Average Error: 0.2 → 0.2
Time: 18.5s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[1 \cdot \left(\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
1 \cdot \left(\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\right)
double f(double B, double x) {
        double r54817 = x;
        double r54818 = 1.0;
        double r54819 = B;
        double r54820 = tan(r54819);
        double r54821 = r54818 / r54820;
        double r54822 = r54817 * r54821;
        double r54823 = -r54822;
        double r54824 = sin(r54819);
        double r54825 = r54818 / r54824;
        double r54826 = r54823 + r54825;
        return r54826;
}
↓
double f(double B, double x) {
        double r54827 = 1.0;
        double r54828 = 1.0;
        double r54829 = B;
        double r54830 = sin(r54829);
        double r54831 = r54828 / r54830;
        double r54832 = x;
        double r54833 = cos(r54829);
        double r54834 = r54832 * r54833;
        double r54835 = r54834 / r54830;
        double r54836 = r54831 - r54835;
        double r54837 = r54827 * r54836;
        return r54837;
}

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

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))