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

Average Error: 14.5 → 0.4

Time: 44.7s

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 r1816717 = r;
        double r1816718 = b;
        double r1816719 = sin(r1816718);
        double r1816720 = r1816717 * r1816719;
        double r1816721 = a;
        double r1816722 = r1816721 + r1816718;
        double r1816723 = cos(r1816722);
        double r1816724 = r1816720 / r1816723;
        return r1816724;
}

↓

double f(double r, double a, double b) {
        double r1816725 = 1.0;
        double r1816726 = b;
        double r1816727 = cos(r1816726);
        double r1816728 = a;
        double r1816729 = cos(r1816728);
        double r1816730 = r1816727 * r1816729;
        double r1816731 = sin(r1816726);
        double r1816732 = sin(r1816728);
        double r1816733 = r1816731 * r1816732;
        double r1816734 = r1816730 - r1816733;
        double r1816735 = r1816725 / r1816734;
        double r1816736 = r;
        double r1816737 = r1816736 * r1816731;
        double r1816738 = r1816735 * r1816737;
        return r1816738;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.5

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 2019149 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))