Tan function

### Program Information

Name: Tan function
Domain: Numerical program
Functionality: Tan function in math
Input: $x$: A certain Real Number
Output: $y$: The value of $\tan x$

#### Reference

 Metamorphic Testing of Programs on Partial Differential Equations: a Case Study https://doi.org/10.1109/CMPSAC.2002.1045022;
Search-Based Inference of Polynomial Metamorphic Relations https://doi.org/10.1145/2642937.2642994; 

### MR Information

#### MR1------

Description:
Property: $\tan x-\tan (x+\pi)=0$
Source input: $x$
Source output: $\tan x$
Follow-up input: $x+\pi$
Follow-up output: $\tan (x+\pi)$
Input relation: $x \Rightarrow (x+\pi)$
Output relation: $\tan x-\tan (x+\pi)=0$
Pattern:

#### MR2------

Description:
Property: $\tan^2(x)-2\tan(x)\tan(2x-3\pi /2)-1=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input: $2x-3\pi /2$
Follow-up output: $\tan(2x-3\pi /2)$
Input relation: $x \Rightarrow (2x-3\pi /2)$
Output relation: $\tan^2(x)-2\tan(x)\tan(2x-3\pi /2)-1=0$
Pattern:

#### MR3------

Description:
Property: $\tan(x)+\tan(-x+\pi)=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input: $-x+\pi$
Follow-up output: $\tan(-x+\pi)$
Input relation: $x \Rightarrow (-x+\pi)$
Output relation: $\tan(x)+\tan(-x+\pi)=0$
Pattern:

#### MR4------

Description:
Property: $\tan(x)-\tan(x-2\pi)=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input:  $x-2\pi$
Follow-up output:  $\tan(x-2\pi)$
Input relation: $x \Rightarrow (x-2\pi)$
Output relation: $\tan(x)-\tan(x-2\pi)=0$
Pattern:

#### MR5------

Description:
Property: $\tan(x)-\tan(x+2\pi)=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input:  $x+2\pi$
Follow-up output: $\tan(x+2\pi)$
Input relation: $x \Rightarrow (x+2\pi)$
Output relation: $\tan(x)-\tan(x+2\pi)=0$
Pattern:

#### MR6------

Description:
Property: $\tan(x)+\tan(-x+2\pi)=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input: $-x+2\pi$
Follow-up output: $\tan(-x+2\pi)$
Input relation: $x \Rightarrow (-x+2\pi)$
Output relation: $\tan(x)+\tan(-x+2\pi)=0$
Pattern:

#### MR7------

Description:
Property: $\tan(x)-\tan(x+\pi)=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input:  $x+\pi$
Follow-up output: $\tan(x+\pi)$
Input relation: $x \Rightarrow (x+\pi)$
Output relation: $\tan(x)-\tan(x+\pi)=0$
Pattern:

#### MR8------

Description:
Property: $\tan(x)+\tan(-x)=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input: $-x$
Follow-up output:  $\tan(-x)$
Input relation: $x \Rightarrow (-x)$
Output relation: $\tan(x)+\tan(-x)=0$
Pattern:

#### MR9------

Description:
Property: $\tan(x)+\tan(-x-2\pi)=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input:  $-x-2\pi$
Follow-up output:  $\tan(-x-2\pi)$
Input relation: $x \Rightarrow (-x-2\pi)$
Output relation: $\tan(x)+\tan(-x-2\pi)=0$
Pattern:

#### MR10------

Description:
Property: $\tan^2(0.5x+1.25\pi)-2\tan(x)\tan(0.5x+1.25\pi)-1=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input:  $0.5x+1.25\pi$
Follow-up output:  $\tan(0.5x+1.25\pi)$
Input relation: $x \Rightarrow (0.5x+1.25\pi)$
Output relation: $\tan^2(0.5x+1.25\pi)-2\tan(x)\tan(0.5x+1.25\pi)-1=0$
Pattern:

#### MR11------

Description:
Property: $\tan^2(x)+\tan^2(-x)+2\tan(x)\tan(-x)=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input:  $-x$
Follow-up output:  $\tan(-x)$
Input relation: $x \Rightarrow (-x)$
Output relation: $\tan^2(x)+\tan^2(-x)+2\tan(x)\tan(-x)=0$
Pattern:

