dl, fracl := dl>>extra, dl&extraMask
	dc, fracc := dc>>extra, dc&extraMask
	du, fracu := du>>extra, du&extraMask
	// Is it allowed to use 'du' as a result?
	// It is always allowed when it is truncated, but also
	// if it is exact and the original binary mantissa is even
	// When disallowed, we can subtract 1.
	uok := !du0 || fracu > 0
	if du0 && fracu == 0 {
		uok = mant&1 == 0
	}

// The pattern matched executes the following computation:
// frac = x - floor(x)
// to_even = (floor(x) - 2 * floor(0.5 * x)) == 1
// if frac > 0.5 || (frac == 0.5 && to_even)
//   return floor(x) + 1
// else
//   return floor(x)
def : Pat<(MHLO_SelectOp
            (MHLO_OrOp
              (MHLO_CompareOp (MHLO_SubtractOp:$frac
                               $input,

		return frac
	}
	frac, e := normalize(frac)
	exp += e
	x := Float64bits(frac)
	exp += int(x>>shift)&mask - bias
	if exp < -1075 {
		return Copysign(0, frac) // underflow
	}
	if exp > 1023 { // overflow
		if frac < 0 {
			return Inf(-1)
		}
		return Inf(1)
	}
	var m float64 = 1
	if exp < -1022 { // denormal
		exp += 53
		m = 1.0 / (1 << 53) // 2**-53
	}
	x &^= mask << shift

// and an integral power of two.
// It returns frac and exp satisfying f == frac × 2**exp,
// with the absolute value of frac in the interval [½, 1).
//
// Special cases are:
//
//	Frexp(±0) = ±0, 0
//	Frexp(±Inf) = ±Inf, 0
//	Frexp(NaN) = NaN, 0
func Frexp(f float64) (frac float64, exp int) {
	if haveArchFrexp {
		return archFrexp(f)
	}
	return frexp(f)
}

func frexp(f float64) (frac float64, exp int) {
	// special cases

var timekeepSharedPage *vdsoTimekeep

//go:nosplit
func (bt *bintime) Add(bt2 *bintime) {
	u := bt.frac
	bt.frac += bt2.frac
	if u > bt.frac {
		bt.sec++
	}
	bt.sec += bt2.sec
}

//go:nosplit
func (bt *bintime) AddX(x uint64) {
	u := bt.frac
	bt.frac += x
	if u > bt.frac {
		bt.sec++
	}
}

var (
	// binuptimeDummy is used in binuptime as the address of an atomic.Load, to simulate

//
//	Modf(±Inf) = ±Inf, NaN
//	Modf(NaN) = NaN, NaN
func Modf(f float64) (int float64, frac float64) {
	if haveArchModf {
		return archModf(f)
	}
	return modf(f)
}

func modf(f float64) (int float64, frac float64) {
	if f < 1 {
		switch {
		case f < 0:
			int, frac = Modf(-f)
			return -int, -frac
		case f == 0:
			return f, f // Return -0, -0 when f == -0
		}
		return 0, f
	}

// license that can be found in the LICENSE file.

#include "textflag.h"

// func archModf(f float64) (int float64, frac float64)
TEXT ·archModf(SB),NOSPLIT,$0
	MOVD	f+0(FP), R0
	FMOVD	R0, F0
	FRINTZD	F0, F1
	FMOVD	F1, int+8(FP)
	FSUBD	F1, F0
	FMOVD	F0, R1
	AND	$(1<<63), R0
	ORR	R0, R1 // must have same sign
	MOVD	R1, frac+16(FP)

	fmt.Printf("%.2f\n", math.Cbrt(8))
	fmt.Printf("%.2f\n", math.Cbrt(27))
	// Output:
	// 2.00
	// 3.00
}

func ExampleModf() {
	int, frac := math.Modf(3.14)
	fmt.Printf("%.2f, %.2f\n", int, frac)

	int, frac = math.Modf(-2.71)
	fmt.Printf("%.2f, %.2f\n", int, frac)
	// Output:
	// 3.00, 0.14
	// -2.00, -0.71

	if haveArchLog2 {
		return archLog2(x)
	}
	return log2(x)
}

func log2(x float64) float64 {
	frac, exp := Frexp(x)
	// Make sure exact powers of two give an exact answer.
	// Don't depend on Log(0.5)*(1/Ln2)+exp being exactly exp-1.
	if frac == 0.5 {
		return float64(exp - 1)
	}
	return Log(frac)*(1/Ln2) + float64(exp)

	if n%3 != 1 {
		term.SetInt64(1)
	} else {
		term.SetInt64((n - 1) / 3 * 2)
	}

	if n > lim {
		return term
	}

	// Directly initialize frac as the fractional
	// inverse of the result of recur.
	frac := new(big.Rat).Inv(recur(n+1, lim))

	return term.Add(term, frac)
}

// This example demonstrates how to use big.Rat to compute the
// first 15 terms in the sequence of rational convergents for