Average Error: 0.2 → 0.2
Time: 21.6s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \frac{\sqrt[3]{1}}{\sin B}, \frac{-1}{\frac{\sin B}{x \cdot 1}} \cdot \cos B\right) + \frac{x \cdot \cos B}{\sin B} \cdot \left(\left(-1\right) + 1\right)\]
\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}, \frac{-1}{\frac{\sin B}{x \cdot 1}} \cdot \cos B\right) + \frac{x \cdot \cos B}{\sin B} \cdot \left(\left(-1\right) + 1\right)
double f(double B, double x) {
        double r25758 = x;
        double r25759 = 1.0;
        double r25760 = B;
        double r25761 = tan(r25760);
        double r25762 = r25759 / r25761;
        double r25763 = r25758 * r25762;
        double r25764 = -r25763;
        double r25765 = sin(r25760);
        double r25766 = r25759 / r25765;
        double r25767 = r25764 + r25766;
        return r25767;
}
↓
double f(double B, double x) {
        double r25768 = 1.0;
        double r25769 = cbrt(r25768);
        double r25770 = r25769 * r25769;
        double r25771 = B;
        double r25772 = sin(r25771);
        double r25773 = r25769 / r25772;
        double r25774 = -1.0;
        double r25775 = x;
        double r25776 = r25775 * r25768;
        double r25777 = r25772 / r25776;
        double r25778 = r25774 / r25777;
        double r25779 = cos(r25771);
        double r25780 = r25778 * r25779;
        double r25781 = fma(r25770, r25773, r25780);
        double r25782 = r25775 * r25779;
        double r25783 = r25782 / r25772;
        double r25784 = -r25768;
        double r25785 = r25784 + r25768;
        double r25786 = r25783 * r25785;
        double r25787 = r25781 + r25786;
        return r25787;
}

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/0.2

    \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x \cdot 1}{\tan B}}\]
  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. Using strategy rm

  15. Applied clear-num0.2

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

  16. Final simplification0.2

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))