Average Error: 0.2 → 0.2
Time: 4.5s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1 \cdot \left(1 - x \cdot \cos B\right)}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1 \cdot \left(1 - x \cdot \cos B\right)}{\sin B}
double f(double B, double x) {
        double r30323 = x;
        double r30324 = 1.0;
        double r30325 = B;
        double r30326 = tan(r30325);
        double r30327 = r30324 / r30326;
        double r30328 = r30323 * r30327;
        double r30329 = -r30328;
        double r30330 = sin(r30325);
        double r30331 = r30324 / r30330;
        double r30332 = r30329 + r30331;
        return r30332;
}
↓
double f(double B, double x) {
        double r30333 = 1.0;
        double r30334 = 1.0;
        double r30335 = x;
        double r30336 = B;
        double r30337 = cos(r30336);
        double r30338 = r30335 * r30337;
        double r30339 = r30334 - r30338;
        double r30340 = r30333 * r30339;
        double r30341 = sin(r30336);
        double r30342 = r30340 / r30341;
        return r30342;
}

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))