Average Error: 0.2 → 0.2
Time: 12.1s
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 r61506 = x;
        double r61507 = 1.0;
        double r61508 = B;
        double r61509 = tan(r61508);
        double r61510 = r61507 / r61509;
        double r61511 = r61506 * r61510;
        double r61512 = -r61511;
        double r61513 = sin(r61508);
        double r61514 = r61507 / r61513;
        double r61515 = r61512 + r61514;
        return r61515;
}
↓
double f(double B, double x) {
        double r61516 = x;
        double r61517 = B;
        double r61518 = cos(r61517);
        double r61519 = r61516 * r61518;
        double r61520 = sin(r61517);
        double r61521 = r61519 / r61520;
        double r61522 = 1.0;
        double r61523 = -r61522;
        double r61524 = r61522 / r61520;
        double r61525 = fma(r61521, r61523, r61524);
        return r61525;
}

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/0.2

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

  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 2019351 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))