VandenBroeck and Keller, Equation (24)

Average Error: 0.2 → 0.2

Time: 10.2s

Precision: 64

Math

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

↓

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

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) (1.0 / ((double) (((double) (((double) tan(B)) / 1.0)) / x)))))) + ((double) (1.0 / ((double) sin(B))))));
}

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. Using strategy rm

3. Applied clear-num 0.2

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

4. Applied un-div-inv 0.2

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

6. Applied clear-num 0.2

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

7. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))