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

Average Error: 14.2 → 0.4

Time: 20.0s

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 r25431 = r;
        double r25432 = b;
        double r25433 = sin(r25432);
        double r25434 = r25431 * r25433;
        double r25435 = a;
        double r25436 = r25435 + r25432;
        double r25437 = cos(r25436);
        double r25438 = r25434 / r25437;
        return r25438;
}

↓

double f(double r, double a, double b) {
        double r25439 = r;
        double r25440 = b;
        double r25441 = sin(r25440);
        double r25442 = r25439 * r25441;
        double r25443 = 1.0;
        double r25444 = a;
        double r25445 = cos(r25444);
        double r25446 = cos(r25440);
        double r25447 = r25445 * r25446;
        double r25448 = sin(r25444);
        double r25449 = r25448 * r25441;
        double r25450 = r25447 - r25449;
        double r25451 = r25443 / r25450;
        double r25452 = r25442 * r25451;
        return r25452;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.2

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 2019208 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))