Average Error: 0.2 → 0.2
Time: 4.9s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\mathsf{fma}\left(-\frac{x \cdot 1}{\sin B}, \cos B, \frac{1}{\sin B}\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\mathsf{fma}\left(-\frac{x \cdot 1}{\sin B}, \cos B, \frac{1}{\sin B}\right)
double code(double B, double x) {
	return (-(x * (1.0 / tan(B))) + (1.0 / sin(B)));
}
↓
double code(double B, double x) {
	return fma(-((x * 1.0) / sin(B)), cos(B), (1.0 / 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. Using strategy rm

  3. Applied associate-*r/0.2

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

  5. Applied tan-quot0.2

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

  6. Applied associate-/r/0.2

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

  7. Applied distribute-lft-neg-in0.2

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

  8. Applied fma-def0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020092 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))