Average Error: 0.2 → 0.2
Time: 19.3s
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 r49197 = x;
        double r49198 = 1.0;
        double r49199 = B;
        double r49200 = tan(r49199);
        double r49201 = r49198 / r49200;
        double r49202 = r49197 * r49201;
        double r49203 = -r49202;
        double r49204 = sin(r49199);
        double r49205 = r49198 / r49204;
        double r49206 = r49203 + r49205;
        return r49206;
}
↓
double f(double B, double x) {
        double r49207 = 1.0;
        double r49208 = cbrt(r49207);
        double r49209 = r49208 * r49208;
        double r49210 = B;
        double r49211 = sin(r49210);
        double r49212 = r49208 / r49211;
        double r49213 = -1.0;
        double r49214 = x;
        double r49215 = r49214 * r49207;
        double r49216 = r49211 / r49215;
        double r49217 = r49213 / r49216;
        double r49218 = cos(r49210);
        double r49219 = r49217 * r49218;
        double r49220 = fma(r49209, r49212, r49219);
        double r49221 = r49214 * r49218;
        double r49222 = r49221 / r49211;
        double r49223 = -r49207;
        double r49224 = r49223 + r49207;
        double r49225 = r49222 * r49224;
        double r49226 = r49220 + r49225;
        return r49226;
}

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))))