Average Error: 12.2 → 12.2
Time: 31.4s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin th \cdot \frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\sin th \cdot \frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}
double f(double kx, double ky, double th) {
        double r34405 = ky;
        double r34406 = sin(r34405);
        double r34407 = kx;
        double r34408 = sin(r34407);
        double r34409 = 2.0;
        double r34410 = pow(r34408, r34409);
        double r34411 = pow(r34406, r34409);
        double r34412 = r34410 + r34411;
        double r34413 = sqrt(r34412);
        double r34414 = r34406 / r34413;
        double r34415 = th;
        double r34416 = sin(r34415);
        double r34417 = r34414 * r34416;
        return r34417;
}
↓
double f(double kx, double ky, double th) {
        double r34418 = th;
        double r34419 = sin(r34418);
        double r34420 = ky;
        double r34421 = sin(r34420);
        double r34422 = kx;
        double r34423 = sin(r34422);
        double r34424 = 2.0;
        double r34425 = pow(r34423, r34424);
        double r34426 = pow(r34421, r34424);
        double r34427 = r34425 + r34426;
        double r34428 = sqrt(r34427);
        double r34429 = r34421 / r34428;
        double r34430 = r34419 * r34429;
        return r34430;
}

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 12.2

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

  3. Applied clear-num12.2

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

  5. Applied div-inv12.3

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

  6. Applied associate-/r*12.2

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

  8. Applied *-un-lft-identity12.2

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

  9. Applied add-cube-cbrt12.2

    \[\leadsto \frac{\frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \sin ky}} \cdot \sin th\]

  10. Applied times-frac12.2

    \[\leadsto \frac{\frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\sin ky}}} \cdot \sin th\]

  11. Applied *-un-lft-identity12.2

    \[\leadsto \frac{\frac{1}{\sqrt{\color{blue}{1 \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\sin ky}} \cdot \sin th\]

  12. Applied sqrt-prod12.2

    \[\leadsto \frac{\frac{1}{\color{blue}{\sqrt{1} \cdot \sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\sin ky}} \cdot \sin th\]

  13. Applied add-cube-cbrt12.2

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{1} \cdot \sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\sin ky}} \cdot \sin th\]

  14. Applied times-frac12.2

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \frac{\sqrt[3]{1}}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\sin ky}} \cdot \sin th\]

  15. Applied times-frac12.2

    \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}{\frac{\sqrt[3]{1}}{\sin ky}}\right)} \cdot \sin th\]

  16. Simplified12.2

    \[\leadsto \left(\color{blue}{1} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}{\frac{\sqrt[3]{1}}{\sin ky}}\right) \cdot \sin th\]

  17. Simplified12.2

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

  18. Final simplification12.2

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

Reproduce

herbie shell --seed 2019347 
(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)))