Average Error: 0.2 → 0.2
Time: 24.7s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1 - \cos B \cdot x}{\sin B}\]
\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 r1789252 = x;
        double r1789253 = 1.0;
        double r1789254 = B;
        double r1789255 = tan(r1789254);
        double r1789256 = r1789253 / r1789255;
        double r1789257 = r1789252 * r1789256;
        double r1789258 = -r1789257;
        double r1789259 = sin(r1789254);
        double r1789260 = r1789253 / r1789259;
        double r1789261 = r1789258 + r1789260;
        return r1789261;
}
↓
double f(double B, double x) {
        double r1789262 = 1.0;
        double r1789263 = B;
        double r1789264 = cos(r1789263);
        double r1789265 = x;
        double r1789266 = r1789264 * r1789265;
        double r1789267 = r1789262 - r1789266;
        double r1789268 = sin(r1789263);
        double r1789269 = r1789267 / r1789268;
        return r1789269;
}

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

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

  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 2019162 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))