Average Error: 0.2 → 0.2
Time: 17.1s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[1 \cdot \frac{1 - x \cdot \cos B}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
1 \cdot \frac{1 - x \cdot \cos B}{\sin B}
double f(double B, double x) {
        double r40743 = x;
        double r40744 = 1.0;
        double r40745 = B;
        double r40746 = tan(r40745);
        double r40747 = r40744 / r40746;
        double r40748 = r40743 * r40747;
        double r40749 = -r40748;
        double r40750 = sin(r40745);
        double r40751 = r40744 / r40750;
        double r40752 = r40749 + r40751;
        return r40752;
}
↓
double f(double B, double x) {
        double r40753 = 1.0;
        double r40754 = 1.0;
        double r40755 = x;
        double r40756 = B;
        double r40757 = cos(r40756);
        double r40758 = r40755 * r40757;
        double r40759 = r40754 - r40758;
        double r40760 = sin(r40756);
        double r40761 = r40759 / r40760;
        double r40762 = r40753 * r40761;
        return r40762;
}

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

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

  4. Applied associate-*r/0.2

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

  5. Taylor expanded around inf 0.2

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

  6. Simplified0.2

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

  8. Applied sub-div0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))