Average Error: 13.0 → 13.0
Time: 15.1s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left({\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}^{\frac{-1}{2}} \cdot \sin ky\right) \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left({\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}^{\frac{-1}{2}} \cdot \sin ky\right) \cdot \sin th
double f(double kx, double ky, double th) {
        double r41826 = ky;
        double r41827 = sin(r41826);
        double r41828 = kx;
        double r41829 = sin(r41828);
        double r41830 = 2.0;
        double r41831 = pow(r41829, r41830);
        double r41832 = pow(r41827, r41830);
        double r41833 = r41831 + r41832;
        double r41834 = sqrt(r41833);
        double r41835 = r41827 / r41834;
        double r41836 = th;
        double r41837 = sin(r41836);
        double r41838 = r41835 * r41837;
        return r41838;
}
↓
double f(double kx, double ky, double th) {
        double r41839 = kx;
        double r41840 = sin(r41839);
        double r41841 = 2.0;
        double r41842 = pow(r41840, r41841);
        double r41843 = ky;
        double r41844 = sin(r41843);
        double r41845 = pow(r41844, r41841);
        double r41846 = r41842 + r41845;
        double r41847 = -0.5;
        double r41848 = pow(r41846, r41847);
        double r41849 = r41848 * r41844;
        double r41850 = th;
        double r41851 = sin(r41850);
        double r41852 = r41849 * r41851;
        return r41852;
}

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 13.0

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

  2. Taylor expanded around inf 13.0

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

  3. Taylor expanded around inf 13.3

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

  5. Applied inv-pow13.3

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

  6. Applied sqrt-pow113.0

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

  7. Simplified13.0

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

  8. Final simplification13.0

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

Reproduce

herbie shell --seed 2020045 
(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)))