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

Average Error: 14.8 → 0.3

Time: 24.6s

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 r990754 = r;
        double r990755 = b;
        double r990756 = sin(r990755);
        double r990757 = r990754 * r990756;
        double r990758 = a;
        double r990759 = r990758 + r990755;
        double r990760 = cos(r990759);
        double r990761 = r990757 / r990760;
        return r990761;
}

↓

double f(double r, double a, double b) {
        double r990762 = b;
        double r990763 = sin(r990762);
        double r990764 = r;
        double r990765 = cos(r990762);
        double r990766 = a;
        double r990767 = cos(r990766);
        double r990768 = r990765 * r990767;
        double r990769 = sin(r990766);
        double r990770 = r990763 * r990769;
        double r990771 = r990768 - r990770;
        double r990772 = r990764 / r990771;
        double r990773 = r990763 * r990772;
        return r990773;
}

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/ 0.3

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

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