Average Error: 12.6 → 11.8
Time: 19.9s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \le 1:\\ \;\;\;\;\sin ky \cdot \left(\frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left(ky + \frac{1}{12} \cdot \left({kx}^{2} \cdot ky\right)\right) - \frac{1}{6} \cdot {ky}^{3}} \cdot \sin th\\ \end{array}\]
\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:\\
\;\;\;\;\sin ky \cdot \left(\frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\right)\\

\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 r32510 = ky;
        double r32511 = sin(r32510);
        double r32512 = kx;
        double r32513 = sin(r32512);
        double r32514 = 2.0;
        double r32515 = pow(r32513, r32514);
        double r32516 = pow(r32511, r32514);
        double r32517 = r32515 + r32516;
        double r32518 = sqrt(r32517);
        double r32519 = r32511 / r32518;
        double r32520 = th;
        double r32521 = sin(r32520);
        double r32522 = r32519 * r32521;
        return r32522;
}
↓
double f(double kx, double ky, double th) {
        double r32523 = ky;
        double r32524 = sin(r32523);
        double r32525 = kx;
        double r32526 = sin(r32525);
        double r32527 = 2.0;
        double r32528 = pow(r32526, r32527);
        double r32529 = pow(r32524, r32527);
        double r32530 = r32528 + r32529;
        double r32531 = sqrt(r32530);
        double r32532 = r32524 / r32531;
        double r32533 = 1.0;
        bool r32534 = r32532 <= r32533;
        double r32535 = 1.0;
        double r32536 = r32535 / r32531;
        double r32537 = th;
        double r32538 = sin(r32537);
        double r32539 = r32536 * r32538;
        double r32540 = r32524 * r32539;
        double r32541 = 0.08333333333333333;
        double r32542 = 2.0;
        double r32543 = pow(r32525, r32542);
        double r32544 = r32543 * r32523;
        double r32545 = r32541 * r32544;
        double r32546 = r32523 + r32545;
        double r32547 = 0.16666666666666666;
        double r32548 = 3.0;
        double r32549 = pow(r32523, r32548);
        double r32550 = r32547 * r32549;
        double r32551 = r32546 - r32550;
        double r32552 = r32524 / r32551;
        double r32553 = r32552 * r32538;
        double r32554 = r32534 ? r32540 : r32553;
        return r32554;
}

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

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

    4. Applied associate-*l*11.3

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

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

    1. Initial program 63.1

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

    2. Taylor expanded around 0 30.0

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

  4. Final simplification11.8

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

Reproduce

herbie shell --seed 2019304 
(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)))