Average Error: 0.2 → 0.2
Time: 6.0s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\mathsf{fma}\left(-\frac{x \cdot 1}{\sin B}, \cos B, \frac{1}{\sin B}\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\mathsf{fma}\left(-\frac{x \cdot 1}{\sin B}, \cos B, \frac{1}{\sin B}\right)
double f(double B, double x) {
        double r13241 = x;
        double r13242 = 1.0;
        double r13243 = B;
        double r13244 = tan(r13243);
        double r13245 = r13242 / r13244;
        double r13246 = r13241 * r13245;
        double r13247 = -r13246;
        double r13248 = sin(r13243);
        double r13249 = r13242 / r13248;
        double r13250 = r13247 + r13249;
        return r13250;
}
↓
double f(double B, double x) {
        double r13251 = x;
        double r13252 = 1.0;
        double r13253 = r13251 * r13252;
        double r13254 = B;
        double r13255 = sin(r13254);
        double r13256 = r13253 / r13255;
        double r13257 = -r13256;
        double r13258 = cos(r13254);
        double r13259 = r13252 / r13255;
        double r13260 = fma(r13257, r13258, r13259);
        return r13260;
}

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. Using strategy rm

  3. Applied associate-*r/0.1

    \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{1}{\sin B}\]
  4. Using strategy rm

  5. Applied tan-quot0.2

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

  6. Applied associate-/r/0.2

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

  7. Applied distribute-lft-neg-in0.2

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

  8. Applied fma-def0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))