Average Error: 3.7 → 0.2
Time: 23.2s
Precision: binary64
\[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
↓
\[\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th
↓
\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))
↓
(FPCore (kx ky th)
 :precision binary64
 (* (sin th) (/ (sin ky) (hypot (sin ky) (sin kx)))))
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 sin(th) * (sin(ky) / hypot(sin(ky), sin(kx)));
}

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 3.7

    \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

  2. Simplified0.2

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

  3. Applied add-sqr-sqrt_binary640.6

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

  4. Applied associate-/r*_binary640.6

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

  5. Applied *-un-lft-identity_binary640.6

    \[\leadsto \color{blue}{\left(1 \cdot \frac{\frac{\sin ky}{\sqrt{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}{\sqrt{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} \cdot \sin th \]

  6. Applied associate-*l*_binary640.6

    \[\leadsto \color{blue}{1 \cdot \left(\frac{\frac{\sin ky}{\sqrt{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}}{\sqrt{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th\right)} \]

  7. Simplified0.2

    \[\leadsto 1 \cdot \color{blue}{\left(\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} \]

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2021291 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  :precision binary64
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))