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

Average Error: 14.7 → 0.4

Time: 17.9s

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 r428703 = r;
        double r428704 = b;
        double r428705 = sin(r428704);
        double r428706 = r428703 * r428705;
        double r428707 = a;
        double r428708 = r428707 + r428704;
        double r428709 = cos(r428708);
        double r428710 = r428706 / r428709;
        return r428710;
}

↓

double f(double r, double a, double b) {
        double r428711 = 1.0;
        double r428712 = b;
        double r428713 = cos(r428712);
        double r428714 = a;
        double r428715 = cos(r428714);
        double r428716 = r428713 * r428715;
        double r428717 = sin(r428712);
        double r428718 = sin(r428714);
        double r428719 = r428717 * r428718;
        double r428720 = r428716 - r428719;
        double r428721 = r428711 / r428720;
        double r428722 = r;
        double r428723 = r428722 * r428717;
        double r428724 = r428721 * r428723;
        return r428724;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.7

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 2019156 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))