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

Average Error: 15.4 → 0.4

Time: 6.4s

Precision: 64

Math

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

↓

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

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 (r / (((cos(a) * cos(b)) - (sin(a) * sin(b))) / sin(b)));
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.4

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020106 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))