Average Error: 12.6 → 11.7
Time: 29.2s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \le 1:\\ \;\;\;\;\left(\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right) \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left(ky + \frac{1}{12} \cdot \left({kx}^{2} \cdot ky\right)\right) - \frac{1}{6} \cdot {ky}^{3}} \cdot \sin th\\ \end{array}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\begin{array}{l}
\mathbf{if}\;\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \le 1:\\
\;\;\;\;\left(\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right) \cdot \sin th\\

\mathbf{else}:\\
\;\;\;\;\frac{\sin ky}{\left(ky + \frac{1}{12} \cdot \left({kx}^{2} \cdot ky\right)\right) - \frac{1}{6} \cdot {ky}^{3}} \cdot \sin th\\

\end{array}
double f(double kx, double ky, double th) {
        double r35432 = ky;
        double r35433 = sin(r35432);
        double r35434 = kx;
        double r35435 = sin(r35434);
        double r35436 = 2.0;
        double r35437 = pow(r35435, r35436);
        double r35438 = pow(r35433, r35436);
        double r35439 = r35437 + r35438;
        double r35440 = sqrt(r35439);
        double r35441 = r35433 / r35440;
        double r35442 = th;
        double r35443 = sin(r35442);
        double r35444 = r35441 * r35443;
        return r35444;
}
↓
double f(double kx, double ky, double th) {
        double r35445 = ky;
        double r35446 = sin(r35445);
        double r35447 = kx;
        double r35448 = sin(r35447);
        double r35449 = 2.0;
        double r35450 = pow(r35448, r35449);
        double r35451 = pow(r35446, r35449);
        double r35452 = r35450 + r35451;
        double r35453 = sqrt(r35452);
        double r35454 = r35446 / r35453;
        double r35455 = 1.0;
        bool r35456 = r35454 <= r35455;
        double r35457 = 1.0;
        double r35458 = r35457 / r35453;
        double r35459 = r35446 * r35458;
        double r35460 = th;
        double r35461 = sin(r35460);
        double r35462 = r35459 * r35461;
        double r35463 = 0.08333333333333333;
        double r35464 = 2.0;
        double r35465 = pow(r35447, r35464);
        double r35466 = r35465 * r35445;
        double r35467 = r35463 * r35466;
        double r35468 = r35445 + r35467;
        double r35469 = 0.16666666666666666;
        double r35470 = 3.0;
        double r35471 = pow(r35445, r35470);
        double r35472 = r35469 * r35471;
        double r35473 = r35468 - r35472;
        double r35474 = r35446 / r35473;
        double r35475 = r35474 * r35461;
        double r35476 = r35456 ? r35462 : r35475;
        return r35476;
}

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. Split input into 2 regimes

  2. if (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) < 1.0

    1. Initial program 11.1

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

    3. Applied div-inv11.2

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

    if 1.0 < (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))

    1. Initial program 62.8

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

    2. Taylor expanded around 0 28.1

      \[\leadsto \frac{\sin ky}{\color{blue}{\left(ky + \frac{1}{12} \cdot \left({kx}^{2} \cdot ky\right)\right) - \frac{1}{6} \cdot {ky}^{3}}} \cdot \sin th\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \le 1:\\ \;\;\;\;\left(\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right) \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left(ky + \frac{1}{12} \cdot \left({kx}^{2} \cdot ky\right)\right) - \frac{1}{6} \cdot {ky}^{3}} \cdot \sin th\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(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)))