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

Average Error: 14.9 → 0.4

Time: 26.1s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r24817 = r;
        double r24818 = b;
        double r24819 = sin(r24818);
        double r24820 = r24817 * r24819;
        double r24821 = a;
        double r24822 = r24821 + r24818;
        double r24823 = cos(r24822);
        double r24824 = r24820 / r24823;
        return r24824;
}

↓

double f(double r, double a, double b) {
        double r24825 = r;
        double r24826 = a;
        double r24827 = cos(r24826);
        double r24828 = b;
        double r24829 = tan(r24828);
        double r24830 = r24827 / r24829;
        double r24831 = sin(r24826);
        double r24832 = r24830 - r24831;
        double r24833 = r24825 / r24832;
        return r24833;
}

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. Simplified 0.4

   \[\leadsto \frac{r}{\color{blue}{\frac{\cos a \cdot \cos b}{\sin b} - \sin a}}\]
7. Using strategy rm

8. Applied associate-/l* 0.4

   \[\leadsto \frac{r}{\color{blue}{\frac{\cos a}{\frac{\sin b}{\cos b}}} - \sin a}\]
9. Using strategy rm

10. Applied quot-tan 0.4

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

11. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019303 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))