Average Error: 0.2 → 0.2
Time: 12.0s
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 r56659 = x;
        double r56660 = 1.0;
        double r56661 = B;
        double r56662 = tan(r56661);
        double r56663 = r56660 / r56662;
        double r56664 = r56659 * r56663;
        double r56665 = -r56664;
        double r56666 = sin(r56661);
        double r56667 = r56660 / r56666;
        double r56668 = r56665 + r56667;
        return r56668;
}
↓
double f(double B, double x) {
        double r56669 = 1.0;
        double r56670 = 1.0;
        double r56671 = x;
        double r56672 = B;
        double r56673 = cos(r56672);
        double r56674 = r56671 * r56673;
        double r56675 = r56670 - r56674;
        double r56676 = sin(r56672);
        double r56677 = r56675 / r56676;
        double r56678 = r56669 * r56677;
        return r56678;
}

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. Using strategy rm

  3. Applied associate-*r/0.2

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

  4. Taylor expanded around inf 0.2

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

  5. Simplified0.2

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

  7. Applied sub-div0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2020047 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))