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 r2427123 = x;
        double r2427124 = 1.0;
        double r2427125 = B;
        double r2427126 = tan(r2427125);
        double r2427127 = r2427124 / r2427126;
        double r2427128 = r2427123 * r2427127;
        double r2427129 = -r2427128;
        double r2427130 = sin(r2427125);
        double r2427131 = r2427124 / r2427130;
        double r2427132 = r2427129 + r2427131;
        return r2427132;
}
↓
double f(double B, double x) {
        double r2427133 = 1.0;
        double r2427134 = B;
        double r2427135 = cos(r2427134);
        double r2427136 = x;
        double r2427137 = r2427135 * r2427136;
        double r2427138 = r2427133 - r2427137;
        double r2427139 = sin(r2427134);
        double r2427140 = r2427138 / r2427139;
        return r2427140;
}


\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))))