Average Error: 0.2 → 0.2
Time: 6.1s
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 r53993 = x;
        double r53994 = 1.0;
        double r53995 = B;
        double r53996 = tan(r53995);
        double r53997 = r53994 / r53996;
        double r53998 = r53993 * r53997;
        double r53999 = -r53998;
        double r54000 = sin(r53995);
        double r54001 = r53994 / r54000;
        double r54002 = r53999 + r54001;
        return r54002;
}
↓
double f(double B, double x) {
        double r54003 = x;
        double r54004 = 1.0;
        double r54005 = r54003 * r54004;
        double r54006 = B;
        double r54007 = sin(r54006);
        double r54008 = r54005 / r54007;
        double r54009 = cos(r54006);
        double r54010 = -r54009;
        double r54011 = r54004 / r54007;
        double r54012 = fma(r54008, r54010, r54011);
        return r54012;
}

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 2020081 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))