Average Error: 14.5 → 0.4
Time: 25.2s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{1}{\cos b \cdot \cos a - \sin b \cdot \sin a} \cdot \left(r \cdot \sin b\right)\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{1}{\cos b \cdot \cos a - \sin b \cdot \sin a} \cdot \left(r \cdot \sin b\right)
double f(double r, double a, double b) {
        double r782742 = r;
        double r782743 = b;
        double r782744 = sin(r782743);
        double r782745 = r782742 * r782744;
        double r782746 = a;
        double r782747 = r782746 + r782743;
        double r782748 = cos(r782747);
        double r782749 = r782745 / r782748;
        return r782749;
}
↓
double f(double r, double a, double b) {
        double r782750 = 1.0;
        double r782751 = b;
        double r782752 = cos(r782751);
        double r782753 = a;
        double r782754 = cos(r782753);
        double r782755 = r782752 * r782754;
        double r782756 = sin(r782751);
        double r782757 = sin(r782753);
        double r782758 = r782756 * r782757;
        double r782759 = r782755 - r782758;
        double r782760 = r782750 / r782759;
        double r782761 = r;
        double r782762 = r782761 * r782756;
        double r782763 = r782760 * r782762;
        return r782763;
}

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 14.5

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 expm1-log1p-u0.3

    \[\leadsto \frac{r \cdot \sin b}{\cos a \cdot \cos b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)}}\]
  6. Using strategy rm

  7. Applied div-inv0.4

    \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos a \cdot \cos b - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)}}\]

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))