Average Error: 12.2 → 11.4
Time: 35.9s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \le 0.999999999998593680494707314210245385766:\\ \;\;\;\;\left(\frac{1}{\sqrt{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \frac{1}{\frac{\sqrt{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}{\sin ky}}\right) \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{6} \cdot {kx}^{2}\right) \cdot \sin th\\ \end{array}\]
\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 0.999999999998593680494707314210245385766:\\
\;\;\;\;\left(\frac{1}{\sqrt{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \frac{1}{\frac{\sqrt{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}{\sin ky}}\right) \cdot \sin th\\

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

\end{array}
double f(double kx, double ky, double th) {
        double r34674 = ky;
        double r34675 = sin(r34674);
        double r34676 = kx;
        double r34677 = sin(r34676);
        double r34678 = 2.0;
        double r34679 = pow(r34677, r34678);
        double r34680 = pow(r34675, r34678);
        double r34681 = r34679 + r34680;
        double r34682 = sqrt(r34681);
        double r34683 = r34675 / r34682;
        double r34684 = th;
        double r34685 = sin(r34684);
        double r34686 = r34683 * r34685;
        return r34686;
}
↓
double f(double kx, double ky, double th) {
        double r34687 = ky;
        double r34688 = sin(r34687);
        double r34689 = kx;
        double r34690 = sin(r34689);
        double r34691 = 2.0;
        double r34692 = pow(r34690, r34691);
        double r34693 = pow(r34688, r34691);
        double r34694 = r34692 + r34693;
        double r34695 = sqrt(r34694);
        double r34696 = r34688 / r34695;
        double r34697 = 0.9999999999985937;
        bool r34698 = r34696 <= r34697;
        double r34699 = 1.0;
        double r34700 = sqrt(r34695);
        double r34701 = r34699 / r34700;
        double r34702 = r34700 / r34688;
        double r34703 = r34699 / r34702;
        double r34704 = r34701 * r34703;
        double r34705 = th;
        double r34706 = sin(r34705);
        double r34707 = r34704 * r34706;
        double r34708 = 0.16666666666666666;
        double r34709 = 2.0;
        double r34710 = pow(r34689, r34709);
        double r34711 = r34708 * r34710;
        double r34712 = r34699 - r34711;
        double r34713 = r34712 * r34706;
        double r34714 = r34698 ? r34707 : r34713;
        return r34714;
}

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)))) < 0.9999999999985937

    1. Initial program 13.1

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

    3. Applied add-sqr-sqrt13.1

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

    4. Applied sqrt-prod13.3

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

    5. Applied *-un-lft-identity13.3

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

    6. Applied times-frac13.3

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

    8. Applied clear-num13.3

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

    if 0.9999999999985937 < (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))

    1. Initial program 8.7

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

    2. Taylor expanded around inf 9.2

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

    3. Taylor expanded around 0 3.8

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

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \le 0.999999999998593680494707314210245385766:\\ \;\;\;\;\left(\frac{1}{\sqrt{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \frac{1}{\frac{\sqrt{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}{\sin ky}}\right) \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{6} \cdot {kx}^{2}\right) \cdot \sin th\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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)))