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

Average Error: 15.1 → 0.3

Time: 6.3s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r17306 = r;
        double r17307 = b;
        double r17308 = sin(r17307);
        double r17309 = r17306 * r17308;
        double r17310 = a;
        double r17311 = r17310 + r17307;
        double r17312 = cos(r17311);
        double r17313 = r17309 / r17312;
        return r17313;
}

↓

double f(double r, double a, double b) {
        double r17314 = r;
        double r17315 = b;
        double r17316 = sin(r17315);
        double r17317 = r17314 * r17316;
        double r17318 = cos(r17315);
        double r17319 = a;
        double r17320 = cos(r17319);
        double r17321 = r17318 * r17320;
        double r17322 = sin(r17319);
        double r17323 = r17322 * r17316;
        double r17324 = r17321 - r17323;
        double r17325 = r17317 / r17324;
        return r17325;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.1

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

5. Applied associate-/l* 0.4

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

7. Applied *-commutative 0.4

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

9. Applied *-un-lft-identity 0.4

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

10. Applied *-un-lft-identity 0.4

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

11. Applied times-frac 0.4

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

12. Applied *-un-lft-identity 0.4

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

13. Applied times-frac 0.4

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

14. Simplified 0.4

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

15. Simplified 0.3

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

16. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020001 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))