Average Error: 12.3 → 12.5
Time: 35.4s
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 r32395 = ky;
        double r32396 = sin(r32395);
        double r32397 = kx;
        double r32398 = sin(r32397);
        double r32399 = 2.0;
        double r32400 = pow(r32398, r32399);
        double r32401 = pow(r32396, r32399);
        double r32402 = r32400 + r32401;
        double r32403 = sqrt(r32402);
        double r32404 = r32396 / r32403;
        double r32405 = th;
        double r32406 = sin(r32405);
        double r32407 = r32404 * r32406;
        return r32407;
}
↓
double f(double kx, double ky, double th) {
        double r32408 = 1.0;
        double r32409 = kx;
        double r32410 = sin(r32409);
        double r32411 = 2.0;
        double r32412 = pow(r32410, r32411);
        double r32413 = ky;
        double r32414 = sin(r32413);
        double r32415 = pow(r32414, r32411);
        double r32416 = r32412 + r32415;
        double r32417 = r32408 / r32416;
        double r32418 = sqrt(r32417);
        double r32419 = r32418 * r32414;
        double r32420 = th;
        double r32421 = sin(r32420);
        double r32422 = r32419 * r32421;
        return r32422;
}

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)))