Average Error: 12.3 → 8.7
Time: 39.8s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin ky}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{\sin ky}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \sin th
double f(double kx, double ky, double th) {
        double r53991 = ky;
        double r53992 = sin(r53991);
        double r53993 = kx;
        double r53994 = sin(r53993);
        double r53995 = 2.0;
        double r53996 = pow(r53994, r53995);
        double r53997 = pow(r53992, r53995);
        double r53998 = r53996 + r53997;
        double r53999 = sqrt(r53998);
        double r54000 = r53992 / r53999;
        double r54001 = th;
        double r54002 = sin(r54001);
        double r54003 = r54000 * r54002;
        return r54003;
}
↓
double f(double kx, double ky, double th) {
        double r54004 = ky;
        double r54005 = sin(r54004);
        double r54006 = kx;
        double r54007 = sin(r54006);
        double r54008 = 2.0;
        double r54009 = 2.0;
        double r54010 = r54008 / r54009;
        double r54011 = pow(r54007, r54010);
        double r54012 = pow(r54005, r54010);
        double r54013 = hypot(r54011, r54012);
        double r54014 = r54005 / r54013;
        double r54015 = th;
        double r54016 = sin(r54015);
        double r54017 = r54014 * r54016;
        return r54017;
}

Error

Bits error versus kx

Bits error versus ky

Bits error versus th

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 12.3

    \[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
  2. Using strategy rm

  3. Applied sqr-pow12.3

    \[\leadsto \frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + \color{blue}{{\left(\sin ky\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}}}} \cdot \sin th\]

  4. Applied sqr-pow12.3

    \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\left(\sin kx\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}} + {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}}} \cdot \sin th\]

  5. Applied hypot-def8.7

    \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}} \cdot \sin th\]

  6. Final simplification8.7

    \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \sin th\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  :precision binary64
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))