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

Average Error: 15.0 → 0.4

Time: 6.1s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r17298 = r;
        double r17299 = b;
        double r17300 = sin(r17299);
        double r17301 = r17298 * r17300;
        double r17302 = a;
        double r17303 = r17302 + r17299;
        double r17304 = cos(r17303);
        double r17305 = r17301 / r17304;
        return r17305;
}

↓

double f(double r, double a, double b) {
        double r17306 = r;
        double r17307 = a;
        double r17308 = cos(r17307);
        double r17309 = b;
        double r17310 = cos(r17309);
        double r17311 = sin(r17307);
        double r17312 = sin(r17309);
        double r17313 = r17311 * r17312;
        double r17314 = -r17313;
        double r17315 = fma(r17308, r17310, r17314);
        double r17316 = r17315 / r17312;
        double r17317 = r17306 / r17316;
        return r17317;
}

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-sum 0.3

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

5. Applied associate-/l* 0.4

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

7. Applied fma-neg 0.4

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

8. Final simplification 0.4

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

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))))