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

Average Error: 15.5 → 0.4

Time: 21.8s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r1437796 = r;
        double r1437797 = b;
        double r1437798 = sin(r1437797);
        double r1437799 = r1437796 * r1437798;
        double r1437800 = a;
        double r1437801 = r1437800 + r1437797;
        double r1437802 = cos(r1437801);
        double r1437803 = r1437799 / r1437802;
        return r1437803;
}

↓

double f(double r, double a, double b) {
        double r1437804 = r;
        double r1437805 = b;
        double r1437806 = sin(r1437805);
        double r1437807 = r1437804 * r1437806;
        double r1437808 = a;
        double r1437809 = cos(r1437808);
        double r1437810 = cos(r1437805);
        double r1437811 = r1437809 * r1437810;
        double r1437812 = sin(r1437808);
        double r1437813 = r1437812 * r1437806;
        double r1437814 = exp(r1437813);
        double r1437815 = log(r1437814);
        double r1437816 = r1437811 - r1437815;
        double r1437817 = r1437807 / r1437816;
        return r1437817;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.5

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 add-log-exp 0.4

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

6. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))