Average Error: 0.2 → 0.2
Time: 20.8s
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 r48373 = x;
        double r48374 = 1.0;
        double r48375 = B;
        double r48376 = tan(r48375);
        double r48377 = r48374 / r48376;
        double r48378 = r48373 * r48377;
        double r48379 = -r48378;
        double r48380 = sin(r48375);
        double r48381 = r48374 / r48380;
        double r48382 = r48379 + r48381;
        return r48382;
}
↓
double f(double B, double x) {
        double r48383 = x;
        double r48384 = 1.0;
        double r48385 = r48383 * r48384;
        double r48386 = B;
        double r48387 = sin(r48386);
        double r48388 = r48385 / r48387;
        double r48389 = -r48388;
        double r48390 = cos(r48386);
        double r48391 = r48384 / r48387;
        double r48392 = fma(r48389, r48390, r48391);
        return r48392;
}

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 2019303 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))