Average Error: 14.9 → 0.4
Time: 34.2s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\frac{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)}{\cos a \cdot \cos b + \sin a \cdot \sin b}}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\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)}{\cos a \cdot \cos b + \sin a \cdot \sin b}}
double f(double r, double a, double b) {
        double r979615 = r;
        double r979616 = b;
        double r979617 = sin(r979616);
        double r979618 = a;
        double r979619 = r979618 + r979616;
        double r979620 = cos(r979619);
        double r979621 = r979617 / r979620;
        double r979622 = r979615 * r979621;
        return r979622;
}
↓
double f(double r, double a, double b) {
        double r979623 = r;
        double r979624 = b;
        double r979625 = sin(r979624);
        double r979626 = r979623 * r979625;
        double r979627 = a;
        double r979628 = cos(r979627);
        double r979629 = cos(r979624);
        double r979630 = r979628 * r979629;
        double r979631 = r979630 * r979630;
        double r979632 = sin(r979627);
        double r979633 = r979632 * r979625;
        double r979634 = r979633 * r979633;
        double r979635 = r979631 - r979634;
        double r979636 = r979630 + r979633;
        double r979637 = r979635 / r979636;
        double r979638 = r979626 / r979637;
        return r979638;
}

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

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

  3. Applied cos-sum0.3

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

  5. Applied associate-*r/0.3

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

  7. Applied flip--0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))