Average Error: 0.2 → 0.2
Time: 5.0s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\mathsf{fma}\left(-x, \frac{1}{\sin B} \cdot \cos B, \frac{1}{\sin B}\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\mathsf{fma}\left(-x, \frac{1}{\sin B} \cdot \cos B, \frac{1}{\sin B}\right)
double code(double B, double x) {
	return ((double) (((double) -(((double) (x * ((double) (1.0 / ((double) tan(B)))))))) + ((double) (1.0 / ((double) sin(B))))));
}
↓
double code(double B, double x) {
	return ((double) fma(((double) -(x)), ((double) (((double) (1.0 / ((double) sin(B)))) * ((double) cos(B)))), ((double) (1.0 / ((double) sin(B))))));
}

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. Using strategy rm

  4. Applied tan-quot0.2

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

  5. Applied associate-/r/0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020123 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))