Average Error: 0.2 → 0.3
Time: 17.9s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\left(\frac{1}{\sin B} - \frac{1}{\frac{\frac{1}{\cos B}}{x} \cdot \sin B}\right) \cdot 1\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\left(\frac{1}{\sin B} - \frac{1}{\frac{\frac{1}{\cos B}}{x} \cdot \sin B}\right) \cdot 1
double f(double B, double x) {
        double r50152 = x;
        double r50153 = 1.0;
        double r50154 = B;
        double r50155 = tan(r50154);
        double r50156 = r50153 / r50155;
        double r50157 = r50152 * r50156;
        double r50158 = -r50157;
        double r50159 = sin(r50154);
        double r50160 = r50153 / r50159;
        double r50161 = r50158 + r50160;
        return r50161;
}
↓
double f(double B, double x) {
        double r50162 = 1.0;
        double r50163 = B;
        double r50164 = sin(r50163);
        double r50165 = r50162 / r50164;
        double r50166 = cos(r50163);
        double r50167 = r50162 / r50166;
        double r50168 = x;
        double r50169 = r50167 / r50168;
        double r50170 = r50169 * r50164;
        double r50171 = r50162 / r50170;
        double r50172 = r50165 - r50171;
        double r50173 = 1.0;
        double r50174 = r50172 * r50173;
        return r50174;
}

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. Taylor expanded around inf 0.2

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

  3. Simplified0.2

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

  5. Applied clear-num0.2

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

  6. Simplified0.2

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

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

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

  9. Applied div-inv0.3

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

  10. Applied times-frac0.3

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

  11. Simplified0.3

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019194 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))