Average Error: 0.2 → 0.2
Time: 17.9s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1 - x \cdot \left(1 \cdot \cos B\right)}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1 - x \cdot \left(1 \cdot \cos B\right)}{\sin B}
double f(double B, double x) {
        double r22131 = x;
        double r22132 = 1.0;
        double r22133 = B;
        double r22134 = tan(r22133);
        double r22135 = r22132 / r22134;
        double r22136 = r22131 * r22135;
        double r22137 = -r22136;
        double r22138 = sin(r22133);
        double r22139 = r22132 / r22138;
        double r22140 = r22137 + r22139;
        return r22140;
}
↓
double f(double B, double x) {
        double r22141 = 1.0;
        double r22142 = x;
        double r22143 = B;
        double r22144 = cos(r22143);
        double r22145 = r22141 * r22144;
        double r22146 = r22142 * r22145;
        double r22147 = r22141 - r22146;
        double r22148 = sin(r22143);
        double r22149 = r22147 / r22148;
        return r22149;
}

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. Taylor expanded around inf 0.2

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

  5. Applied associate-*r/0.2

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

  6. Applied associate-*r/0.2

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

  7. Applied sub-div0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))