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

Average Error: 14.8 → 0.3

Time: 17.2s

Precision: 64

Math

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

↓

\[r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}\]

C

double f(double r, double a, double b) {
        double r24823 = r;
        double r24824 = b;
        double r24825 = sin(r24824);
        double r24826 = r24823 * r24825;
        double r24827 = a;
        double r24828 = r24827 + r24824;
        double r24829 = cos(r24828);
        double r24830 = r24826 / r24829;
        return r24830;
}

↓

double f(double r, double a, double b) {
        double r24831 = r;
        double r24832 = b;
        double r24833 = sin(r24832);
        double r24834 = a;
        double r24835 = cos(r24834);
        double r24836 = cos(r24832);
        double r24837 = r24835 * r24836;
        double r24838 = sin(r24834);
        double r24839 = r24838 * r24833;
        double r24840 = r24837 - r24839;
        double r24841 = r24833 / r24840;
        double r24842 = r24831 * r24841;
        return r24842;
}

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 *-un-lft-identity 0.3

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

6. Applied times-frac 0.3

   \[\leadsto \color{blue}{\frac{r}{1} \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]

7. Simplified 0.3

   \[\leadsto \color{blue}{r} \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}\]

8. Final simplification 0.3

   \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}\]

Reproduce

herbie shell --seed 2019305 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))