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

Average Error: 14.8 → 0.3

Time: 18.4s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r25258 = r;
        double r25259 = b;
        double r25260 = sin(r25259);
        double r25261 = r25258 * r25260;
        double r25262 = a;
        double r25263 = r25262 + r25259;
        double r25264 = cos(r25263);
        double r25265 = r25261 / r25264;
        return r25265;
}

↓

double f(double r, double a, double b) {
        double r25266 = r;
        double r25267 = b;
        double r25268 = sin(r25267);
        double r25269 = r25266 * r25268;
        double r25270 = a;
        double r25271 = cos(r25270);
        double r25272 = cos(r25267);
        double r25273 = -r25268;
        double r25274 = sin(r25270);
        double r25275 = r25273 * r25274;
        double r25276 = fma(r25271, r25272, r25275);
        double r25277 = r25269 / r25276;
        return r25277;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.8

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 fma-neg 0.3

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

6. Simplified 0.3

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

7. Final simplification 0.3

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

Reproduce

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