Average Error: 15.0 → 0.4
Time: 7.2s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\sin b}}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\sin b}}
double f(double r, double a, double b) {
        double r18833 = r;
        double r18834 = b;
        double r18835 = sin(r18834);
        double r18836 = r18833 * r18835;
        double r18837 = a;
        double r18838 = r18837 + r18834;
        double r18839 = cos(r18838);
        double r18840 = r18836 / r18839;
        return r18840;
}
↓
double f(double r, double a, double b) {
        double r18841 = r;
        double r18842 = a;
        double r18843 = cos(r18842);
        double r18844 = b;
        double r18845 = cos(r18844);
        double r18846 = r18843 * r18845;
        double r18847 = sin(r18842);
        double r18848 = sin(r18844);
        double r18849 = r18847 * r18848;
        double r18850 = r18846 - r18849;
        double r18851 = r18850 / r18848;
        double r18852 = r18841 / r18851;
        return r18852;
}

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 15.0

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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. Final simplification0.4

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

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))