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

Average Error: 15.0 → 0.4

Time: 6.2s

Precision: 64

Math

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

↓

\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin a \cdot \sin b\right)}^{3}}}\]

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 * sin(b)) / ((cos(a) * cos(b)) - cbrt(pow((sin(a) * sin(b)), 3.0))));
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.0

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 add-cbrt-cube 0.4

   \[\leadsto \frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin a \cdot \color{blue}{\sqrt[3]{\left(\sin b \cdot \sin b\right) \cdot \sin b}}}\]

6. Applied add-cbrt-cube 0.4

   \[\leadsto \frac{r \cdot \sin b}{\cos a \cdot \cos b - \color{blue}{\sqrt[3]{\left(\sin a \cdot \sin a\right) \cdot \sin a}} \cdot \sqrt[3]{\left(\sin b \cdot \sin b\right) \cdot \sin b}}\]

7. Applied cbrt-unprod 0.4

   \[\leadsto \frac{r \cdot \sin b}{\cos a \cdot \cos b - \color{blue}{\sqrt[3]{\left(\left(\sin a \cdot \sin a\right) \cdot \sin a\right) \cdot \left(\left(\sin b \cdot \sin b\right) \cdot \sin b\right)}}}\]

8. Simplified 0.4

   \[\leadsto \frac{r \cdot \sin b}{\cos a \cdot \cos b - \sqrt[3]{\color{blue}{{\left(\sin a \cdot \sin b\right)}^{3}}}}\]

9. Final simplification 0.4

   \[\leadsto \frac{r \cdot \sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin a \cdot \sin b\right)}^{3}}}\]

Reproduce

herbie shell --seed 2020092 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))