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

Average Error: 14.9 → 0.4

Time: 6.4s

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 r17235 = r;
        double r17236 = b;
        double r17237 = sin(r17236);
        double r17238 = r17235 * r17237;
        double r17239 = a;
        double r17240 = r17239 + r17236;
        double r17241 = cos(r17240);
        double r17242 = r17238 / r17241;
        return r17242;
}

↓

double f(double r, double a, double b) {
        double r17243 = r;
        double r17244 = b;
        double r17245 = sin(r17244);
        double r17246 = r17243 * r17245;
        double r17247 = 1.0;
        double r17248 = a;
        double r17249 = cos(r17248);
        double r17250 = cos(r17244);
        double r17251 = r17249 * r17250;
        double r17252 = sin(r17248);
        double r17253 = r17252 * r17245;
        double r17254 = r17251 - r17253;
        double r17255 = r17247 / r17254;
        double r17256 = r17246 * r17255;
        return r17256;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.9

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 2019353 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))