Average Error: 14.6 → 0.4
Time: 5.9s
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 r16413 = r;
        double r16414 = b;
        double r16415 = sin(r16414);
        double r16416 = r16413 * r16415;
        double r16417 = a;
        double r16418 = r16417 + r16414;
        double r16419 = cos(r16418);
        double r16420 = r16416 / r16419;
        return r16420;
}
↓
double f(double r, double a, double b) {
        double r16421 = r;
        double r16422 = a;
        double r16423 = cos(r16422);
        double r16424 = b;
        double r16425 = cos(r16424);
        double r16426 = r16423 * r16425;
        double r16427 = sin(r16422);
        double r16428 = sin(r16424);
        double r16429 = r16427 * r16428;
        double r16430 = r16426 - r16429;
        double r16431 = r16430 / r16428;
        double r16432 = r16421 / r16431;
        return r16432;
}

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 14.6

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 2020062 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))