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

Average Error: 15.5 → 0.4

Time: 8.3s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r24322 = r;
        double r24323 = b;
        double r24324 = sin(r24323);
        double r24325 = r24322 * r24324;
        double r24326 = a;
        double r24327 = r24326 + r24323;
        double r24328 = cos(r24327);
        double r24329 = r24325 / r24328;
        return r24329;
}

↓

double f(double r, double a, double b) {
        double r24330 = r;
        double r24331 = b;
        double r24332 = sin(r24331);
        double r24333 = r24330 * r24332;
        double r24334 = 1.0;
        double r24335 = a;
        double r24336 = cos(r24335);
        double r24337 = cos(r24331);
        double r24338 = r24336 * r24337;
        double r24339 = sin(r24335);
        double r24340 = r24339 * r24332;
        double r24341 = r24338 - r24340;
        double r24342 = r24334 / r24341;
        double r24343 = r24333 * r24342;
        return r24343;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.5

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 div-inv 0.4

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

6. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019322 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))