Average Error: 15.0 → 0.3
Time: 48.1s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot \sin b\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot \sin b
double f(double r, double a, double b) {
        double r749702 = r;
        double r749703 = b;
        double r749704 = sin(r749703);
        double r749705 = a;
        double r749706 = r749705 + r749703;
        double r749707 = cos(r749706);
        double r749708 = r749704 / r749707;
        double r749709 = r749702 * r749708;
        return r749709;
}
↓
double f(double r, double a, double b) {
        double r749710 = r;
        double r749711 = a;
        double r749712 = cos(r749711);
        double r749713 = b;
        double r749714 = cos(r749713);
        double r749715 = r749712 * r749714;
        double r749716 = sin(r749713);
        double r749717 = sin(r749711);
        double r749718 = r749716 * r749717;
        double r749719 = r749715 - r749718;
        double r749720 = r749710 / r749719;
        double r749721 = r749720 * r749716;
        return r749721;
}

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 15.0

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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. Taylor expanded around inf 0.3

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

  6. Applied *-un-lft-identity0.3

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

  7. Applied times-frac0.3

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))