Toniolo and Linder, Equation (3b), real

Average Error: 3.9 → 4.2

Time: 10.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;kx \le -2.1002849544539553 \cdot 10^{-256}:\\ \;\;\;\;\left(\sqrt{\frac{1}{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;kx \le -1.87767470466880026 \cdot 10^{-272}:\\ \;\;\;\;\left(1 - \frac{1}{6} \cdot {kx}^{2}\right) \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right) \cdot \sin th\\ \end{array}\]

C

double code(double kx, double ky, double th) {
	return ((double) (((double) (((double) sin(ky)) / ((double) sqrt(((double) (((double) pow(((double) sin(kx)), 2.0)) + ((double) pow(((double) sin(ky)), 2.0)))))))) * ((double) sin(th))));
}

↓

double code(double kx, double ky, double th) {
	double VAR;
	if ((kx <= -2.1002849544539553e-256)) {
		VAR = ((double) (((double) (((double) sqrt(((double) (1.0 / ((double) (((double) pow(((double) sin(kx)), 2.0)) + ((double) pow(((double) sin(ky)), 2.0)))))))) * ((double) sin(ky)))) * ((double) sin(th))));
	} else {
		double VAR_1;
		if ((kx <= -1.8776747046688003e-272)) {
			VAR_1 = ((double) (((double) (1.0 - ((double) (0.16666666666666666 * ((double) pow(kx, 2.0)))))) * ((double) sin(th))));
		} else {
			VAR_1 = ((double) (((double) (((double) sin(ky)) * ((double) (1.0 / ((double) sqrt(((double) (((double) pow(((double) sin(kx)), 2.0)) + ((double) pow(((double) sin(ky)), 2.0)))))))))) * ((double) sin(th))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus kx

Bits error versus ky

Bits error versus th

Derivation

1. Split input into 3 regimes

2. if kx < -2.1002849544539553e-256

   1. Initial program 2.9

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

   2. Taylor expanded around inf 3.1

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin ky\right)} \cdot \sin th\]

   if -2.1002849544539553e-256 < kx < -1.87767470466880026e-272

   1. Initial program 18.6

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

   2. Taylor expanded around inf 18.7

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin ky\right)} \cdot \sin th\]

   3. Taylor expanded around 0 32.4

      \[\leadsto \color{blue}{\left(1 - \frac{1}{6} \cdot {kx}^{2}\right)} \cdot \sin th\]

   if -1.87767470466880026e-272 < kx

   1. Initial program 4.3

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

   3. Applied div-inv 4.4

      \[\leadsto \color{blue}{\left(\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right)} \cdot \sin th\]
3. Recombined 3 regimes into one program.

4. Final simplification 4.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \le -2.1002849544539553 \cdot 10^{-256}:\\ \;\;\;\;\left(\sqrt{\frac{1}{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;kx \le -1.87767470466880026 \cdot 10^{-272}:\\ \;\;\;\;\left(1 - \frac{1}{6} \cdot {kx}^{2}\right) \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right) \cdot \sin th\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(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)))