Average Error: 12.4 → 8.9
Time: 45.2s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin ky}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right)} \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{\sin ky}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right)} \cdot \sin th
double f(double kx, double ky, double th) {
        double r59571 = ky;
        double r59572 = sin(r59571);
        double r59573 = kx;
        double r59574 = sin(r59573);
        double r59575 = 2.0;
        double r59576 = pow(r59574, r59575);
        double r59577 = pow(r59572, r59575);
        double r59578 = r59576 + r59577;
        double r59579 = sqrt(r59578);
        double r59580 = r59572 / r59579;
        double r59581 = th;
        double r59582 = sin(r59581);
        double r59583 = r59580 * r59582;
        return r59583;
}
↓
double f(double kx, double ky, double th) {
        double r59584 = ky;
        double r59585 = sin(r59584);
        double r59586 = kx;
        double r59587 = sin(r59586);
        double r59588 = 2.0;
        double r59589 = 2.0;
        double r59590 = r59588 / r59589;
        double r59591 = pow(r59587, r59590);
        double r59592 = pow(r59585, r59590);
        double r59593 = hypot(r59591, r59592);
        double r59594 = log1p(r59593);
        double r59595 = expm1(r59594);
        double r59596 = r59585 / r59595;
        double r59597 = th;
        double r59598 = sin(r59597);
        double r59599 = r59596 * r59598;
        return r59599;
}

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 sqr-pow12.4

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

  4. Applied sqr-pow12.4

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

  5. Applied hypot-def8.8

    \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)}} \cdot \sin th\]
  6. Using strategy rm

  7. Applied expm1-log1p-u8.9

    \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right)}} \cdot \sin th\]

  8. Final simplification8.9

    \[\leadsto \frac{\sin ky}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right)} \cdot \sin th\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(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)))