Average Error: 0.2 → 0.3
Time: 4.8s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\frac{\sin B}{1 \cdot \left(1 - x \cdot \cos B\right)}}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1}{\frac{\sin B}{1 \cdot \left(1 - x \cdot \cos B\right)}}
double f(double B, double x) {
        double r44277 = x;
        double r44278 = 1.0;
        double r44279 = B;
        double r44280 = tan(r44279);
        double r44281 = r44278 / r44280;
        double r44282 = r44277 * r44281;
        double r44283 = -r44282;
        double r44284 = sin(r44279);
        double r44285 = r44278 / r44284;
        double r44286 = r44283 + r44285;
        return r44286;
}
↓
double f(double B, double x) {
        double r44287 = 1.0;
        double r44288 = B;
        double r44289 = sin(r44288);
        double r44290 = 1.0;
        double r44291 = x;
        double r44292 = cos(r44288);
        double r44293 = r44291 * r44292;
        double r44294 = r44287 - r44293;
        double r44295 = r44290 * r44294;
        double r44296 = r44289 / r44295;
        double r44297 = r44287 / r44296;
        return r44297;
}

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. Simplified0.2

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

  6. Applied associate-*l/0.2

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

  8. Applied clear-num0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))