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

Average Error: 13.9 → 0.4

Time: 8.3s

Precision: 64

Math

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

↓

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

C

double code(double r, double a, double b) {
	return (r * (sin(b) / cos((a + b))));
}

↓

double code(double r, double a, double b) {
	return ((1.0 / (((cos(a) * cos(b)) / sin(b)) + -sin(a))) * r);
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 13.9

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

3. Applied cos-sum 0.3

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

5. Applied clear-num 0.4

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

6. Applied un-div-inv 0.4

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

8. Applied *-un-lft-identity 0.4

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

9. Applied *-un-lft-identity 0.4

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

10. Applied times-frac 0.4

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

11. Simplified 0.4

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

12. Simplified 0.4

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

14. Applied *-commutative 0.4

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

15. Applied *-un-lft-identity 0.4

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

16. Applied times-frac 0.4

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

17. Simplified 0.4

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

18. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))