Average Error: 15.2 → 0.4
Time: 6.6s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\frac{\cos b \cdot \cos a - \sin a \cdot \sin b}{\sin b}}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\frac{\cos b \cdot \cos a - \sin a \cdot \sin b}{\sin b}}
double f(double r, double a, double b) {
        double r17446 = r;
        double r17447 = b;
        double r17448 = sin(r17447);
        double r17449 = a;
        double r17450 = r17449 + r17447;
        double r17451 = cos(r17450);
        double r17452 = r17448 / r17451;
        double r17453 = r17446 * r17452;
        return r17453;
}
↓
double f(double r, double a, double b) {
        double r17454 = r;
        double r17455 = b;
        double r17456 = cos(r17455);
        double r17457 = a;
        double r17458 = cos(r17457);
        double r17459 = r17456 * r17458;
        double r17460 = sin(r17457);
        double r17461 = sin(r17455);
        double r17462 = r17460 * r17461;
        double r17463 = r17459 - r17462;
        double r17464 = r17463 / r17461;
        double r17465 = r17454 / r17464;
        return r17465;
}

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 15.2

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 fma-neg0.3

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

  7. Applied associate-*r/0.3

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

  9. Applied associate-/l*0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))