Average Error: 0.2 → 0.2
Time: 9.3m
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\sin B} - \frac{1}{\frac{\frac{\sin B}{x}}{\cos B}}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1}{\sin B} - \frac{1}{\frac{\frac{\sin B}{x}}{\cos B}}
double f(double B, double x) {
        double r20753694 = x;
        double r20753695 = 1.0;
        double r20753696 = B;
        double r20753697 = tan(r20753696);
        double r20753698 = r20753695 / r20753697;
        double r20753699 = r20753694 * r20753698;
        double r20753700 = -r20753699;
        double r20753701 = sin(r20753696);
        double r20753702 = r20753695 / r20753701;
        double r20753703 = r20753700 + r20753702;
        return r20753703;
}
↓
double f(double B, double x) {
        double r20753704 = 1.0;
        double r20753705 = B;
        double r20753706 = sin(r20753705);
        double r20753707 = r20753704 / r20753706;
        double r20753708 = x;
        double r20753709 = r20753706 / r20753708;
        double r20753710 = cos(r20753705);
        double r20753711 = r20753709 / r20753710;
        double r20753712 = r20753704 / r20753711;
        double r20753713 = r20753707 - r20753712;
        return r20753713;
}

Error

Bits error versus B

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]

  2. Simplified0.2

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

  3. Taylor expanded around inf 0.2

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

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

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

  6. Applied times-frac0.2

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

  7. Applied add-cube-cbrt0.9

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

  8. Applied add-sqr-sqrt0.9

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

  9. Applied times-frac0.9

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

  10. Applied prod-diff0.9

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

  11. Simplified0.2

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

  12. Simplified0.2

    \[\leadsto \left(\frac{1}{\sin B} - \frac{\cos B}{\frac{\sin B}{x}}\right) + \color{blue}{0}\]
  13. Using strategy rm

  14. Applied clear-num0.2

    \[\leadsto \left(\frac{1}{\sin B} - \color{blue}{\frac{1}{\frac{\frac{\sin B}{x}}{\cos B}}}\right) + 0\]

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))