Average Error: 0.2 → 0.2
Time: 17.3s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\mathsf{fma}\left(\frac{x}{\sin B} \cdot \cos B, -1, \frac{1}{\sin B}\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\mathsf{fma}\left(\frac{x}{\sin B} \cdot \cos B, -1, \frac{1}{\sin B}\right)
double f(double B, double x) {
        double r23112 = x;
        double r23113 = 1.0;
        double r23114 = B;
        double r23115 = tan(r23114);
        double r23116 = r23113 / r23115;
        double r23117 = r23112 * r23116;
        double r23118 = -r23117;
        double r23119 = sin(r23114);
        double r23120 = r23113 / r23119;
        double r23121 = r23118 + r23120;
        return r23121;
}
↓
double f(double B, double x) {
        double r23122 = x;
        double r23123 = B;
        double r23124 = sin(r23123);
        double r23125 = r23122 / r23124;
        double r23126 = cos(r23123);
        double r23127 = r23125 * r23126;
        double r23128 = 1.0;
        double r23129 = -r23128;
        double r23130 = r23128 / r23124;
        double r23131 = fma(r23127, r23129, r23130);
        return r23131;
}

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. Simplified0.1

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

  4. Applied tan-quot0.2

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

  5. Applied associate-/r/0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))