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

Average Error: 15.1 → 0.4

Time: 24.5s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r734821 = r;
        double r734822 = b;
        double r734823 = sin(r734822);
        double r734824 = a;
        double r734825 = r734824 + r734822;
        double r734826 = cos(r734825);
        double r734827 = r734823 / r734826;
        double r734828 = r734821 * r734827;
        return r734828;
}

↓

double f(double r, double a, double b) {
        double r734829 = b;
        double r734830 = sin(r734829);
        double r734831 = a;
        double r734832 = cos(r734831);
        double r734833 = cos(r734829);
        double r734834 = r734832 * r734833;
        double r734835 = sin(r734831);
        double r734836 = r734835 * r734830;
        double r734837 = r734834 - r734836;
        double r734838 = r;
        double r734839 = r734837 / r734838;
        double r734840 = r734830 / r734839;
        return r734840;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.1

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
2. Using strategy rm

3. Applied cos-sum 0.3

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

4. Taylor expanded around -inf 0.4

   \[\leadsto \color{blue}{\frac{\sin b \cdot r}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]
5. Using strategy rm

6. Applied associate-/l* 0.4

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

7. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019129 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))