Average Error: 0.2 → 0.2
Time: 19.7s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\mathsf{fma}\left(\frac{x \cdot \cos B}{\sin B}, -1, \frac{1}{\sin B}\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\mathsf{fma}\left(\frac{x \cdot \cos B}{\sin B}, -1, \frac{1}{\sin B}\right)
double f(double B, double x) {
        double r40352 = x;
        double r40353 = 1.0;
        double r40354 = B;
        double r40355 = tan(r40354);
        double r40356 = r40353 / r40355;
        double r40357 = r40352 * r40356;
        double r40358 = -r40357;
        double r40359 = sin(r40354);
        double r40360 = r40353 / r40359;
        double r40361 = r40358 + r40360;
        return r40361;
}
↓
double f(double B, double x) {
        double r40362 = x;
        double r40363 = B;
        double r40364 = cos(r40363);
        double r40365 = r40362 * r40364;
        double r40366 = sin(r40363);
        double r40367 = r40365 / r40366;
        double r40368 = 1.0;
        double r40369 = -r40368;
        double r40370 = r40368 / r40366;
        double r40371 = fma(r40367, r40369, r40370);
        return r40371;
}

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. Using strategy rm

  4. Applied associate-*r/0.1

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

  5. Taylor expanded around inf 0.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))