Average Error: 12.1 → 11.6
Time: 11.6s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \le 1:\\ \;\;\;\;\left(\sqrt[3]{\frac{\sin ky}{\left|\sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right| \cdot \sqrt{\sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \left(\sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left(ky + \frac{1}{12} \cdot \left({kx}^{2} \cdot ky\right)\right) - \frac{1}{6} \cdot {ky}^{3}} \cdot \sin th\\ \end{array}\]
\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:\\
\;\;\;\;\left(\sqrt[3]{\frac{\sin ky}{\left|\sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right| \cdot \sqrt{\sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \left(\sqrt[3]{\frac{\sin ky}{\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 r39578 = ky;
        double r39579 = sin(r39578);
        double r39580 = kx;
        double r39581 = sin(r39580);
        double r39582 = 2.0;
        double r39583 = pow(r39581, r39582);
        double r39584 = pow(r39579, r39582);
        double r39585 = r39583 + r39584;
        double r39586 = sqrt(r39585);
        double r39587 = r39579 / r39586;
        double r39588 = th;
        double r39589 = sin(r39588);
        double r39590 = r39587 * r39589;
        return r39590;
}
↓
double f(double kx, double ky, double th) {
        double r39591 = ky;
        double r39592 = sin(r39591);
        double r39593 = kx;
        double r39594 = sin(r39593);
        double r39595 = 2.0;
        double r39596 = pow(r39594, r39595);
        double r39597 = pow(r39592, r39595);
        double r39598 = r39596 + r39597;
        double r39599 = sqrt(r39598);
        double r39600 = r39592 / r39599;
        double r39601 = 1.0;
        bool r39602 = r39600 <= r39601;
        double r39603 = cbrt(r39598);
        double r39604 = fabs(r39603);
        double r39605 = sqrt(r39603);
        double r39606 = r39604 * r39605;
        double r39607 = r39592 / r39606;
        double r39608 = cbrt(r39607);
        double r39609 = cbrt(r39600);
        double r39610 = r39608 * r39609;
        double r39611 = th;
        double r39612 = sin(r39611);
        double r39613 = r39609 * r39612;
        double r39614 = r39610 * r39613;
        double r39615 = 0.08333333333333333;
        double r39616 = 2.0;
        double r39617 = pow(r39593, r39616);
        double r39618 = r39617 * r39591;
        double r39619 = r39615 * r39618;
        double r39620 = r39591 + r39619;
        double r39621 = 0.16666666666666666;
        double r39622 = 3.0;
        double r39623 = pow(r39591, r39622);
        double r39624 = r39621 * r39623;
        double r39625 = r39620 - r39624;
        double r39626 = r39592 / r39625;
        double r39627 = r39626 * r39612;
        double r39628 = r39602 ? r39614 : r39627;
        return r39628;
}

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 10.8

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

    3. Applied add-cube-cbrt11.1

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right)} \cdot \sin th\]

    4. Applied associate-*l*11.1

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \left(\sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt11.1

      \[\leadsto \left(\sqrt[3]{\frac{\sin ky}{\sqrt{\color{blue}{\left(\sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}} \cdot \sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \left(\sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\right)\]

    7. Applied sqrt-prod11.1

      \[\leadsto \left(\sqrt[3]{\frac{\sin ky}{\color{blue}{\sqrt{\sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}} \cdot \sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sqrt{\sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \left(\sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\right)\]

    8. Simplified11.1

      \[\leadsto \left(\sqrt[3]{\frac{\sin ky}{\color{blue}{\left|\sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right|} \cdot \sqrt{\sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \left(\sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\right)\]

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

    1. Initial program 63.2

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

    2. Taylor expanded around 0 29.7

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

  4. Final simplification11.6

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

Reproduce

herbie shell --seed 2020021 
(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)))