Average Error: 15.0 → 0.3
Time: 14.2s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, -\sin a \cdot \sin b\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, -\sin a \cdot \sin b\right)}
double f(double r, double a, double b) {
        double r16887 = r;
        double r16888 = b;
        double r16889 = sin(r16888);
        double r16890 = a;
        double r16891 = r16890 + r16888;
        double r16892 = cos(r16891);
        double r16893 = r16889 / r16892;
        double r16894 = r16887 * r16893;
        return r16894;
}
↓
double f(double r, double a, double b) {
        double r16895 = r;
        double r16896 = b;
        double r16897 = sin(r16896);
        double r16898 = a;
        double r16899 = cos(r16898);
        double r16900 = cos(r16896);
        double r16901 = sin(r16898);
        double r16902 = r16901 * r16897;
        double r16903 = -r16902;
        double r16904 = fma(r16899, r16900, r16903);
        double r16905 = r16897 / r16904;
        double r16906 = r16895 * r16905;
        return r16906;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

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. Using strategy rm

  5. Applied associate-*r/0.3

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

  7. Applied fma-neg0.3

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

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

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

  10. Applied times-frac0.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))