VandenBroeck and Keller, Equation (24)

Average Error: 0.2 → 0.2

Time: 5.4s

Precision: binary64

Math

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

↓

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

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

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

3. Taylor expanded around inf 0.2

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

4. Simplified 0.2

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

5. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020185 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (neg (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))