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

Average Error: 14.8 → 0.4

Time: 21.8s

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 r1076831 = r;
        double r1076832 = b;
        double r1076833 = sin(r1076832);
        double r1076834 = r1076831 * r1076833;
        double r1076835 = a;
        double r1076836 = r1076835 + r1076832;
        double r1076837 = cos(r1076836);
        double r1076838 = r1076834 / r1076837;
        return r1076838;
}

↓

double f(double r, double a, double b) {
        double r1076839 = r;
        double r1076840 = b;
        double r1076841 = cos(r1076840);
        double r1076842 = a;
        double r1076843 = cos(r1076842);
        double r1076844 = r1076841 * r1076843;
        double r1076845 = sin(r1076840);
        double r1076846 = sin(r1076842);
        double r1076847 = r1076845 * r1076846;
        double r1076848 = r1076844 - r1076847;
        double r1076849 = r1076848 / r1076845;
        double r1076850 = r1076839 / r1076849;
        return r1076850;
}

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. Final simplification 0.4

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

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))))