Sin function

### Program Information

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

#### Reference

 Metamorphic Testing and Beyond https://doi.org/10.1109/STEP.2003.18;
An Effective Testing Method for End-User Programmers https://doi.org/10.1145/1082983.1083236 ;
Search-Based Inference of Polynomial Metamorphic Relations https://doi.org/10.1145/2642937.2642994  

### MR Information

#### MR1------

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

#### MR2------

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

#### MR3------

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

#### MR4------

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

#### MR5------

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

#### MR6------

Description:
Property:    $\sin x+\sin y +\sin z-\sin(x+y+z)-4\sin ((x+y)/2) \sin ((x+z)/2) \sin ((y+z)/2)$
Source input:  $x,y,z$
Source output:  $\sin x,\sin y,\sin z$
Follow-up input:   $x+y+z,(x+y)/2,(x+z)/2,(y+z)/2$
Follow-up output:    $\sin(x+y+z),\sin((x+y)/2),\sin((x+z)/2),\sin((y+z)/2)$
Input relation:   $x,y,z \Rightarrow (x+y+z),(x+y)/2,(x+z)/2,(y+z)/2$
Output relation:   $\sin x+\sin y +\sin z-\sin(x+y+z)-4\sin ((x+y)/2) \sin ((x+z)/2) \sin ((y+z)/2)$
Pattern:

#### MR7------

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

#### MR8------

Description:
Property:    $\sin 3x-3\sin x+4\sin^3 x=0$
Source input:   $x$
Source output:   $\sin x$
Follow-up input:   $3x$
Follow-up output:    $\sin (3x)$
Input relation:   $x \Rightarrow (3x)$
Output relation:   $\sin 3x-3\sin x+4\sin^3 x=0$
Pattern:

#### MR9------

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

#### MR10------

Description:
Property:    $\sin 5x-16\sin5x-5\sin 3x+10\sin x=0$
Source input:   $x$
Source output:   $\sin x$
Follow-up input:  $5x,3x$
Follow-up output:    $\sin (3x),\sin (5x)$
Input relation:   $x,y \Rightarrow (3x),(5x)$
Output relation:  $\sin 5x-16\sin5x-5\sin 3x+10\sin x=0$
Pattern:

#### MR11------

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

#### MR12------

Description:
Property:    $\sin^2(-0.5x-0.75\pi)+0.5\sin(x)-0.5=0$
Source input:   $x$
Source output:   $\sin x$
Follow-up input:  $-0.5x-0.75\pi$
Follow-up output:   $\sin(-0.5x-0.75\pi)$
Input relation:  $x \Rightarrow (-0.5x-0.75\pi)$
Output relation:  $\sin^2(-0.5x-0.75\pi)+0.5\sin(x)-0.5=0$
Pattern:

#### MR13------

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

#### MR14------

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

#### MR15------

Description:
Property:    $\sin^2(x)+\sin^2(x-2.5\pi)-1=0$
Source input:   $x$
Source output:   $\sin x$
Follow-up input:  $x-2.5\pi$
Follow-up output:   $\sin(x-2.5\pi)$
Input relation:   $x \Rightarrow (x-5.5\pi)$
Output relation:  $\sin^2(x)+\sin^2(x-2.5\pi)-1=0$
Pattern:

#### MR16------

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

#### MR17------

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

#### MR18------

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

#### MR19------

Description:
Property: $\sin(x)+\sin(-x-4\pi)=0$
Source input:   $x$
Source output:  $\sin x$
Follow-up input: $-x-4\pi$
Follow-up output: $\sin(-x-4\pi)$
Input relation: $x \Rightarrow (-x-4\pi)$
Output relation: $\sin(x)+\sin(-x-4\pi)=0$
Pattern:

#### MR20------

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

#### MR21------

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

#### MR22------

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

#### MR23------

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

#### MR24------

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

#### MR25------

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

#### MR26------

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

#### MR27------

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

#### MR28------

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