Average Error: 0.2 → 0.2
Time: 4.8s
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 r28433 = x;
        double r28434 = 1.0;
        double r28435 = B;
        double r28436 = tan(r28435);
        double r28437 = r28434 / r28436;
        double r28438 = r28433 * r28437;
        double r28439 = -r28438;
        double r28440 = sin(r28435);
        double r28441 = r28434 / r28440;
        double r28442 = r28439 + r28441;
        return r28442;
}
↓
double f(double B, double x) {
        double r28443 = 1.0;
        double r28444 = 1.0;
        double r28445 = x;
        double r28446 = B;
        double r28447 = cos(r28446);
        double r28448 = r28445 * r28447;
        double r28449 = r28444 - r28448;
        double r28450 = sin(r28446);
        double r28451 = r28449 / r28450;
        double r28452 = r28443 * r28451;
        return r28452;
}

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2020027 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))