Average Error: 0.2 → 0.2
Time: 4.7s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[1 \cdot \frac{1}{\frac{\sin B}{1 - x \cdot \cos B}}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
1 \cdot \frac{1}{\frac{\sin B}{1 - x \cdot \cos B}}
double f(double B, double x) {
        double r9428 = x;
        double r9429 = 1.0;
        double r9430 = B;
        double r9431 = tan(r9430);
        double r9432 = r9429 / r9431;
        double r9433 = r9428 * r9432;
        double r9434 = -r9433;
        double r9435 = sin(r9430);
        double r9436 = r9429 / r9435;
        double r9437 = r9434 + r9436;
        return r9437;
}
↓
double f(double B, double x) {
        double r9438 = 1.0;
        double r9439 = 1.0;
        double r9440 = B;
        double r9441 = sin(r9440);
        double r9442 = x;
        double r9443 = cos(r9440);
        double r9444 = r9442 * r9443;
        double r9445 = r9439 - r9444;
        double r9446 = r9441 / r9445;
        double r9447 = r9439 / r9446;
        double r9448 = r9438 * r9447;
        return r9448;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{1}{\tan B}, \frac{1}{\sin B}\right)}\]

  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. Simplified0.3

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

  6. Applied div-inv0.3

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

  7. Applied associate-*l*0.3

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

  8. Simplified0.2

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

  10. Applied clear-num0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))