Average Error: 12.1 → 12.4
Time: 2.1m
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{1}{\frac{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin th}}{\sin ky}}\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{1}{\frac{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin th}}{\sin ky}}
double f(double kx, double ky, double th) {
        double r237739 = ky;
        double r237740 = sin(r237739);
        double r237741 = kx;
        double r237742 = sin(r237741);
        double r237743 = 2.0;
        double r237744 = pow(r237742, r237743);
        double r237745 = pow(r237740, r237743);
        double r237746 = r237744 + r237745;
        double r237747 = sqrt(r237746);
        double r237748 = r237740 / r237747;
        double r237749 = th;
        double r237750 = sin(r237749);
        double r237751 = r237748 * r237750;
        return r237751;
}
↓
double f(double kx, double ky, double th) {
        double r237752 = 1.0;
        double r237753 = kx;
        double r237754 = sin(r237753);
        double r237755 = 2.0;
        double r237756 = pow(r237754, r237755);
        double r237757 = ky;
        double r237758 = sin(r237757);
        double r237759 = pow(r237758, r237755);
        double r237760 = r237756 + r237759;
        double r237761 = sqrt(r237760);
        double r237762 = th;
        double r237763 = sin(r237762);
        double r237764 = r237761 / r237763;
        double r237765 = r237764 / r237758;
        double r237766 = r237752 / r237765;
        return r237766;
}

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 clear-num12.1

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

  5. Applied div-inv12.2

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

  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

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

  9. Applied unpow-prod-down12.2

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

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

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

  11. Applied unpow-prod-down12.2

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

  12. Applied distribute-lft-out12.2

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

  13. Applied sqrt-prod12.2

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

  14. Applied add-cube-cbrt12.2

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

  15. Applied times-frac12.2

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

  16. Applied associate-/l*12.2

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

  18. Applied associate-/r/12.2

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

  19. Applied associate-/r*12.2

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

  20. Final simplification12.4

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

Reproduce

herbie shell --seed 2019303 
(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)))