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

Average Error: 14.8 → 0.8

Time: 6.3s

Precision: 64

Math

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

↓

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

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 ((1.0 / ((cos(a) * cos(b)) - (sin(a) * sin(b)))) / (1.0 / (r * sin(b))));
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.8

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 clear-num 0.7

   \[\leadsto \color{blue}{\frac{1}{\frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{r \cdot \sin b}}}\]
6. Using strategy rm

7. Applied div-inv 0.8

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

8. Applied associate-/r* 0.8

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

9. Final simplification 0.8

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

Reproduce

herbie shell --seed 2020103 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))