Average Error: 14.8 → 0.4
Time: 19.6s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\frac{\mathsf{fma}\left(\cos a, \cos b, \sin b \cdot \sin a\right) \cdot \left(\cos a \cdot \cos b - \sin b \cdot \sin a\right)}{\mathsf{fma}\left(\cos a, \cos b, \sin b \cdot \sin a\right)}}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\frac{\mathsf{fma}\left(\cos a, \cos b, \sin b \cdot \sin a\right) \cdot \left(\cos a \cdot \cos b - \sin b \cdot \sin a\right)}{\mathsf{fma}\left(\cos a, \cos b, \sin b \cdot \sin a\right)}}
double f(double r, double a, double b) {
        double r27250 = r;
        double r27251 = b;
        double r27252 = sin(r27251);
        double r27253 = a;
        double r27254 = r27253 + r27251;
        double r27255 = cos(r27254);
        double r27256 = r27252 / r27255;
        double r27257 = r27250 * r27256;
        return r27257;
}
↓
double f(double r, double a, double b) {
        double r27258 = r;
        double r27259 = b;
        double r27260 = sin(r27259);
        double r27261 = r27258 * r27260;
        double r27262 = a;
        double r27263 = cos(r27262);
        double r27264 = cos(r27259);
        double r27265 = sin(r27262);
        double r27266 = r27260 * r27265;
        double r27267 = fma(r27263, r27264, r27266);
        double r27268 = r27263 * r27264;
        double r27269 = r27268 - r27266;
        double r27270 = r27267 * r27269;
        double r27271 = r27270 / r27267;
        double r27272 = r27261 / r27271;
        return r27272;
}

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. Simplified0.3

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

  6. Applied associate-*r/0.3

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

  8. Applied flip--0.4

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

  9. Simplified0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))