Average Error: 14.7 → 0.4
Time: 24.5s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\mathsf{fma}\left(\cos b \cdot \cos a, \frac{1}{\sin b}, -\sin a\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\mathsf{fma}\left(\cos b \cdot \cos a, \frac{1}{\sin b}, -\sin a\right)}
double f(double r, double a, double b) {
        double r25316 = r;
        double r25317 = b;
        double r25318 = sin(r25317);
        double r25319 = a;
        double r25320 = r25319 + r25317;
        double r25321 = cos(r25320);
        double r25322 = r25318 / r25321;
        double r25323 = r25316 * r25322;
        return r25323;
}
↓
double f(double r, double a, double b) {
        double r25324 = r;
        double r25325 = b;
        double r25326 = cos(r25325);
        double r25327 = a;
        double r25328 = cos(r25327);
        double r25329 = r25326 * r25328;
        double r25330 = 1.0;
        double r25331 = sin(r25325);
        double r25332 = r25330 / r25331;
        double r25333 = sin(r25327);
        double r25334 = -r25333;
        double r25335 = fma(r25329, r25332, r25334);
        double r25336 = r25324 / r25335;
        return r25336;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.7

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 associate-/l*0.4

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

  8. Simplified0.4

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

  10. Applied div-inv0.4

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

  11. Applied fma-neg0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))