[Ether]

Instead of using arctangent and exponential functions, why not just multiply the X axis by a factor of 4, clip it at 1, and normalize the resulting X,Y pair?

void adjust(float Xj, float Yj, float *Xa, float *Ya){

	float mag;

	*Ya=Yj;
	
	if (Xj<=-0.25) *Xa=-1; 
	else if (Xj<0.25) *Xa=4.0*Xj;
	else *Xa=1.0;

	// normalize adjusted values:
	mag=sqrt(*Xa**Xa+*Ya**Ya);
	if (mag>1) {*Xa/=mag; *Ya/=mag;}
}

See attached Excel spreadsheet. The 100 tab shows an overview of adjusted XY pairs (cyan cells) for the entire first quadrant. The 25 tab zooms in for a closer look for small joystick values.

The attached plot shows a family of Xa vs Xj curves for Yj=0 to 1.


Or you could make the X gain a function of Yj, like so:

void adjust(float Xj, float Yj, float *Xa, float *Ya){

	float mag, gain;

        gain=1+3*Yj;

	*Ya=Yj;
	
        *Xa=gain*Xj;
        if (*Xa>1) *Xa=1;
        else if (*Xa<-1) *Xa=-1;

	// normalize adjusted values:
	mag=sqrt(*Xa**Xa+*Ya**Ya);
	if (mag>1) {*Xa/=mag; *Ya/=mag;}
}

See attached plots for Xa vs Xj for Yj=0 to 1
and Ya vs Yj for Xj=0 to 1