Why does the FMA _mm256_fmadd_pd() intrinsic have 3 asm mnemonics, "vfmadd132pd", "231" and "213"?

The fused multiply-add instructions multiply two (packed) values, add a third value, and then overwrite one of the values with the result. Only one of the three values can be a memory operand rather than a register.

The way it works is that all three instructions overwrite ymm0 and allow only ymm2 to be a memory operand. The choice of instruction determines which two operands are multiplied and which is added.

Assuming that ymm0 is the first operand in Intel syntax (or the last in AT&T syntax):

vfmadd132pd:  ymm0 = ymm0 * ymm2/mem + ymm1
vfmadd231pd:  ymm0 = ymm1 * ymm2/mem + ymm0
vfmadd213pd:  ymm0 = ymm1 * ymm0 + ymm2/mem 

When using the C intrinsics, this choice isn't necessary: The intrinsic does not overwrite a value but returns its result instead, and it allows all three values to be read from memory. The compiler will add memory reads/writes if needed, and will allocate a temporary register to store the result if it does not want any of the three values to be overwritten. It will choose one of the three instructions as it sees fit.

This is in the assembly instruction set reference, and also in HTML extracts of it, like the entry for VFMADD*PD:

VFMADD132PD: Multiplies the two or four packed double-precision floating-point values from the first source operand to the two or four packed double-precision floating-point values in the third source operand, adds the infinite precision intermediate result to the two or four packed double-precision floating-point values in the second source operand, performs rounding and stores the resulting two or four packed double-precision floating-point values to the destination operand (first source operand).

VFMADD213PD: Multiplies the two or four packed double-precision floating-point values from the second source operand to the two or four packed double-precision floating-point values in the first source operand, adds the infinite precision intermediate result to the two or four packed double-precision floating-point values in the third source operand, performs rounding and stores the resulting two or four packed double-precision floating-point values to the destination operand (first source operand).

VFMADD231PD: Multiplies the two or four packed double-precision floating-point values from the second source to the two or four packed double-precision floating-point values in the third source operand, adds the infinite precision intermediate result to the two or four packed double-precision floating-point values in the first source operand, performs rounding and stores the resulting two or four packed double-precision floating-point values to the desti- nation operand (first source operand).