Average Error: 0.2 → 0.2
Time: 5.5s
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 r36765 = x;
        double r36766 = 1.0;
        double r36767 = B;
        double r36768 = tan(r36767);
        double r36769 = r36766 / r36768;
        double r36770 = r36765 * r36769;
        double r36771 = -r36770;
        double r36772 = sin(r36767);
        double r36773 = r36766 / r36772;
        double r36774 = r36771 + r36773;
        return r36774;
}
↓
double f(double B, double x) {
        double r36775 = x;
        double r36776 = 1.0;
        double r36777 = r36775 * r36776;
        double r36778 = B;
        double r36779 = sin(r36778);
        double r36780 = r36777 / r36779;
        double r36781 = cos(r36778);
        double r36782 = -r36781;
        double r36783 = r36776 / r36779;
        double r36784 = fma(r36780, r36782, r36783);
        return r36784;
}

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/0.2

    \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{1}{\sin B}\]
  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-rgt-neg-in0.2

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

  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 2019346 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))