Average Error: 14.9 → 0.4
Time: 24.2s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\sin b \cdot \frac{r}{\cos a \cdot \cos b - \sin b \cdot \sin a}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\sin b \cdot \frac{r}{\cos a \cdot \cos b - \sin b \cdot \sin a}
double f(double r, double a, double b) {
        double r494292 = r;
        double r494293 = b;
        double r494294 = sin(r494293);
        double r494295 = r494292 * r494294;
        double r494296 = a;
        double r494297 = r494296 + r494293;
        double r494298 = cos(r494297);
        double r494299 = r494295 / r494298;
        return r494299;
}
↓
double f(double r, double a, double b) {
        double r494300 = b;
        double r494301 = sin(r494300);
        double r494302 = r;
        double r494303 = a;
        double r494304 = cos(r494303);
        double r494305 = cos(r494300);
        double r494306 = r494304 * r494305;
        double r494307 = sin(r494303);
        double r494308 = r494301 * r494307;
        double r494309 = r494306 - r494308;
        double r494310 = r494302 / r494309;
        double r494311 = r494301 * r494310;
        return r494311;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.9

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  2. Using strategy rm

  3. Applied cos-sum0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]
  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-inv0.4

    \[\leadsto \frac{r}{\color{blue}{\left(\cos a \cdot \cos b - \sin a \cdot \sin b\right) \cdot \frac{1}{\sin b}}}\]
  8. Using strategy rm

  9. Applied associate-*r/0.4

    \[\leadsto \frac{r}{\color{blue}{\frac{\left(\cos a \cdot \cos b - \sin a \cdot \sin b\right) \cdot 1}{\sin b}}}\]

  10. Applied associate-/r/0.4

    \[\leadsto \color{blue}{\frac{r}{\left(\cos a \cdot \cos b - \sin a \cdot \sin b\right) \cdot 1} \cdot \sin b}\]

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))