Average Error: 0.2 → 0.2
Time: 19.5s
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 r47391 = x;
        double r47392 = 1.0;
        double r47393 = B;
        double r47394 = tan(r47393);
        double r47395 = r47392 / r47394;
        double r47396 = r47391 * r47395;
        double r47397 = -r47396;
        double r47398 = sin(r47393);
        double r47399 = r47392 / r47398;
        double r47400 = r47397 + r47399;
        return r47400;
}
↓
double f(double B, double x) {
        double r47401 = 1.0;
        double r47402 = 1.0;
        double r47403 = x;
        double r47404 = B;
        double r47405 = cos(r47404);
        double r47406 = r47403 * r47405;
        double r47407 = r47402 - r47406;
        double r47408 = sin(r47404);
        double r47409 = r47407 / r47408;
        double r47410 = r47401 * r47409;
        return r47410;
}

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. Using strategy rm

  6. Applied sub-div0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))