Average Error: 12.5 → 12.9
Time: 11.3s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \left(\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\right)\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \left(\frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\right)
double code(double kx, double ky, double th) {
	return ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th));
}
↓
double code(double kx, double ky, double th) {
	return (((cbrt(sin(ky)) * cbrt(sin(ky))) / (cbrt(sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * cbrt(sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))))) * ((cbrt(sin(ky)) / cbrt(sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0))))) * sin(th)));
}

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 12.5

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

  3. Applied add-cube-cbrt13.3

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

  4. Applied add-cube-cbrt12.9

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

  5. Applied times-frac12.9

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

  6. Applied associate-*l*12.9

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

  7. Final simplification12.9

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

Reproduce

herbie shell --seed 2020049 
(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)))