Published:12/28/2015 Last Updated:12/28/2015

Author: Marius Cornea, Intel Corporation

We are providing two files, RECIP14.c and RECIP28EXP2.c, containing reference implementations for the scalar versions of 10 approximation instructions introduced in the "Intel® Architecture Instruction Set Extensions Programming Reference" document (see /content/www/us/en/develop/tools/isa-extensions.html). The files can be downloaded from the links provided above.

RECIP14.c contains emulation routines for the underlying algorithms of:

- VRCP14PD - Compute Approximate Reciprocals of Packed Float64 Values with relative error of less than 2
^{-14} - VRCP14SD - Compute Approximate Reciprocal of Scalar Float64 Value with relative error of less than 2
^{-14} - VRCP14PS - Compute Approximate Reciprocals of Packed Float32 Values with relative error of less than 2
^{-14} - VRCP14SS - Compute Approximate Reciprocal of Scalar Float32 Value with relative error of less than 2
^{-14} - VRSQRT14PD - Compute Approximate Reciprocals of Square Roots of Packed Float64 Values with relative error of less than 2
^{-14} - VRSQRT14SD - Compute Approximate Reciprocal of Square Root of Scalar Float64 Value with relative error of less than 2
^{-14} - VRSQRT14PS - Compute Approximate Reciprocals of Square Roots of PackedFloat32 Values with relative error of less than 2
^{-14} - VRSQRT14SS - Compute Approximate Reciprocal of Square Root of Scalar Float32 Value with relative error of less than 2
^{-14}

The corresponding emulation routines (only scalar versions) are:

- RCP14S - reciprocal approximation for Float32
- RCP14D - reciprocal approximation for Float64
- RSQRT14S - reciprocal square root approximation for Float32
- RSQRT14D - reciprocal square root approximation for Float64

RECIP28EXP2.c contains emulation routines for the underlying algorithms of:

- VRCP28PD - Approximation to the Reciprocal of Packed Double Precision Floating-Point Values with Less Than 2
^{-28}Relative Error - VRCP28SD - Approximation to the Reciprocal of Scalar Double Precision Floating-Point Value with Less Than 2
^{-28}Relative Error - VRCP28PS - Approximation to the Reciprocal of Packed Single Precision Floating-Point Values with Less Than 2
^{-28}Relative Error - VRCP28SS - Approximation to the Reciprocal of Scalar Single Precision Floating-Point Value with Less Than 2
^{-28}Relative Error - VRSQRT28PD - Approximation to the Reciprocal Square Root of Packed Double Precision Floating-Point Values with Less Than 2
^{-28}Relative Error - VRSQRT28SD - Approximation to the Reciprocal Square Root of Scalar Double Precision Floating-Point Value with Less Than 2^-28 Relative Error
- VRSQRT28PS - Approximation to the Reciprocal Square Root of Packed Single Precision Floating-Point Values with Less Than 2
^{-28}Relative Error - VRSQRT28SS - Approximation to the Reciprocal Square Root of Scalar Single Precision Floating-Point Value with Less Than 2
^{-28}Relative Error - VEXP2PD - Approximation to the Exponential 2
^{x}of Packed Double Precision Floating-Point Values with Less Than 2^{-23 }Relative Error - VEXP2PS - Approximation to the Exponential 2
^{x}of Packed Single Precision Floating-Point Values with Less Than 2^{-23 }Relative Error

The corresponding emulation routines (only scalar versions) are:

- RCP28S - reciprocal approximation for Float32
- RCP28D - reciprocal approximation for Float64
- RSQRT28S - reciprocal square root approximation for Float32
- RSQRT28D - reciprocal square root approximation for Float64
- EXP2S - Base-2 exponential approximation for Float32
- EXP2D - Base-2 exponential approximation for Float64

The reference functions have to be compiled with the DAZ and FTZ mode turned off (e.g. with the Intel compiler for Linux, using the -no-ftz option), and have to be run with the rounding mode set to round-to-nearest, and with floating-point exceptions masked.

**Usage example for RCP14S and RCP14D **

The following example may be compiled with any of the following (or other, equivalent) commands:

```
icc -no-ftz -Wall -Werror main.c RECIP14.c
gcc -m32 -Wall -Werror main.c RECIP14.c -lm
gcc -Wall -Werror main.c RECIP14.c -lm
```

where main.c is shown below:

```
#include <stdio.h>
typedef union {
unsigned int u;
float f;
} type32;
typedef union {
unsigned long long u;
double f;
} type64;
extern void RCP14S (unsigned int mxcsr, type32 *dst, type32 src);
extern void RCP14D (unsigned int mxcsr, type64 *dst, type64 src);
int main () {
type32 dst32, src32;
type64 dst64, src64;
unsigned int mxcsr = 0x00000000;
printf ("MXCSR = %8.8x\n", mxcsr);
src32.f = 3.0;
RCP14S (mxcsr, &dst32, src32);
printf ("RCP14S(%f = %8.8x HEX) = (%f = %8.8x HEX)\n", src32.f, src32.u,
dst32.f, dst32.u);
src64.f = 3.0;
RCP14D (mxcsr, &dst64, src64);
printf ("RCP14D(%f = %16.16llx HEX) = (%f = %16.16llx HEX)\n", src64.f,
src64.u, dst64.f, dst64.u);
return (0);
}
```

