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

Average Error: 14.8 → 0.3

Time: 10.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\frac{r}{\cos a \cdot \cos b - \sin a \cdot \sin b} \cdot \sin b\]

C

double f(double r, double a, double b) {
        double r18312 = r;
        double r18313 = b;
        double r18314 = sin(r18313);
        double r18315 = r18312 * r18314;
        double r18316 = a;
        double r18317 = r18316 + r18313;
        double r18318 = cos(r18317);
        double r18319 = r18315 / r18318;
        return r18319;
}

↓

double f(double r, double a, double b) {
        double r18320 = r;
        double r18321 = a;
        double r18322 = cos(r18321);
        double r18323 = b;
        double r18324 = cos(r18323);
        double r18325 = r18322 * r18324;
        double r18326 = sin(r18321);
        double r18327 = sin(r18323);
        double r18328 = r18326 * r18327;
        double r18329 = r18325 - r18328;
        double r18330 = r18320 / r18329;
        double r18331 = r18330 * r18327;
        return r18331;
}

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 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 associate-/r/ 0.3

   \[\leadsto \color{blue}{\frac{r}{\cos a \cdot \cos b - \sin a \cdot \sin b} \cdot \sin b}\]

8. Final simplification 0.3

   \[\leadsto \frac{r}{\cos a \cdot \cos b - \sin a \cdot \sin b} \cdot \sin b\]

Reproduce

herbie shell --seed 2020046 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))