Average Error: 0.2 → 0.2
Time: 20.0s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \frac{\sqrt[3]{1}}{\sin B}, -\cos B \cdot \frac{x \cdot 1}{\sin B}\right) + \frac{x \cdot \cos B}{\sin B} \cdot \left(\left(-1\right) + 1\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \frac{\sqrt[3]{1}}{\sin B}, -\cos B \cdot \frac{x \cdot 1}{\sin B}\right) + \frac{x \cdot \cos B}{\sin B} \cdot \left(\left(-1\right) + 1\right)
double f(double B, double x) {
        double r41609 = x;
        double r41610 = 1.0;
        double r41611 = B;
        double r41612 = tan(r41611);
        double r41613 = r41610 / r41612;
        double r41614 = r41609 * r41613;
        double r41615 = -r41614;
        double r41616 = sin(r41611);
        double r41617 = r41610 / r41616;
        double r41618 = r41615 + r41617;
        return r41618;
}
↓
double f(double B, double x) {
        double r41619 = 1.0;
        double r41620 = cbrt(r41619);
        double r41621 = r41620 * r41620;
        double r41622 = B;
        double r41623 = sin(r41622);
        double r41624 = r41620 / r41623;
        double r41625 = cos(r41622);
        double r41626 = x;
        double r41627 = r41626 * r41619;
        double r41628 = r41627 / r41623;
        double r41629 = r41625 * r41628;
        double r41630 = -r41629;
        double r41631 = fma(r41621, r41624, r41630);
        double r41632 = r41626 * r41625;
        double r41633 = r41632 / r41623;
        double r41634 = -r41619;
        double r41635 = r41634 + r41619;
        double r41636 = r41633 * r41635;
        double r41637 = r41631 + r41636;
        return r41637;
}

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. Using strategy rm

  6. Applied tan-quot0.2

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

  7. Applied associate-/r/0.2

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

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

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

  9. Applied add-cube-cbrt0.2

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

  10. Applied times-frac0.2

    \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\sin B}} - \frac{x \cdot 1}{\sin B} \cdot \cos B\]

  11. Applied prod-diff0.2

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

  12. Simplified0.2

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

  13. Simplified0.2

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

  14. Final simplification0.2

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

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))))