Average Error: 12.3 → 12.5
Time: 34.9s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\sqrt{\frac{1}{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin ky\right) \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left(\sqrt{\frac{1}{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin ky\right) \cdot \sin th
double f(double kx, double ky, double th) {
        double r32433 = ky;
        double r32434 = sin(r32433);
        double r32435 = kx;
        double r32436 = sin(r32435);
        double r32437 = 2.0;
        double r32438 = pow(r32436, r32437);
        double r32439 = pow(r32434, r32437);
        double r32440 = r32438 + r32439;
        double r32441 = sqrt(r32440);
        double r32442 = r32434 / r32441;
        double r32443 = th;
        double r32444 = sin(r32443);
        double r32445 = r32442 * r32444;
        return r32445;
}
↓
double f(double kx, double ky, double th) {
        double r32446 = 1.0;
        double r32447 = kx;
        double r32448 = sin(r32447);
        double r32449 = 2.0;
        double r32450 = pow(r32448, r32449);
        double r32451 = ky;
        double r32452 = sin(r32451);
        double r32453 = pow(r32452, r32449);
        double r32454 = r32450 + r32453;
        double r32455 = r32446 / r32454;
        double r32456 = sqrt(r32455);
        double r32457 = r32456 * r32452;
        double r32458 = th;
        double r32459 = sin(r32458);
        double r32460 = r32457 * r32459;
        return r32460;
}

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 add-sqr-sqrt12.3

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

  4. Applied sqrt-prod12.5

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

  5. Applied *-un-lft-identity12.5

    \[\leadsto \frac{\color{blue}{1 \cdot \sin ky}}{\sqrt{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sqrt{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\]

  6. Applied times-frac12.6

    \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \frac{\sin ky}{\sqrt{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right)} \cdot \sin th\]

  7. Taylor expanded around inf 12.5

    \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin ky\right)} \cdot \sin th\]

  8. Final simplification12.5

    \[\leadsto \left(\sqrt{\frac{1}{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin ky\right) \cdot \sin th\]

Reproduce

herbie shell --seed 2019323 
(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)))