Average Error: 0.2 → 0.2
Time: 2.2m
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1 - \cos B \cdot x}{\sin B}\]
double f(double B, double x) {
        double r3506410 = x;
        double r3506411 = 1.0;
        double r3506412 = B;
        double r3506413 = tan(r3506412);
        double r3506414 = r3506411 / r3506413;
        double r3506415 = r3506410 * r3506414;
        double r3506416 = -r3506415;
        double r3506417 = sin(r3506412);
        double r3506418 = r3506411 / r3506417;
        double r3506419 = r3506416 + r3506418;
        return r3506419;
}
↓
double f(double B, double x) {
        double r3506420 = 1.0;
        double r3506421 = B;
        double r3506422 = cos(r3506421);
        double r3506423 = x;
        double r3506424 = r3506422 * r3506423;
        double r3506425 = r3506420 - r3506424;
        double r3506426 = sin(r3506421);
        double r3506427 = r3506425 / r3506426;
        return r3506427;
}


\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1 - \cos B \cdot x}{\sin B}

Error

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  3. Taylor expanded around inf 0.2

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

  5. Applied sub-div0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))