Average Error: 0.2 → 0.2
Time: 7.2m
Precision: 64
Internal Precision: 128
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\sin B} - \frac{1}{\frac{\tan B}{x}}\]

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. Simplified0.1

    \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.1

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

  5. Applied associate-/l*0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019091 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))