Average Error: 15.7 → 0.4
Time: 22.5s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{-r}{\mathsf{fma}\left(-\frac{\cos a}{\sin b}, \cos b, \sin a\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{-r}{\mathsf{fma}\left(-\frac{\cos a}{\sin b}, \cos b, \sin a\right)}
double f(double r, double a, double b) {
        double r988461 = r;
        double r988462 = b;
        double r988463 = sin(r988462);
        double r988464 = a;
        double r988465 = r988464 + r988462;
        double r988466 = cos(r988465);
        double r988467 = r988463 / r988466;
        double r988468 = r988461 * r988467;
        return r988468;
}
↓
double f(double r, double a, double b) {
        double r988469 = r;
        double r988470 = -r988469;
        double r988471 = a;
        double r988472 = cos(r988471);
        double r988473 = b;
        double r988474 = sin(r988473);
        double r988475 = r988472 / r988474;
        double r988476 = -r988475;
        double r988477 = cos(r988473);
        double r988478 = sin(r988471);
        double r988479 = fma(r988476, r988477, r988478);
        double r988480 = r988470 / r988479;
        return r988480;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.7

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

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

  6. Applied associate-*l*0.3

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

  7. Simplified0.4

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

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

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

  10. Applied times-frac0.4

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

  11. Applied fma-neg0.4

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

  13. Applied frac-2neg0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

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