Average Error: 0.2 → 0.2
Time: 19.5s
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 r42252 = x;
        double r42253 = 1.0;
        double r42254 = B;
        double r42255 = tan(r42254);
        double r42256 = r42253 / r42255;
        double r42257 = r42252 * r42256;
        double r42258 = -r42257;
        double r42259 = sin(r42254);
        double r42260 = r42253 / r42259;
        double r42261 = r42258 + r42260;
        return r42261;
}
↓
double f(double B, double x) {
        double r42262 = 1.0;
        double r42263 = cbrt(r42262);
        double r42264 = r42263 * r42263;
        double r42265 = B;
        double r42266 = sin(r42265);
        double r42267 = r42263 / r42266;
        double r42268 = -1.0;
        double r42269 = x;
        double r42270 = r42269 * r42262;
        double r42271 = r42266 / r42270;
        double r42272 = r42268 / r42271;
        double r42273 = cos(r42265);
        double r42274 = r42272 * r42273;
        double r42275 = fma(r42264, r42267, r42274);
        double r42276 = r42269 * r42273;
        double r42277 = r42276 / r42266;
        double r42278 = -r42262;
        double r42279 = r42278 + r42262;
        double r42280 = r42277 * r42279;
        double r42281 = r42275 + r42280;
        return r42281;
}

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))))