r*sin(b)/cos(a+b), B

Average Error: 14.9 → 0.3

Time: 6.9s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r17285 = r;
        double r17286 = b;
        double r17287 = sin(r17286);
        double r17288 = a;
        double r17289 = r17288 + r17286;
        double r17290 = cos(r17289);
        double r17291 = r17287 / r17290;
        double r17292 = r17285 * r17291;
        return r17292;
}

↓

double f(double r, double a, double b) {
        double r17293 = r;
        double r17294 = b;
        double r17295 = sin(r17294);
        double r17296 = r17293 * r17295;
        double r17297 = a;
        double r17298 = cos(r17297);
        double r17299 = cos(r17294);
        double r17300 = sin(r17297);
        double r17301 = r17300 * r17295;
        double r17302 = -r17301;
        double r17303 = fma(r17298, r17299, r17302);
        double r17304 = r17296 / r17303;
        return r17304;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.9

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

3. Applied cos-sum 0.3

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

5. Applied fma-neg 0.3

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

7. Applied associate-*r/ 0.3

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

8. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))