Average Error: 12.1 → 8.7
Time: 11.9s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \sqrt{1}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left(\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \sqrt{1}
double f(double kx, double ky, double th) {
        double r58521 = ky;
        double r58522 = sin(r58521);
        double r58523 = kx;
        double r58524 = sin(r58523);
        double r58525 = 2.0;
        double r58526 = pow(r58524, r58525);
        double r58527 = pow(r58522, r58525);
        double r58528 = r58526 + r58527;
        double r58529 = sqrt(r58528);
        double r58530 = r58522 / r58529;
        double r58531 = th;
        double r58532 = sin(r58531);
        double r58533 = r58530 * r58532;
        return r58533;
}
↓
double f(double kx, double ky, double th) {
        double r58534 = th;
        double r58535 = sin(r58534);
        double r58536 = ky;
        double r58537 = sin(r58536);
        double r58538 = kx;
        double r58539 = sin(r58538);
        double r58540 = 2.0;
        double r58541 = 2.0;
        double r58542 = r58540 / r58541;
        double r58543 = pow(r58539, r58542);
        double r58544 = pow(r58537, r58542);
        double r58545 = hypot(r58543, r58544);
        double r58546 = r58537 / r58545;
        double r58547 = r58535 * r58546;
        double r58548 = 1.0;
        double r58549 = sqrt(r58548);
        double r58550 = r58547 * r58549;
        return r58550;
}

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 12.1

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

  3. Applied sqr-pow12.1

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

  4. Applied sqr-pow12.1

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

  5. Applied hypot-def8.7

    \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}} \cdot \sin th\]
  6. Using strategy rm

  7. Applied clear-num8.7

    \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}{\sin ky}}} \cdot \sin th\]
  8. Using strategy rm

  9. Applied *-un-lft-identity8.7

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

  10. Applied *-un-lft-identity8.7

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

  11. Applied times-frac8.7

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

  12. Applied add-sqr-sqrt8.7

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

  13. Applied times-frac8.7

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

  14. Applied associate-*l*8.7

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

  15. Simplified8.7

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

  17. Applied div-inv8.7

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

  18. Simplified8.7

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

  19. Final simplification8.7

    \[\leadsto \left(\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \sqrt{1}\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(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)))