Average Error: 12.3 → 12.5
Time: 41.6s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\sqrt{\frac{1}{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin ky\right) \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\left(\sqrt{\frac{1}{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin ky\right) \cdot \sin th
double f(double kx, double ky, double th) {
        double r32103 = ky;
        double r32104 = sin(r32103);
        double r32105 = kx;
        double r32106 = sin(r32105);
        double r32107 = 2.0;
        double r32108 = pow(r32106, r32107);
        double r32109 = pow(r32104, r32107);
        double r32110 = r32108 + r32109;
        double r32111 = sqrt(r32110);
        double r32112 = r32104 / r32111;
        double r32113 = th;
        double r32114 = sin(r32113);
        double r32115 = r32112 * r32114;
        return r32115;
}
↓
double f(double kx, double ky, double th) {
        double r32116 = 1.0;
        double r32117 = kx;
        double r32118 = sin(r32117);
        double r32119 = 2.0;
        double r32120 = pow(r32118, r32119);
        double r32121 = ky;
        double r32122 = sin(r32121);
        double r32123 = pow(r32122, r32119);
        double r32124 = r32120 + r32123;
        double r32125 = r32116 / r32124;
        double r32126 = sqrt(r32125);
        double r32127 = r32126 * r32122;
        double r32128 = th;
        double r32129 = sin(r32128);
        double r32130 = r32127 * r32129;
        return r32130;
}

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 12.3

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

  3. Applied add-sqr-sqrt12.3

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

  4. Applied sqrt-prod12.5

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

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

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

  6. Applied times-frac12.6

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

  7. Taylor expanded around inf 12.5

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

  8. Final simplification12.5

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

Reproduce

herbie shell --seed 2019323 
(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)))