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

Average Error: 14.8 → 0.4

Time: 20.8s

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 r1036763 = r;
        double r1036764 = b;
        double r1036765 = sin(r1036764);
        double r1036766 = r1036763 * r1036765;
        double r1036767 = a;
        double r1036768 = r1036767 + r1036764;
        double r1036769 = cos(r1036768);
        double r1036770 = r1036766 / r1036769;
        return r1036770;
}

↓

double f(double r, double a, double b) {
        double r1036771 = r;
        double r1036772 = b;
        double r1036773 = cos(r1036772);
        double r1036774 = a;
        double r1036775 = cos(r1036774);
        double r1036776 = r1036773 * r1036775;
        double r1036777 = sin(r1036772);
        double r1036778 = r1036776 / r1036777;
        double r1036779 = sin(r1036774);
        double r1036780 = r1036778 - r1036779;
        double r1036781 = r1036771 / r1036780;
        return r1036781;
}

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 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 2019171 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))