Average Error: 0.2 → 0.2
Time: 23.3s
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 f(double B, double x) {
        double r22595 = x;
        double r22596 = 1.0;
        double r22597 = B;
        double r22598 = tan(r22597);
        double r22599 = r22596 / r22598;
        double r22600 = r22595 * r22599;
        double r22601 = -r22600;
        double r22602 = sin(r22597);
        double r22603 = r22596 / r22602;
        double r22604 = r22601 + r22603;
        return r22604;
}
↓
double f(double B, double x) {
        double r22605 = x;
        double r22606 = 1.0;
        double r22607 = r22605 * r22606;
        double r22608 = B;
        double r22609 = sin(r22608);
        double r22610 = r22607 / r22609;
        double r22611 = -r22610;
        double r22612 = cos(r22608);
        double r22613 = r22606 / r22609;
        double r22614 = fma(r22611, r22612, r22613);
        return r22614;
}

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. Using strategy rm

  3. Applied associate-*r/0.1

    \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{1}{\sin B}\]
  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 2019303 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))