Average Error: 0.2 → 0.2
Time: 20.6s
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 r52706 = x;
        double r52707 = 1.0;
        double r52708 = B;
        double r52709 = tan(r52708);
        double r52710 = r52707 / r52709;
        double r52711 = r52706 * r52710;
        double r52712 = -r52711;
        double r52713 = sin(r52708);
        double r52714 = r52707 / r52713;
        double r52715 = r52712 + r52714;
        return r52715;
}
↓
double f(double B, double x) {
        double r52716 = x;
        double r52717 = B;
        double r52718 = cos(r52717);
        double r52719 = r52716 * r52718;
        double r52720 = sin(r52717);
        double r52721 = r52719 / r52720;
        double r52722 = 1.0;
        double r52723 = -r52722;
        double r52724 = r52722 / r52720;
        double r52725 = fma(r52721, r52723, r52724);
        return r52725;
}

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))))