r*sin(b)/cos(a+b), A

Average Error: 14.5 → 0.3

Time: 18.8s

Precision: 64

Math

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

↓

\[\sin b \cdot \frac{r}{\cos b \cdot \cos a - \sin b \cdot \sin a}\]

C

double f(double r, double a, double b) {
        double r24919 = r;
        double r24920 = b;
        double r24921 = sin(r24920);
        double r24922 = r24919 * r24921;
        double r24923 = a;
        double r24924 = r24923 + r24920;
        double r24925 = cos(r24924);
        double r24926 = r24922 / r24925;
        return r24926;
}

↓

double f(double r, double a, double b) {
        double r24927 = b;
        double r24928 = sin(r24927);
        double r24929 = r;
        double r24930 = cos(r24927);
        double r24931 = a;
        double r24932 = cos(r24931);
        double r24933 = r24930 * r24932;
        double r24934 = sin(r24931);
        double r24935 = r24928 * r24934;
        double r24936 = r24933 - r24935;
        double r24937 = r24929 / r24936;
        double r24938 = r24928 * r24937;
        return r24938;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.5

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

3. Applied cos-sum 0.3

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

4. Simplified 0.3

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

6. Applied associate-/l* 0.4

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

7. Simplified 0.4

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

9. Applied div-inv 0.4

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

10. Applied *-un-lft-identity 0.4

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

11. Applied times-frac 0.4

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

12. Simplified 0.4

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

14. Applied associate-*l/ 0.4

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

15. Applied frac-sub 0.4

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

16. Applied associate-/r/ 0.3

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

17. Simplified 0.3

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

18. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019194 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))