Average Error: 0.2 → 0.2
Time: 7.9s
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 r204 = x;
        double r205 = 1.0;
        double r206 = B;
        double r207 = tan(r206);
        double r208 = r205 / r207;
        double r209 = r204 * r208;
        double r210 = -r209;
        double r211 = sin(r206);
        double r212 = r205 / r211;
        double r213 = r210 + r212;
        return r213;
}
↓
double f(double B, double x) {
        double r214 = x;
        double r215 = -r214;
        double r216 = 1.0;
        double r217 = B;
        double r218 = cos(r217);
        double r219 = sin(r217);
        double r220 = r218 / r219;
        double r221 = r216 * r220;
        double r222 = r216 / r219;
        double r223 = fma(r215, r221, r222);
        return r223;
}

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))))