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

Average Error: 14.9 → 0.4

Time: 6.3s

Precision: 64

Math

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

↓

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

C

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

↓

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

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.9

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 *-un-lft-identity 0.3

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

6. Applied times-frac 0.3

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

7. Simplified 0.3

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

8. Simplified 0.3

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

10. Applied div-inv 0.4

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

11. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020126 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))