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

Average Error: 14.8 → 0.4

Time: 22.2s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r997128 = r;
        double r997129 = b;
        double r997130 = sin(r997129);
        double r997131 = r997128 * r997130;
        double r997132 = a;
        double r997133 = r997132 + r997129;
        double r997134 = cos(r997133);
        double r997135 = r997131 / r997134;
        return r997135;
}

↓

double f(double r, double a, double b) {
        double r997136 = r;
        double r997137 = b;
        double r997138 = cos(r997137);
        double r997139 = a;
        double r997140 = cos(r997139);
        double r997141 = r997138 * r997140;
        double r997142 = sin(r997137);
        double r997143 = sin(r997139);
        double r997144 = r997142 * r997143;
        double r997145 = r997141 - r997144;
        double r997146 = r997145 / r997142;
        double r997147 = r997136 / r997146;
        return r997147;
}

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 associate-/l* 0.4

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

6. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019163 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))