#### MR12------

Description:
Property: $\tan^2(x)+\tan^2(x+3\pi)-2\tan(x)\tan(x+3\pi)=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input:  $x+3\pi$
Follow-up output:  $\tan(x+3\pi)$
Input relation: $x \Rightarrow (x+3\pi)$
Output relation: $\tan^2(x)+\tan^2(x+3\pi)-2\tan(x)\tan(x+3\pi)=0$
Pattern:

#### MR13------

Description:
Property: $\tan^2(x)-\tan^2(x+\pi)-\tan(x)+\tan(x+\pi)=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input:  $x+\pi$
Follow-up output:  $\tan(x+\pi)$
Input relation: $x \Rightarrow (x+\pi)$
Output relation: $\tan^2(x)-\tan^2(x+\pi)-\tan(x)+\tan(x+\pi)=0$
Pattern:

#### MR14------

Description:
Property: $\tan^2(x)-\tan^2(-x)=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input: $-x$
Follow-up output:  $\tan(-x)$
Input relation: $x \Rightarrow (-x)$
Output relation: $\tan^2(x)-\tan^2(-x)=0$
Pattern:

#### MR15------

Description:
Property: $\tan^2(x)-\tan^2(x+\pi)+\tan(x)-\tan(x+\pi)=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input: $x+\pi$
Follow-up output:  $\tan(x+\pi)$
Input relation: $x \Rightarrow (x+\pi)$
Output relation: $\tan^2(x)-\tan^2(x+\pi)+\tan(x)-\tan(x+\pi)=0$
Pattern:

#### MR16------

Description:
Property: $\tan^2(x)+\tan^2(-x-2\pi)+2\tan(x)\tan(-x-2\pi)=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input: $-x-2\pi$
Follow-up output:  $\tan(-x-2\pi)$
Input relation: $x \Rightarrow (-x-2\pi)$
Output relation: $\tan^2(x)+\tan^2(-x-2\pi)+2\tan(x)\tan(-x-2\pi)=0$
Pattern:

#### MR17------

Description:
Property: $\tan^2(x)+\tan^2(-x+\pi)+2\tan(x)\tan(-x+\pi)=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input: $-x+\pi$
Follow-up output: $\tan(-x+\pi)$
Input relation: $x \Rightarrow (-x+\pi)$
Output relation: $\tan^2(x)+\tan^2(-x+\pi)+2\tan(x)\tan(-x+\pi)=0$
Pattern:

#### MR18------

Description:
Property: $\tan^2(x)+0.5\tan^2(-x)+1.5\tan(x)\tan(-x)-0.5\tan(x)-0.5\tan(-x)=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input:  $-x$
Follow-up output:  $\tan(-x)$
Input relation: $x \Rightarrow (-x)$
Output relation: $\tan^2(x)+0.5\tan^2(-x)+1.5\tan(x)\tan(-x)-0.5\tan(x)-0.5\tan(-x)=0$
Pattern:

#### MR19------

Description:
Property: $\tan^2(x)+\tan^2(-x+2\pi)+2\tan(x)\tan(-x+2\pi)=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input: $-x+2\pi$
Follow-up output:  $\tan(-x+2\pi)$
Input relation: $x \Rightarrow (-x+2\pi)$
Output relation: $\tan^2(x)+\tan^2(-x+2\pi)+2\tan(x)\tan(-x+2\pi)=0$
Pattern:

#### MR20------

Description:
Property: $\tan^2(0.5x-1.75\pi)-2\tan(x)\tan(0.5x-1.75\pi)-1=0$
Source input: $x$
Source output: $\tan(x)$
Follow-up input:  $0.5x-1.75\pi$
Follow-up output: $\tan(0.5x-1.75\pi)$
Input relation: $x \Rightarrow (0.5x-1.75\pi)$
Output relation: $\tan^2(0.5x-1.75\pi)-2\tan(x)\tan(0.5x-1.75\pi)-1=0$
Pattern:
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