Average Error: 14.9 → 0.3
Time: 23.7s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, \left(-\sin b\right) \cdot \sin a\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, \left(-\sin b\right) \cdot \sin a\right)}
double f(double r, double a, double b) {
        double r930992 = r;
        double r930993 = b;
        double r930994 = sin(r930993);
        double r930995 = r930992 * r930994;
        double r930996 = a;
        double r930997 = r930996 + r930993;
        double r930998 = cos(r930997);
        double r930999 = r930995 / r930998;
        return r930999;
}
↓
double f(double r, double a, double b) {
        double r931000 = r;
        double r931001 = b;
        double r931002 = sin(r931001);
        double r931003 = a;
        double r931004 = cos(r931003);
        double r931005 = cos(r931001);
        double r931006 = -r931002;
        double r931007 = sin(r931003);
        double r931008 = r931006 * r931007;
        double r931009 = fma(r931004, r931005, r931008);
        double r931010 = r931002 / r931009;
        double r931011 = r931000 * r931010;
        return r931011;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.9

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

  5. Applied *-un-lft-identity0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{1 \cdot \left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)}}\]

  6. Applied times-frac0.3

    \[\leadsto \color{blue}{\frac{r}{1} \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]

  7. Simplified0.3

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

  9. Applied fma-neg0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))