**Usage example for RSQRT14S and RSQRT14D**

The following example may be compiled with any of the following (or other, equivalent) commands:

```
icc -no-ftz -Wall -Werror main.c RECIP14.c
gcc -m32 -Wall -Werror main.c RECIP14.c -lm
gcc -Wall -Werror main.c RECIP14.c -lm
```

where main.c is shown below:

```
#include <stdio.h>
typedef union {
unsigned int u;
float f;
} type32;
typedef union {
unsigned long long u;
double f;
} type64;
extern void RSQRT14S (unsigned int mxcsr, type32 *dst, type32 src);
extern void RSQRT14D (unsigned int mxcsr, type64 *dst, type64 src);
int main () {
type32 dst32, src32;
type64 dst64, src64;
unsigned int mxcsr = 0x00000000;
printf ("MXCSR = %8.8x\n", mxcsr);
src32.f = 2.0;
RSQRT14S (mxcsr, &dst32, src32);
printf ("RSQRT14S(%f = %8.8x HEX) = (%f = %8.8x HEX)\n", src32.f, src32.u,
dst32.f, dst32.u);
src64.f = 2.0;
RSQRT14D (mxcsr, &dst64, src64);
printf ("RSQRT14D(%f = %16.16llx HEX) = (%f = %16.16llx HEX)\n", src64.f,
src64.u, dst64.f, dst64.u);
return (0);
}
```

**Usage example for RCP28S and RCP28D**

The following example may be compiled with any of the following (or other, equivalent) commands:

```
icc -no-ftz -Wall -Werror main.c RECIP28EXP2.c
gcc -m32 -Wall -Werror main.c RECIP28EXP2.c -lm
gcc -Wall -Werror main.c RECIP28EXP2.c -lm
```

where main.c is shown below:

```
#include <stdio.h>
typedef union {
unsigned int u;
float f;
} type32;
typedef union {
unsigned long long u;
double f;
} type64;
extern unsigned int RCP28S (type32 *dst, type32 src);
extern unsigned int RCP28D (type64 *dst, type64 src);
int main () {
type32 dst32, src32;
type64 dst64, src64;
unsigned int flags = 0x00000000; // PUOZDI
printf ("FLAGS = %2.2x\n", flags);
src32.f = 3.0;
flags = RCP28S (&dst32, src32);
printf ("RCP28S(%f = %8.8x HEX) = (%f = %8.8x HEX) flags = %2.2x\n",
src32.f, src32.u, dst32.f, dst32.u, flags);
src64.f = 3.0;
flags = RCP28D (&dst64, src64);
printf ("RCP28D(%f = %16.16llx HEX) = (%f = %16.16llx HEX) flags = %2.2x\n",
src64.f, src64.u, dst64.f, dst64.u, flags);
return (0);
}
```

**Usage example for RSQRT28S and RSQRT28D**

The following example may be compiled with any of the following (or other, equivalent) commands:

```
icc -no-ftz -Wall -Werror main.c RECIP28EXP2.c
gcc -m32 -Wall -Werror main.c RECIP28EXP2.c -lm
gcc -Wall -Werror main.c RECIP28EXP2.c -lm
```

where main.c is shown below:

```
#include <stdio.h>
typedef union {
unsigned int u;
float f;
} type32;
typedef union {
unsigned long long u;
double f;
} type64;
extern unsigned int RSQRT28S (type32 *dst, type32 src);
extern unsigned int RSQRT28D (type64 *dst, type64 src);
int main () {
type32 dst32, src32;
type64 dst64, src64;
unsigned int flags = 0x00000000; // PUOZDI
printf ("FLAGS = %2.2x\n", flags);
src32.f = 2.0;
flags = RSQRT28S (&dst32, src32);
printf ("RSQRT28S(%f = %8.8x HEX) = (%f = %8.8x HEX) flags = %2.2x\n",
src32.f, src32.u, dst32.f, dst32.u, flags);
src64.f = 2.0;
flags = RSQRT28D (&dst64, src64);
printf ("RSQRT28D(%f = %16.16llx HEX) = (%f = %16.16llx HEX) flags = %2.2x\n",
src64.f, src64.u, dst64.f, dst64.u, flags);
return (0);
}
```

**Usage example for EXP2S and EXP2D**

The following example may be compiled with any of the following (or other, equivalent) commands:

```
icc -no-ftz -Wall -Werror main.c RECIP28EXP2.c
gcc -m32 -Wall -Werror main.c RECIP28EXP2.c -lm
gcc -Wall -Werror main.c RECIP28EXP2.c -lm
```

where main.c is shown below:

```
#include <stdio.h>
typedef union {
unsigned int u;
float f;
} type32;
typedef union {
unsigned long long u;
double f;
} type64;
extern unsigned int EXP2S (type32 *dst, type32 src);
extern unsigned int EXP2D (type64 *dst, type64 src);
int main () {
type32 dst32, src32;
type64 dst64, src64;
unsigned int flags = 0x00000000; // PUOZDI
printf ("FLAGS = %2.2x\n", flags);
src32.f = 1.5;
flags = EXP2S (&dst32, src32);
printf ("EXP2S(%f = %8.8x HEX) = (%f = %8.8x HEX) flags = %2.2x\n",
src32.f, src32.u, dst32.f, dst32.u, flags);
src64.f = 1.5;
flags = EXP2D (&dst64, src64);
printf ("EXP2D(%f = %16.16llx HEX) = (%f = %16.16llx HEX) flags = %2.2x\n",
src64.f, src64.u, dst64.f, dst64.u, flags);
return (0);
}
```

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804