Toniolo and Linder, Equation (3b), real

Average Error: 4.0 → 0.2

Time: 10.3s

Precision: 64

Math

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

↓

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

C

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(ky) / hypot(sin(ky), sin(kx))) * sin(th));
}

Error

Bits error versus kx

Bits error versus ky

Bits error versus th

Derivation

1. Initial program 4.0

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

2. Taylor expanded around inf 4.0

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

3. Simplified 0.2

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

5. Applied clear-num 0.3

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

7. Applied *-un-lft-identity 0.3

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

8. Applied associate-*r* 0.3

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

9. Simplified 0.2

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

10. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020065 +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)))