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

Average Error: 14.9 → 0.3

Time: 5.5s

Precision: 64

Math

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

↓

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

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

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.9

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

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

6. Applied fma-neg 0.3

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, -\sin a \cdot \sin b\right)}}\]
7. Using strategy rm

8. Applied clear-num 0.4

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

9. Applied un-div-inv 0.4

   \[\leadsto \color{blue}{\frac{r}{\frac{\mathsf{fma}\left(\cos b, \cos a, -\sin a \cdot \sin b\right)}{\sin b}}}\]
10. Using strategy rm

11. Applied associate-/r/ 0.3

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

12. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))