FYL2X instruction (x87)

FYL2X computes y * log2(x), with x in ST(0) and y in ST(1): it stores the product in ST(1) and pops, so the result ends up in ST(0). The factor y is what makes it useful. The x87 has only the binary logarithm, and the others follow from it by a constant that a load instruction pushes first:

the natural logarithm: FLDLN2, x, FYL2X, since ln(x) = ln(2) * log2(x);
the common logarithm: FLDLG2 instead, log10(2);
log2(x) itself: FLD1.

The product is rounded once, so only the constant has been rounded before it. For x close to 1, where x - 1 has lost bits by the time FYL2X sees x, FYL2XP1 takes x - 1 instead.

The edges

x = 0 of either sign is a division by zero: ZE, and y times -infinity. Zen 5 gave -infinity for y = 1, and +infinity for y = -2.5 and x = -0.
A negative x, -infinity included, is an invalid operation: IE, and with it masked the QNaN indefinite. So are the products of a zero and an infinity: y = 0 with x = 0 or +infinity, and an infinite y with x = 1. An infinite y with x = 0 gives an infinity without ZE.
log2(1) is an exact +0 for y = 2. Other exact results still raise PE on Zen 5: log2(8) * 3 gave 9, and log2(0.5) * -1 gave 1, both with PE.
A denormal x sets DE: the smallest, 2^-16445, gave -16445.
The largest x times the largest y overflows: OE.

For ordinary operands the page’s pseudocode checks Zen 5’s results within 1 ulp (1.5 when not rounding to nearest), and the worst of its 20 recorded results was 0.42 ulp to nearest and 0.94 in the other modes.

From C

GCC 14 and Clang 23 at -O2 compile logl, log2l and log10l on a long double to calls of the library. With -ffast-math, GCC emits the three sequences above, a constant, an fld of x and fyl2x; Clang 23 still calls the library.

AMD’s Zen 5 guide lists FYL2X as microcoded, and Intel’s optimization manual suggests software implementations where 80 bits are not needed (Section 3.9.1).

Forms

Every encoding of FYL2X that Intel XED knows, one row per XED instruction form (iform). How to read the notation.

Instruction	Encoding	Requires
`FYL2X` FYL2X	`D9 F1`	FPU

Operation

What the forms do, in pseudocode that our tests run against results recorded on real processors. How to read.

x = st0
y = st1
if empty(st0) or empty(st1):
    underflow()                  # IE, SF, C1 = 0
    st1 = INDEFINITE
else:
    st1 = ylog2(y, x)            # y * log2(x), rounded once, within
                                 # 1 ulp (1.5 when not rounding to
                                 # nearest); Zen 5's 20 recorded
                                 # results: 0.42 at most to nearest,
                                 # 0.94 in the other modes. x = 0
                                 # raises ZE; x < 0, 0 *
                                 # log2(0), inf * log2(1) and 0 * inf
                                 # raise IE
pop()                            # the result is ST(0) now
c0 = c2 = c3 = UNDEFINED         # Zen 5 keeps them

Run on AMD Zen 5 (EPYC 9645): 52 cases.

Operands

What each form reads and writes, as Intel XED records it: the operands of its syntax, in order, and the implicit ones that the syntax does not name. How to read.

Operand	Kind	Action
`FYL2X`
`ST(0)`	implicit	read
`ST(1)`	implicit	read, write
`x87 register stack`	implicit	pop
`x87 status word`	implicit	write

Flags

M modified, U undefined; blank: not affected. How to read.

C0	C1	C2	C3
All forms
U	M	U	U

Attributes

What else XED records about the forms. How to read.

All forms: may abort a TSX transaction

CPUID and processors

Each form belongs to an XED ISA set. A form can be used when the processor reports every CPUID bit of one of its ISA set's alternatives.

CPUID requirements and processors of the XED ISA sets above. Alternatives are separated by “or”; processors follow XED’s chip model, whose AMD processors end with Zen 2, and the CPU data of this site for Zen 3 to Zen 6.

ISA set	CPUID	Processors
`X87` x87	FPU `CPUID.01H:EDX[0]` FPU	Every x86-64 processor in XED’s model and Zen 3 to Zen 6

More x87 instructions

All 90 x87 instructions

The edges#

From C#