Average Error: 12.2 → 8.5
Time: 11.5s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin ky}{\mathsf{hypot}\left({\left(\sin kx\right)}^{\left(\frac{2}{2}\right)}, {\left(\sin ky\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \sin th\right)\right)\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin ky}{\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\right)\right)
double f(double kx, double ky, double th) {
        double r46543 = ky;
        double r46544 = sin(r46543);
        double r46545 = kx;
        double r46546 = sin(r46545);
        double r46547 = 2.0;
        double r46548 = pow(r46546, r46547);
        double r46549 = pow(r46544, r46547);
        double r46550 = r46548 + r46549;
        double r46551 = sqrt(r46550);
        double r46552 = r46544 / r46551;
        double r46553 = th;
        double r46554 = sin(r46553);
        double r46555 = r46552 * r46554;
        return r46555;
}
↓
double f(double kx, double ky, double th) {
        double r46556 = ky;
        double r46557 = sin(r46556);
        double r46558 = kx;
        double r46559 = sin(r46558);
        double r46560 = 2.0;
        double r46561 = 2.0;
        double r46562 = r46560 / r46561;
        double r46563 = pow(r46559, r46562);
        double r46564 = pow(r46557, r46562);
        double r46565 = hypot(r46563, r46564);
        double r46566 = r46557 / r46565;
        double r46567 = th;
        double r46568 = sin(r46567);
        double r46569 = r46566 * r46568;
        double r46570 = expm1(r46569);
        double r46571 = log1p(r46570);
        return r46571;
}

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 sqr-pow12.2

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

  4. Applied sqr-pow12.2

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

  5. Applied hypot-def8.5

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

  7. Applied log1p-expm1-u8.5

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

  8. Final simplification8.5

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(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)))