, we have about 28 binary
// digits of accuracy.
let mut correction32: u32 =
negate_u32(((recip32 as u64).wrapping_mul(q31b as u64) >> 32) as u32);
recip32 = ((recip32 as u64).wrapping_mul(correction32 as u64) >> 31) as u32;
correction32 = negate_u32(((recip32 as u64).wrapping_mul(q31b as u64) >> 32) as u32);
recip32 = ((recip32 as u64).wrapping_mul(correction32 as u64) >> 31) as u32;
correction32 = negate_u32(((recip32 as u64).wrapping_mul(q31b as u64) >> 32) as u32);
recip32 = ((recip32 as u64).wrapping_mul(correction32 as u64) >> 31) as u32;
// recip32 might have overflowed to exactly zero in the preceeding
// computation if the high word of b is exactly 1.0. This would sabotage
// the full-width final stage of the computation that follows, so we adjust
// recip32 downward by one bit.
recip32 = recip32.wrapping_sub(1);
// We need to perform one more iteration to get us to 56 binary digits;
// The last iteration needs to happen with extra precision.
let q63blo = CastInto::<u32>::cast(b_significand << 11.cast());
let correction: u64 = negate_u64(
(recip32 as u64)
.wrapping_mul(q31b as u64)
.wrapping_add((recip32 as u64).wrapping_mul(q63blo as u64) >> 32),
);
let c_hi = (correction >> 32) as u32;
let c_lo = correction as u32;
let mut reciprocal: u64 = (recip32 as u64)
.wrapping_mul(c_hi as u64)
.wrapping_add((recip32 as u64).wrapping_mul(c_lo as u64) >> 32);
// We already adjusted the 32-bit estimate, now we need to adjust the final
// 64-bit reciprocal estimate downward to ensure that it is strictly smaller
// than the infinitely precise exact reciprocal. Because the computation
// of the Newton-Raphson step is truncating at every step, this adjustment
// is small; most of the work is already done.
reciprocal = reciprocal.wrapping_sub(2);
// The numerical reciprocal is accurate to within 2^-56, lies in the
// interval [0.5, 1.0), and is strictly smaller than the true reciprocal
// of b. Multiplying a by this reciprocal thus gives a numerical q = a/b
// in Q53 with the following properties:
//
// 1. q < a/b
// 2. q is in the interval [0.5, 2.0)
// 3. the error in q is bounded away from 2^-53 (actually, we have a
// couple of bits to spare, but this is all we need).
// We need a 64 x 64 multiply high to compute q, which isn't a basic
// operation in C, so we need to be a little bit fussy.
// let mut quotient: F::Int = ((((reciprocal as u64)
// .wrapping_mul(CastInto::<u32>::cast(a_significand << 1) as u64))
// >> 32) as u32)
// .cast();
// We need a 64 x 64 multiply high to compute q, which isn't a basic
// operation in C, so we need to be a little bit fussy.
let mut quotient = (a_significand << 2).widen_mul(reciprocal.cast()).hi();
// Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0).
// In either case, we are going to compute a residual of the form
//
// r = a - q*b
//
// We know from the construction of q that r satisfies:
//
// 0 <= r < ulp(q)*b
//
// if r is greater than 1/2 ulp(q)*b, then q rounds up. Otherwise, we
// already have the correct result. The exact halfway case cannot occur.
// We also take this time to right shift quotient if it falls in the [1,2)
// range and adjust the exponent accordingly.
let residual = if quotient < (implicit_bit << 1) {
quotient_exponent = quotient_exponent.wrapping_sub(1);
(a_significand << (significand_bits + 1)).wrapping_sub(quotient.wrapping_mul(b_significand))
} else {
quotient >>= 1;
(a_significand << significand_bits).wrapping_sub(quotient.wrapping_mul(b_significand))
};
let written_exponent = quotient_exponent.wrapping_add(exponent_bias as i32);
if written_exponent >= max_exponent as i32 {
// If we have overflowed the exponent, return infinity.
return F::from_repr(inf_rep | quotient_sign);
} else if written_exponent < 1 {
// Flush denormals to zero. In the future, it would be nice to add
// code to round them correctly.
return F::from_repr(quotient_sign);
} else {
let round = ((residual << 1) > b_significand) as u32;
// Clear the implicit bits
let mut abs_result = quotient & significand_mask;
// Insert the exponent
abs_result |= written_exponent.cast() << significand_bits;
// Round
abs_result = abs_result.wrapping_add(round.cast());
// Insert the sign and return
return F::from_repr(abs_result | quotient_sign);
}
}
intrinsics! {
#[arm_aeabi_alias = __aeabi_fdiv]
pub extern "C" fn __divsf3(a: f32, b: f32) -> f32 {
div32(a, b)
}
#[arm_aeabi_alias = __aeabi_ddiv]
pub extern "C" fn __divdf3(a: f64, b: f64) -> f64 {
div64(a, b)
}
#[cfg(target_arch = "arm")]
pub extern "C" fn __divsf3vfp(a: f32, b: f32) -> f32 {
a / b
}
#[cfg(target_arch = "arm")]
pub extern "C" fn __divdf3vfp(a: f64, b: f64) -> f64 {
a / b
}
}