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

Average Error: 14.2 → 0.4

Time: 18.9s

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 r24337 = r;
        double r24338 = b;
        double r24339 = sin(r24338);
        double r24340 = r24337 * r24339;
        double r24341 = a;
        double r24342 = r24341 + r24338;
        double r24343 = cos(r24342);
        double r24344 = r24340 / r24343;
        return r24344;
}

↓

double f(double r, double a, double b) {
        double r24345 = r;
        double r24346 = b;
        double r24347 = sin(r24346);
        double r24348 = r24345 * r24347;
        double r24349 = 1.0;
        double r24350 = a;
        double r24351 = cos(r24350);
        double r24352 = cos(r24346);
        double r24353 = r24351 * r24352;
        double r24354 = sin(r24350);
        double r24355 = r24354 * r24347;
        double r24356 = r24353 - r24355;
        double r24357 = r24349 / r24356;
        double r24358 = r24348 * r24357;
        return r24358;
}

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 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))