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

Average Error: 14.4 → 0.4

Time: 22.9s

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 r1008839 = r;
        double r1008840 = b;
        double r1008841 = sin(r1008840);
        double r1008842 = r1008839 * r1008841;
        double r1008843 = a;
        double r1008844 = r1008843 + r1008840;
        double r1008845 = cos(r1008844);
        double r1008846 = r1008842 / r1008845;
        return r1008846;
}

↓

double f(double r, double a, double b) {
        double r1008847 = r;
        double r1008848 = b;
        double r1008849 = cos(r1008848);
        double r1008850 = a;
        double r1008851 = cos(r1008850);
        double r1008852 = r1008849 * r1008851;
        double r1008853 = sin(r1008848);
        double r1008854 = sin(r1008850);
        double r1008855 = r1008853 * r1008854;
        double r1008856 = r1008852 - r1008855;
        double r1008857 = r1008856 / r1008853;
        double r1008858 = r1008847 / r1008857;
        return r1008858;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.4

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 2019164 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))