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

Average Error: 15.1 → 0.4

Time: 22.9s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r982146 = r;
        double r982147 = b;
        double r982148 = sin(r982147);
        double r982149 = a;
        double r982150 = r982149 + r982147;
        double r982151 = cos(r982150);
        double r982152 = r982148 / r982151;
        double r982153 = r982146 * r982152;
        return r982153;
}

↓

double f(double r, double a, double b) {
        double r982154 = r;
        double r982155 = b;
        double r982156 = cos(r982155);
        double r982157 = sin(r982155);
        double r982158 = a;
        double r982159 = cos(r982158);
        double r982160 = r982157 / r982159;
        double r982161 = r982156 / r982160;
        double r982162 = sin(r982158);
        double r982163 = r982161 - r982162;
        double r982164 = r982154 / r982163;
        return r982164;
}

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. Using strategy rm

5. Applied *-un-lft-identity 0.3

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

6. Applied associate-*l* 0.3

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

7. Simplified 0.4

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

8. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019172 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))