# FPATAN

Partial Arctangent

## Opcodes

Hex Mnemonic Encoding Long Mode Legacy Mode Description
D9 F3 FPATAN None Valid Valid Replace ST(1) with arctan(ST(1)ST(0)) and pop the register stack.

## Description

Computes the arctangent of the source operand in register ST(1) divided by the source operand in register ST(0), stores the result in ST(1), and pops the FPU register stack. The result in register ST(0) has the same sign as the source operand ST(1) and a magnitude less than +π.

The FPATAN instruction returns the angle between the X axis and the line from the origin to the point (X,Y), where Y (the ordinate) is ST(1) and X (the abscissa) is ST(0). The angle depends on the sign of X and Y independently, not just on the sign of the ratio Y/X. This is because a point (-X,Y) is in the second quadrant, resulting in an angle between π/2 and π, while a point (X,-Y) is in the fourth quadrant, resulting in an angle between 0 and -π/2. A point (-X,-Y) is in the third quadrant, giving an angle between -π/2 and -π.

The following table shows the results obtained when computing the arctangent of various classes of numbers, assuming that underflow does not occur.

FPATAN Results
DEST
SRC
 -∞ -F -0 +0 +F +∞ nan -∞ -3π/4* -π/2 -π/2 -π/2 -π/2 -π/4* nan -F -p -π to -π/2 -π/2 -π/2 -π/2 to -0 -0 nan -0 -p -p -p* -0* -0 -0 nan +0 +p +p +π* +0* +0 +0 nan +F +p +π to +π/2 +π/2 +π/2 +π/2 to +0 +0 nan +∞ +3π/4* +π/2 +π/2 +π/2 +π/2 +π/4* nan NaN NaN NaN NaN NaN NaN NaN nan
• Notes:
• F means finite floating-point value
• * Indicates floating-point invalid-arithmetic-operand (#IA) exception.

If the source operand is outside the acceptable range, the C2 flag in the FPU status word is set, and the value in register ST(0) remains unchanged. The instruction does not raise an exception when the source operand is out of range. It is up to the program to check the C2 flag for out-of-range conditions. Source values outside the range 263 to 263 can be reduced to the range of the instruction by subtracting an appropriate integer multiple of 2 or by using the FPREM instruction with a divisor of 2. See the section titled “Pi” in Chapter 8 of the Intel® 64 and IA-32 Architectures Software Developer’s Manual, Volume 1, for a discussion of the proper value to use for  in performing such reductions.

The value 1.0 is pushed onto the register stack after the tangent has been computed to maintain compatibility with the Intel 8087 and Intel287 math coprocessors. This operation also simplifies the calculation of other trigonometric functions. For instance, the cotangent (which is the reciprocal of the tangent) can be computed by executing a FDIVR instruction after the FPTAN instruction.

This instruction’s operation is the same in non-64-bit modes and 64-bit mode.

## Pseudo Code

```ST(1) = arctan(ST(1) / ST(0));
PopRegisterStack;
```

## FPU Flags Affected

C1: Set to 0 if stack underflow occurred. Set if result was rounded up; cleared otherwise. C0, C2, C3 are undefined.

## Exceptions

### Floating-Point Exceptions

Exception Description
#P Value cannot be represented exactly in destination format.
#U Result is too small for destination format.
#D Source operand is a denormal value.
#IA Source operand is an SNaN value or unsupported format.
#IS Stack underflow occurred.

### 64-Bit Mode Exceptions

Same exceptions as in protected mode.

### Compatibility Mode Exceptions

Same exceptions as in protected mode.

### Virtual-8086 Mode Exceptions

Same exceptions as in protected mode.

### Real-Address Mode Exceptions

Same exceptions as in protected mode.

### Protected Mode Exceptions

Exception Description
#UD If the LOCK prefix is used.
#MF If there is a pending x87 FPU exception.
#NM CR0.EM[bit 2] or CR0.TS[bit 3] = 1.