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

Average Error: 15.1 → 0.4

Time: 22.3s

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 r982144 = r;
        double r982145 = b;
        double r982146 = sin(r982145);
        double r982147 = a;
        double r982148 = r982147 + r982145;
        double r982149 = cos(r982148);
        double r982150 = r982146 / r982149;
        double r982151 = r982144 * r982150;
        return r982151;
}

↓

double f(double r, double a, double b) {
        double r982152 = r;
        double r982153 = b;
        double r982154 = cos(r982153);
        double r982155 = sin(r982153);
        double r982156 = a;
        double r982157 = cos(r982156);
        double r982158 = r982155 / r982157;
        double r982159 = r982154 / r982158;
        double r982160 = sin(r982156);
        double r982161 = r982159 - r982160;
        double r982162 = r982152 / r982161;
        return r982162;
}

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)))))