Average Error: 0.2 → 0.5
Time: 23.2s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[1 \cdot \mathsf{fma}\left(1, \frac{1}{\sin B}, x \cdot \left(-\frac{\sqrt[3]{\cos B} \cdot \sqrt[3]{\cos B}}{\frac{\sin B}{\sqrt[3]{\cos B}}}\right)\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
1 \cdot \mathsf{fma}\left(1, \frac{1}{\sin B}, x \cdot \left(-\frac{\sqrt[3]{\cos B} \cdot \sqrt[3]{\cos B}}{\frac{\sin B}{\sqrt[3]{\cos B}}}\right)\right)
double f(double B, double x) {
        double r21634 = x;
        double r21635 = 1.0;
        double r21636 = B;
        double r21637 = tan(r21636);
        double r21638 = r21635 / r21637;
        double r21639 = r21634 * r21638;
        double r21640 = -r21639;
        double r21641 = sin(r21636);
        double r21642 = r21635 / r21641;
        double r21643 = r21640 + r21642;
        return r21643;
}
↓
double f(double B, double x) {
        double r21644 = 1.0;
        double r21645 = 1.0;
        double r21646 = B;
        double r21647 = sin(r21646);
        double r21648 = r21645 / r21647;
        double r21649 = x;
        double r21650 = cos(r21646);
        double r21651 = cbrt(r21650);
        double r21652 = r21651 * r21651;
        double r21653 = r21647 / r21651;
        double r21654 = r21652 / r21653;
        double r21655 = -r21654;
        double r21656 = r21649 * r21655;
        double r21657 = fma(r21645, r21648, r21656);
        double r21658 = r21644 * r21657;
        return r21658;
}

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. Taylor expanded around inf 0.2

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

  4. Simplified0.2

    \[\leadsto \color{blue}{1 \cdot \left(\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\right)}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity0.2

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

  7. Applied fma-neg0.2

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

  8. Simplified0.2

    \[\leadsto 1 \cdot \mathsf{fma}\left(1, \frac{1}{\sin B}, \color{blue}{x \cdot \left(-\frac{\cos B}{\sin B}\right)}\right)\]
  9. Using strategy rm

  10. Applied add-cube-cbrt0.5

    \[\leadsto 1 \cdot \mathsf{fma}\left(1, \frac{1}{\sin B}, x \cdot \left(-\frac{\color{blue}{\left(\sqrt[3]{\cos B} \cdot \sqrt[3]{\cos B}\right) \cdot \sqrt[3]{\cos B}}}{\sin B}\right)\right)\]

  11. Applied associate-/l*0.5

    \[\leadsto 1 \cdot \mathsf{fma}\left(1, \frac{1}{\sin B}, x \cdot \left(-\color{blue}{\frac{\sqrt[3]{\cos B} \cdot \sqrt[3]{\cos B}}{\frac{\sin B}{\sqrt[3]{\cos B}}}}\right)\right)\]

  12. Final simplification0.5

    \[\leadsto 1 \cdot \mathsf{fma}\left(1, \frac{1}{\sin B}, x \cdot \left(-\frac{\sqrt[3]{\cos B} \cdot \sqrt[3]{\cos B}}{\frac{\sin B}{\sqrt[3]{\cos B}}}\right)\right)\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))