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

Average Error: 14.9 → 0.3

Time: 25.6s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r1129484 = r;
        double r1129485 = b;
        double r1129486 = sin(r1129485);
        double r1129487 = r1129484 * r1129486;
        double r1129488 = a;
        double r1129489 = r1129488 + r1129485;
        double r1129490 = cos(r1129489);
        double r1129491 = r1129487 / r1129490;
        return r1129491;
}

↓

double f(double r, double a, double b) {
        double r1129492 = b;
        double r1129493 = sin(r1129492);
        double r1129494 = r;
        double r1129495 = cos(r1129492);
        double r1129496 = a;
        double r1129497 = cos(r1129496);
        double r1129498 = r1129495 * r1129497;
        double r1129499 = sin(r1129496);
        double r1129500 = r1129493 * r1129499;
        double r1129501 = r1129498 - r1129500;
        double r1129502 = r1129494 / r1129501;
        double r1129503 = r1129493 * r1129502;
        return r1129503;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.9

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

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

8. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019169 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))