Average Error: 0.2 → 0.2
Time: 8.2s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\mathsf{fma}\left(-x, 1 \cdot \frac{\cos B}{\sin B}, \frac{1}{\sin B}\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\mathsf{fma}\left(-x, 1 \cdot \frac{\cos B}{\sin B}, \frac{1}{\sin B}\right)
double f(double B, double x) {
        double r64 = x;
        double r65 = 1.0;
        double r66 = B;
        double r67 = tan(r66);
        double r68 = r65 / r67;
        double r69 = r64 * r68;
        double r70 = -r69;
        double r71 = sin(r66);
        double r72 = r65 / r71;
        double r73 = r70 + r72;
        return r73;
}
↓
double f(double B, double x) {
        double r74 = x;
        double r75 = -r74;
        double r76 = 1.0;
        double r77 = B;
        double r78 = cos(r77);
        double r79 = sin(r77);
        double r80 = r78 / r79;
        double r81 = r76 * r80;
        double r82 = r76 / r79;
        double r83 = fma(r75, r81, r82);
        return r83;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{1}{\tan B}, \frac{1}{\sin B}\right)}\]

  3. Taylor expanded around inf 0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))