Average Error: 15.3 → 0.4
Time: 6.8s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\mathsf{fma}\left(\cos b, \cos a, \sin a \cdot \sin b\right)} \cdot \frac{\sin b}{\frac{\cos b \cdot \cos a - \sin a \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin a \cdot \sin b\right)}}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\mathsf{fma}\left(\cos b, \cos a, \sin a \cdot \sin b\right)} \cdot \frac{\sin b}{\frac{\cos b \cdot \cos a - \sin a \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin a \cdot \sin b\right)}}
double f(double r, double a, double b) {
        double r19499 = r;
        double r19500 = b;
        double r19501 = sin(r19500);
        double r19502 = r19499 * r19501;
        double r19503 = a;
        double r19504 = r19503 + r19500;
        double r19505 = cos(r19504);
        double r19506 = r19502 / r19505;
        return r19506;
}
↓
double f(double r, double a, double b) {
        double r19507 = r;
        double r19508 = b;
        double r19509 = cos(r19508);
        double r19510 = a;
        double r19511 = cos(r19510);
        double r19512 = sin(r19510);
        double r19513 = sin(r19508);
        double r19514 = r19512 * r19513;
        double r19515 = fma(r19509, r19511, r19514);
        double r19516 = r19507 / r19515;
        double r19517 = r19509 * r19511;
        double r19518 = r19517 - r19514;
        double r19519 = r19518 / r19515;
        double r19520 = r19513 / r19519;
        double r19521 = r19516 * r19520;
        return r19521;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.3

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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. Simplified0.3

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

  10. Applied flip--0.4

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

  11. Simplified0.4

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

  12. Simplified0.4

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

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

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

  15. Applied times-frac0.4

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

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

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

  17. Applied times-frac0.4

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

  18. Applied associate-*r*0.4

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

  19. Simplified0.4

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

  20. Final simplification0.4

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))