Average Error: 12.3 → 8.7
Time: 54.8s
Precision: 64
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin th\]
\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th
↓
\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin th
double f(double kx, double ky, double th) {
        double r30487 = ky;
        double r30488 = sin(r30487);
        double r30489 = kx;
        double r30490 = sin(r30489);
        double r30491 = 2.0;
        double r30492 = pow(r30490, r30491);
        double r30493 = pow(r30488, r30491);
        double r30494 = r30492 + r30493;
        double r30495 = sqrt(r30494);
        double r30496 = r30488 / r30495;
        double r30497 = th;
        double r30498 = sin(r30497);
        double r30499 = r30496 * r30498;
        return r30499;
}
↓
double f(double kx, double ky, double th) {
        double r30500 = ky;
        double r30501 = sin(r30500);
        double r30502 = kx;
        double r30503 = sin(r30502);
        double r30504 = hypot(r30503, r30501);
        double r30505 = r30501 / r30504;
        double r30506 = th;
        double r30507 = sin(r30506);
        double r30508 = r30505 * r30507;
        return r30508;
}

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. Taylor expanded around inf 12.3

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

  3. Simplified8.7

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

  4. Taylor expanded around inf 12.3

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

  5. Simplified8.7

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

  6. Final simplification8.7

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

Reproduce

herbie shell --seed 2019323 +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)))