Average Error: 15.0 → 0.4
Time: 7.3s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\frac{\mathsf{fma}\left(\cos a, \cos b, -\sin a \cdot \sin b\right)}{\sin b}}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\frac{\mathsf{fma}\left(\cos a, \cos b, -\sin a \cdot \sin b\right)}{\sin b}}
double f(double r, double a, double b) {
        double r18580 = r;
        double r18581 = b;
        double r18582 = sin(r18581);
        double r18583 = a;
        double r18584 = r18583 + r18581;
        double r18585 = cos(r18584);
        double r18586 = r18582 / r18585;
        double r18587 = r18580 * r18586;
        return r18587;
}
↓
double f(double r, double a, double b) {
        double r18588 = r;
        double r18589 = a;
        double r18590 = cos(r18589);
        double r18591 = b;
        double r18592 = cos(r18591);
        double r18593 = sin(r18589);
        double r18594 = sin(r18591);
        double r18595 = r18593 * r18594;
        double r18596 = -r18595;
        double r18597 = fma(r18590, r18592, r18596);
        double r18598 = r18597 / r18594;
        double r18599 = r18588 / r18598;
        return r18599;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.0

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

  3. Applied cos-sum0.3

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

  5. Applied associate-*r/0.3

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

  7. Applied associate-/l*0.4

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

  9. Applied fma-neg0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))