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

Average Error: 15.1 → 0.4

Time: 27.1s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r660989 = r;
        double r660990 = b;
        double r660991 = sin(r660990);
        double r660992 = r660989 * r660991;
        double r660993 = a;
        double r660994 = r660993 + r660990;
        double r660995 = cos(r660994);
        double r660996 = r660992 / r660995;
        return r660996;
}

↓

double f(double r, double a, double b) {
        double r660997 = r;
        double r660998 = b;
        double r660999 = cos(r660998);
        double r661000 = a;
        double r661001 = cos(r661000);
        double r661002 = r660999 * r661001;
        double r661003 = sin(r660998);
        double r661004 = sin(r661000);
        double r661005 = r661003 * r661004;
        double r661006 = r661002 - r661005;
        double r661007 = r661006 / r661003;
        double r661008 = r660997 / r661007;
        return r661008;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.1

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019151 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))