Average Error: 15.0 → 0.3
Time: 12.5s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\mathsf{fma}\left(\cos b, \cos a, -\sin b \cdot \sin a\right)} \cdot \sin b\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\mathsf{fma}\left(\cos b, \cos a, -\sin b \cdot \sin a\right)} \cdot \sin b
double f(double r, double a, double b) {
        double r17297 = r;
        double r17298 = b;
        double r17299 = sin(r17298);
        double r17300 = r17297 * r17299;
        double r17301 = a;
        double r17302 = r17301 + r17298;
        double r17303 = cos(r17302);
        double r17304 = r17300 / r17303;
        return r17304;
}
↓
double f(double r, double a, double b) {
        double r17305 = r;
        double r17306 = b;
        double r17307 = cos(r17306);
        double r17308 = a;
        double r17309 = cos(r17308);
        double r17310 = sin(r17306);
        double r17311 = sin(r17308);
        double r17312 = r17310 * r17311;
        double r17313 = -r17312;
        double r17314 = fma(r17307, r17309, r17313);
        double r17315 = r17305 / r17314;
        double r17316 = r17315 * r17310;
        return r17316;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.0

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  2. Using strategy rm

  3. Applied cos-sum0.3

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

  4. Taylor expanded around inf 0.3

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

  5. Simplified0.3

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

  7. Applied associate-/l*0.4

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

  9. Applied associate-/r/0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))