Average Error: 0.2 → 0.2
Time: 1.8m
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1 - \cos B \cdot x}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1 - \cos B \cdot x}{\sin B}
double f(double B, double x) {
        double r768628 = x;
        double r768629 = 1.0;
        double r768630 = B;
        double r768631 = tan(r768630);
        double r768632 = r768629 / r768631;
        double r768633 = r768628 * r768632;
        double r768634 = -r768633;
        double r768635 = sin(r768630);
        double r768636 = r768629 / r768635;
        double r768637 = r768634 + r768636;
        return r768637;
}
↓
double f(double B, double x) {
        double r768638 = 1.0;
        double r768639 = B;
        double r768640 = cos(r768639);
        double r768641 = x;
        double r768642 = r768640 * r768641;
        double r768643 = r768638 - r768642;
        double r768644 = sin(r768639);
        double r768645 = r768643 / r768644;
        return r768645;
}

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. Simplified0.1

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

  3. Taylor expanded around inf 0.2

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

  5. Applied sub-div0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))