VandenBroeck and Keller, Equation (24)

Average Error: 0.2 → 0.2

Time: 6.5s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

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. 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. Using strategy rm

5. Applied associate-*r/ 0.2

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

6. Applied sub-div 0.2

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

7. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020182 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (neg (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))