VandenBroeck and Keller, Equation (24)

Average Error: 0.2 → 0.1

Time: 6.5s

Precision: binary64

Math

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

↓

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

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 associate-*r/ 0.1

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

4. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020181 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (neg (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))