Average Error: 15.5 → 0.3
Time: 17.5s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}
double f(double r, double a, double b) {
        double r24432 = r;
        double r24433 = b;
        double r24434 = sin(r24433);
        double r24435 = r24432 * r24434;
        double r24436 = a;
        double r24437 = r24436 + r24433;
        double r24438 = cos(r24437);
        double r24439 = r24435 / r24438;
        return r24439;
}
↓
double f(double r, double a, double b) {
        double r24440 = r;
        double r24441 = b;
        double r24442 = sin(r24441);
        double r24443 = r24440 * r24442;
        double r24444 = cos(r24441);
        double r24445 = a;
        double r24446 = cos(r24445);
        double r24447 = -r24442;
        double r24448 = sin(r24445);
        double r24449 = r24447 * r24448;
        double r24450 = fma(r24444, r24446, r24449);
        double r24451 = r24443 / r24450;
        return r24451;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.5

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  2. Using strategy rm

  3. Applied cos-sum0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]

  4. Simplified0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} - \sin a \cdot \sin b}\]
  5. Using strategy rm

  6. Applied fma-neg0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, -\sin a \cdot \sin b\right)}}\]

  7. Simplified0.3

    \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{-\sin b \cdot \sin a}\right)}\]

  8. Final simplification0.3

    \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))