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

Average Error: 15.7 → 0.3

Time: 23.6s

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 r1002847 = r;
        double r1002848 = b;
        double r1002849 = sin(r1002848);
        double r1002850 = r1002847 * r1002849;
        double r1002851 = a;
        double r1002852 = r1002851 + r1002848;
        double r1002853 = cos(r1002852);
        double r1002854 = r1002850 / r1002853;
        return r1002854;
}

↓

double f(double r, double a, double b) {
        double r1002855 = b;
        double r1002856 = sin(r1002855);
        double r1002857 = r;
        double r1002858 = cos(r1002855);
        double r1002859 = a;
        double r1002860 = cos(r1002859);
        double r1002861 = r1002858 * r1002860;
        double r1002862 = sin(r1002859);
        double r1002863 = r1002856 * r1002862;
        double r1002864 = r1002861 - r1002863;
        double r1002865 = r1002857 / r1002864;
        double r1002866 = r1002856 * r1002865;
        return r1002866;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.7

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 2019162 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))