Average Error: 14.8 → 0.4
Time: 24.4s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\mathsf{fma}\left(\cos a, \frac{\cos b}{\sin b}, -\sin a\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\mathsf{fma}\left(\cos a, \frac{\cos b}{\sin b}, -\sin a\right)}
double f(double r, double a, double b) {
        double r905808 = r;
        double r905809 = b;
        double r905810 = sin(r905809);
        double r905811 = a;
        double r905812 = r905811 + r905809;
        double r905813 = cos(r905812);
        double r905814 = r905810 / r905813;
        double r905815 = r905808 * r905814;
        return r905815;
}
↓
double f(double r, double a, double b) {
        double r905816 = r;
        double r905817 = a;
        double r905818 = cos(r905817);
        double r905819 = b;
        double r905820 = cos(r905819);
        double r905821 = sin(r905819);
        double r905822 = r905820 / r905821;
        double r905823 = sin(r905817);
        double r905824 = -r905823;
        double r905825 = fma(r905818, r905822, r905824);
        double r905826 = r905816 / r905825;
        return r905826;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.8

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 *-un-lft-identity0.3

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

  6. Applied associate-*l*0.3

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

  7. Simplified0.4

    \[\leadsto 1 \cdot \color{blue}{\frac{r}{\frac{\cos a \cdot \cos b}{\sin b} - \sin a}}\]
  8. Using strategy rm

  9. Applied *-un-lft-identity0.4

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

  10. Applied times-frac0.4

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

  11. Applied fma-neg0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))