Average Error: 0.2 → 0.2
Time: 35.2s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\left(\frac{1}{\sin B} - \frac{\cos B}{\frac{\sin B}{x}}\right) + \left(\frac{\cos B}{\frac{\sin B}{x}} - \frac{\cos B}{\frac{\sin B}{x}}\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\left(\frac{1}{\sin B} - \frac{\cos B}{\frac{\sin B}{x}}\right) + \left(\frac{\cos B}{\frac{\sin B}{x}} - \frac{\cos B}{\frac{\sin B}{x}}\right)
double f(double B, double x) {
        double r595856 = x;
        double r595857 = 1.0;
        double r595858 = B;
        double r595859 = tan(r595858);
        double r595860 = r595857 / r595859;
        double r595861 = r595856 * r595860;
        double r595862 = -r595861;
        double r595863 = sin(r595858);
        double r595864 = r595857 / r595863;
        double r595865 = r595862 + r595864;
        return r595865;
}
↓
double f(double B, double x) {
        double r595866 = 1.0;
        double r595867 = B;
        double r595868 = sin(r595867);
        double r595869 = r595866 / r595868;
        double r595870 = cos(r595867);
        double r595871 = x;
        double r595872 = r595868 / r595871;
        double r595873 = r595870 / r595872;
        double r595874 = r595869 - r595873;
        double r595875 = r595873 - r595873;
        double r595876 = r595874 + r595875;
        return r595876;
}

Error

Bits error versus B

Bits error versus x

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Results

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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. Using strategy rm

  4. Applied tan-quot0.2

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

  5. Applied associate-/r/0.2

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

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

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

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

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

  8. Applied times-frac0.2

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

  9. Applied prod-diff0.2

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

  10. Simplified0.2

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

  11. Simplified0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))