VandenBroeck and Keller, Equation (24)

Average Error: 0.2 → 0.2

Time: 5.0s

Precision: 64

Math

\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]

↓

\[\frac{\sqrt{1} \cdot \left(1 - x \cdot \cos B\right)}{\frac{\sin B}{\sqrt{1}}}\]

C

double code(double B, double x) {
	return ((double) (((double) -(((double) (x * ((double) (1.0 / ((double) tan(B)))))))) + ((double) (1.0 / ((double) sin(B))))));
}

↓

double code(double B, double x) {
	return ((double) (((double) (((double) sqrt(1.0)) * ((double) (1.0 - ((double) (x * ((double) cos(B)))))))) / ((double) (((double) sin(B)) / ((double) sqrt(1.0))))));
}

Error

Bits error versus B

Bits error versus x

Derivation

1. Initial program 0.2

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]

2. Simplified 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{1}{\tan B}, \frac{1}{\sin B}\right)}\]

3. Taylor expanded around inf 0.2

   \[\leadsto \color{blue}{1 \cdot \frac{1}{\sin B} - 1 \cdot \frac{x \cdot \cos B}{\sin B}}\]

4. Simplified 0.3

   \[\leadsto \color{blue}{\frac{1}{\sin B} \cdot \left(1 - x \cdot \cos B\right)}\]
5. Using strategy rm

6. Applied add-sqr-sqrt 0.3

   \[\leadsto \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\sin B} \cdot \left(1 - x \cdot \cos B\right)\]

7. Applied associate-/l* 0.3

   \[\leadsto \color{blue}{\frac{\sqrt{1}}{\frac{\sin B}{\sqrt{1}}}} \cdot \left(1 - x \cdot \cos B\right)\]

8. Applied associate-*l/ 0.2

   \[\leadsto \color{blue}{\frac{\sqrt{1} \cdot \left(1 - x \cdot \cos B\right)}{\frac{\sin B}{\sqrt{1}}}}\]

9. Final simplification 0.2

   \[\leadsto \frac{\sqrt{1} \cdot \left(1 - x \cdot \cos B\right)}{\frac{\sin B}{\sqrt{1}}}\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))