Average Error: 0.2 → 0.2
Time: 4.7s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[1 \cdot \frac{1}{\frac{\sin B}{1 - x \cdot \cos B}}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
1 \cdot \frac{1}{\frac{\sin B}{1 - x \cdot \cos B}}
double f(double B, double x) {
        double r53975 = x;
        double r53976 = 1.0;
        double r53977 = B;
        double r53978 = tan(r53977);
        double r53979 = r53976 / r53978;
        double r53980 = r53975 * r53979;
        double r53981 = -r53980;
        double r53982 = sin(r53977);
        double r53983 = r53976 / r53982;
        double r53984 = r53981 + r53983;
        return r53984;
}
↓
double f(double B, double x) {
        double r53985 = 1.0;
        double r53986 = 1.0;
        double r53987 = B;
        double r53988 = sin(r53987);
        double r53989 = x;
        double r53990 = cos(r53987);
        double r53991 = r53989 * r53990;
        double r53992 = r53986 - r53991;
        double r53993 = r53988 / r53992;
        double r53994 = r53986 / r53993;
        double r53995 = r53985 * r53994;
        return r53995;
}

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

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

  3. Taylor expanded around inf 0.2

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

  4. Simplified0.2

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

  6. Applied div-inv0.2

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

  7. Applied associate-*l*0.2

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

  8. Simplified0.2

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

  10. Applied clear-num0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))