Average Error: 0.2 → 0.2
Time: 5.2s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1 - \cos B \cdot \left(1 \cdot x\right)}{\sin B}\]

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. Simplified0.2

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

  3. Taylor expanded around inf 0.2

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

  4. Simplified0.2

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

  6. Applied associate-*r/0.2

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

  7. Applied associate-*r/0.2

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

  8. Applied sub-div0.2

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2020184 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (neg (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))