Average Error: 15.1 → 0.4
Time: 29.4s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\sin b}{\frac{\cos b \cdot \cos a - \sin b \cdot \sin a}{\cos b \cdot \cos a + \sin b \cdot \sin a}} \cdot \frac{r}{\cos b \cdot \cos a + \sin b \cdot \sin a}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{\sin b}{\frac{\cos b \cdot \cos a - \sin b \cdot \sin a}{\cos b \cdot \cos a + \sin b \cdot \sin a}} \cdot \frac{r}{\cos b \cdot \cos a + \sin b \cdot \sin a}
double f(double r, double a, double b) {
        double r561674 = r;
        double r561675 = b;
        double r561676 = sin(r561675);
        double r561677 = r561674 * r561676;
        double r561678 = a;
        double r561679 = r561678 + r561675;
        double r561680 = cos(r561679);
        double r561681 = r561677 / r561680;
        return r561681;
}
↓
double f(double r, double a, double b) {
        double r561682 = b;
        double r561683 = sin(r561682);
        double r561684 = cos(r561682);
        double r561685 = a;
        double r561686 = cos(r561685);
        double r561687 = r561684 * r561686;
        double r561688 = sin(r561685);
        double r561689 = r561683 * r561688;
        double r561690 = r561687 - r561689;
        double r561691 = r561687 + r561689;
        double r561692 = r561690 / r561691;
        double r561693 = r561683 / r561692;
        double r561694 = r;
        double r561695 = r561694 / r561691;
        double r561696 = r561693 * r561695;
        return r561696;
}

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 15.1

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 flip--0.4

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

  7. Applied *-un-lft-identity0.4

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

  8. Applied difference-of-squares0.4

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

  9. Applied times-frac0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\frac{\cos a \cdot \cos b + \sin a \cdot \sin b}{1} \cdot \frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\cos a \cdot \cos b + \sin a \cdot \sin b}}}\]

  10. Applied times-frac0.4

    \[\leadsto \color{blue}{\frac{r}{\frac{\cos a \cdot \cos b + \sin a \cdot \sin b}{1}} \cdot \frac{\sin b}{\frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\cos a \cdot \cos b + \sin a \cdot \sin b}}}\]

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))