Average Error: 12.4 → 12.4
Time: 46.0s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{1}{1} \cdot \left(\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\right)\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{1}{1} \cdot \left(\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\right)
double f(double kx, double ky, double th) {
        double r36801 = ky;
        double r36802 = sin(r36801);
        double r36803 = kx;
        double r36804 = sin(r36803);
        double r36805 = 2.0;
        double r36806 = pow(r36804, r36805);
        double r36807 = pow(r36802, r36805);
        double r36808 = r36806 + r36807;
        double r36809 = sqrt(r36808);
        double r36810 = r36802 / r36809;
        double r36811 = th;
        double r36812 = sin(r36811);
        double r36813 = r36810 * r36812;
        return r36813;
}
↓
double f(double kx, double ky, double th) {
        double r36814 = 1.0;
        double r36815 = r36814 / r36814;
        double r36816 = ky;
        double r36817 = sin(r36816);
        double r36818 = kx;
        double r36819 = sin(r36818);
        double r36820 = 2.0;
        double r36821 = pow(r36819, r36820);
        double r36822 = pow(r36817, r36820);
        double r36823 = r36821 + r36822;
        double r36824 = sqrt(r36823);
        double r36825 = r36817 / r36824;
        double r36826 = th;
        double r36827 = sin(r36826);
        double r36828 = r36825 * r36827;
        double r36829 = r36815 * r36828;
        return r36829;
}

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 12.4

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

  3. Applied *-un-lft-identity12.4

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

  4. Applied unpow-prod-down12.4

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

  5. Applied *-un-lft-identity12.4

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

  6. Applied unpow-prod-down12.4

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

  7. Applied distribute-lft-out12.4

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

  8. Applied sqrt-prod12.4

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

  9. Applied *-un-lft-identity12.4

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

  10. Applied times-frac12.4

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

  11. Applied associate-*l*12.4

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

  12. Final simplification12.4

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

Reproduce

herbie shell --seed 2019199 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))