Average Error: 12.5 → 12.8
Time: 34.6s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\sqrt[3]{\sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \left(\sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}\right)} \cdot \left(\sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}\right)\right) \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left(\sqrt[3]{\sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \left(\sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}\right)} \cdot \left(\sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}\right)\right) \cdot \sin th
double f(double kx, double ky, double th) {
        double r1208428 = ky;
        double r1208429 = sin(r1208428);
        double r1208430 = kx;
        double r1208431 = sin(r1208430);
        double r1208432 = 2.0;
        double r1208433 = pow(r1208431, r1208432);
        double r1208434 = pow(r1208429, r1208432);
        double r1208435 = r1208433 + r1208434;
        double r1208436 = sqrt(r1208435);
        double r1208437 = r1208429 / r1208436;
        double r1208438 = th;
        double r1208439 = sin(r1208438);
        double r1208440 = r1208437 * r1208439;
        return r1208440;
}
↓
double f(double kx, double ky, double th) {
        double r1208441 = ky;
        double r1208442 = sin(r1208441);
        double r1208443 = r1208442 * r1208442;
        double r1208444 = kx;
        double r1208445 = sin(r1208444);
        double r1208446 = r1208445 * r1208445;
        double r1208447 = r1208443 + r1208446;
        double r1208448 = sqrt(r1208447);
        double r1208449 = r1208442 / r1208448;
        double r1208450 = cbrt(r1208449);
        double r1208451 = r1208450 * r1208450;
        double r1208452 = r1208450 * r1208451;
        double r1208453 = cbrt(r1208452);
        double r1208454 = r1208453 * r1208451;
        double r1208455 = th;
        double r1208456 = sin(r1208455);
        double r1208457 = r1208454 * r1208456;
        return r1208457;
}

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 12.5

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

  2. Taylor expanded around inf 12.5

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

  3. Simplified12.5

    \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}} \cdot \sin th\]
  4. Using strategy rm

  5. Applied add-cube-cbrt12.8

    \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}\right) \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}\right)} \cdot \sin th\]
  6. Using strategy rm

  7. Applied add-cbrt-cube12.8

    \[\leadsto \left(\left(\sqrt[3]{\frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}\right) \cdot \color{blue}{\sqrt[3]{\left(\sqrt[3]{\frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}\right) \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}}}\right) \cdot \sin th\]

  8. Final simplification12.8

    \[\leadsto \left(\sqrt[3]{\sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \left(\sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}\right)} \cdot \left(\sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}\right)\right) \cdot \sin th\]

Reproduce

herbie shell --seed 2019162 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))