Average Error: 15.7 → 0.4
Time: 22.6s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\mathsf{fma}\left(\cos a, \frac{\cos b}{\sin b}, -\sin a\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\mathsf{fma}\left(\cos a, \frac{\cos b}{\sin b}, -\sin a\right)}
double f(double r, double a, double b) {
        double r1014220 = r;
        double r1014221 = b;
        double r1014222 = sin(r1014221);
        double r1014223 = r1014220 * r1014222;
        double r1014224 = a;
        double r1014225 = r1014224 + r1014221;
        double r1014226 = cos(r1014225);
        double r1014227 = r1014223 / r1014226;
        return r1014227;
}
↓
double f(double r, double a, double b) {
        double r1014228 = r;
        double r1014229 = a;
        double r1014230 = cos(r1014229);
        double r1014231 = b;
        double r1014232 = cos(r1014231);
        double r1014233 = sin(r1014231);
        double r1014234 = r1014232 / r1014233;
        double r1014235 = sin(r1014229);
        double r1014236 = -r1014235;
        double r1014237 = fma(r1014230, r1014234, r1014236);
        double r1014238 = r1014228 / r1014237;
        return r1014238;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.7

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 *-un-lft-identity0.3

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

  10. Applied associate-*l*0.3

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

  11. Simplified0.4

    \[\leadsto 1 \cdot \color{blue}{\frac{r}{\frac{\cos a \cdot \cos b}{\sin b} - \sin a}}\]
  12. Using strategy rm

  13. Applied *-un-lft-identity0.4

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

  14. Applied times-frac0.4

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

  15. Applied fma-neg0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))