Average Error: 12.2 → 12.5
Time: 18.4s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\left(\sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \sqrt[3]{\frac{\sin ky}{\left|\sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right| \cdot \sqrt{\sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}}\right) \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left(\left(\sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right) \cdot \sqrt[3]{\frac{\sin ky}{\left|\sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right| \cdot \sqrt{\sqrt[3]{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}}\right) \cdot \sin th
double f(double kx, double ky, double th) {
        double r44117 = ky;
        double r44118 = sin(r44117);
        double r44119 = kx;
        double r44120 = sin(r44119);
        double r44121 = 2.0;
        double r44122 = pow(r44120, r44121);
        double r44123 = pow(r44118, r44121);
        double r44124 = r44122 + r44123;
        double r44125 = sqrt(r44124);
        double r44126 = r44118 / r44125;
        double r44127 = th;
        double r44128 = sin(r44127);
        double r44129 = r44126 * r44128;
        return r44129;
}
↓
double f(double kx, double ky, double th) {
        double r44130 = ky;
        double r44131 = sin(r44130);
        double r44132 = kx;
        double r44133 = sin(r44132);
        double r44134 = 2.0;
        double r44135 = pow(r44133, r44134);
        double r44136 = pow(r44131, r44134);
        double r44137 = r44135 + r44136;
        double r44138 = sqrt(r44137);
        double r44139 = r44131 / r44138;
        double r44140 = cbrt(r44139);
        double r44141 = r44140 * r44140;
        double r44142 = cbrt(r44137);
        double r44143 = fabs(r44142);
        double r44144 = sqrt(r44142);
        double r44145 = r44143 * r44144;
        double r44146 = r44131 / r44145;
        double r44147 = cbrt(r44146);
        double r44148 = r44141 * r44147;
        double r44149 = th;
        double r44150 = sin(r44149);
        double r44151 = r44148 * r44150;
        return r44151;
}

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 add-cube-cbrt12.5

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

  5. Applied add-cube-cbrt12.5

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

  6. Applied sqrt-prod12.5

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

  7. Simplified12.5

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

  8. Final simplification12.5

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

Reproduce

herbie shell --seed 2020046 
(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)))