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

Average Error: 14.8 → 0.4

Time: 20.0s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r453439 = r;
        double r453440 = b;
        double r453441 = sin(r453440);
        double r453442 = r453439 * r453441;
        double r453443 = a;
        double r453444 = r453443 + r453440;
        double r453445 = cos(r453444);
        double r453446 = r453442 / r453445;
        return r453446;
}

↓

double f(double r, double a, double b) {
        double r453447 = 1.0;
        double r453448 = b;
        double r453449 = cos(r453448);
        double r453450 = a;
        double r453451 = cos(r453450);
        double r453452 = r453449 * r453451;
        double r453453 = sin(r453448);
        double r453454 = sin(r453450);
        double r453455 = r453453 * r453454;
        double r453456 = r453452 - r453455;
        double r453457 = r453447 / r453456;
        double r453458 = r;
        double r453459 = r453458 * r453453;
        double r453460 = r453457 * r453459;
        return r453460;
}

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 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

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

Reproduce

herbie shell --seed 2019154 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))