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

Average Error: 14.8 → 0.4

Time: 23.7s

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 r953722 = r;
        double r953723 = b;
        double r953724 = sin(r953723);
        double r953725 = r953722 * r953724;
        double r953726 = a;
        double r953727 = r953726 + r953723;
        double r953728 = cos(r953727);
        double r953729 = r953725 / r953728;
        return r953729;
}

↓

double f(double r, double a, double b) {
        double r953730 = b;
        double r953731 = sin(r953730);
        double r953732 = r;
        double r953733 = cos(r953730);
        double r953734 = a;
        double r953735 = cos(r953734);
        double r953736 = r953733 * r953735;
        double r953737 = sin(r953734);
        double r953738 = r953731 * r953737;
        double r953739 = r953736 - r953738;
        double r953740 = r953732 / r953739;
        double r953741 = r953731 * r953740;
        return r953741;
}

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 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/ 0.4

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

8. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))