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

Average Error: 15.5 → 0.4

Time: 20.7s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r1077302 = r;
        double r1077303 = b;
        double r1077304 = sin(r1077303);
        double r1077305 = r1077302 * r1077304;
        double r1077306 = a;
        double r1077307 = r1077306 + r1077303;
        double r1077308 = cos(r1077307);
        double r1077309 = r1077305 / r1077308;
        return r1077309;
}

↓

double f(double r, double a, double b) {
        double r1077310 = r;
        double r1077311 = b;
        double r1077312 = cos(r1077311);
        double r1077313 = a;
        double r1077314 = cos(r1077313);
        double r1077315 = r1077312 * r1077314;
        double r1077316 = sin(r1077311);
        double r1077317 = r1077315 / r1077316;
        double r1077318 = sin(r1077313);
        double r1077319 = r1077317 - r1077318;
        double r1077320 = r1077310 / r1077319;
        return r1077320;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.5

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

7. Applied div-sub 0.4

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

8. Simplified 0.4

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

9. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019165 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))