Text, speed improvements:

Introduced
const float halfPi    = PI / 2.0f; and
const float triHalfPi = halfPi * 3;
and used those to draw the rounded background.
Introduced
GLfloat xPlusWidth   = x + width; and
GLfloat yMinusHeight = y - height;
and used those coordinates for the vertexData array.

Text, cleanup:

Declaration of local variables outside of loops.
Fixed indentation.

Approved by PS Jenkins bot, Sam Spilsbury.