Average Error: 0.2 → 0.2
Time: 3.3s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1 - x \cdot \cos B}{\frac{\sin B}{1}}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1 - x \cdot \cos B}{\frac{\sin B}{1}}
double code(double B, double x) {
	return (-(x * (1.0 / tan(B))) + (1.0 / sin(B)));
}
↓
double code(double B, double x) {
	return ((1.0 - (x * cos(B))) / (sin(B) / 1.0));
}

Error

Bits error versus B

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{1}{\tan B}, \frac{1}{\sin B}\right)}\]

  3. Taylor expanded around inf 0.2

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

  4. Simplified0.3

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

  6. Applied clear-num0.3

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

  7. Applied associate-*l/0.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))