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

Average Error: 15.2 → 0.4

Time: 19.9s

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 r1068115 = r;
        double r1068116 = b;
        double r1068117 = sin(r1068116);
        double r1068118 = r1068115 * r1068117;
        double r1068119 = a;
        double r1068120 = r1068119 + r1068116;
        double r1068121 = cos(r1068120);
        double r1068122 = r1068118 / r1068121;
        return r1068122;
}

↓

double f(double r, double a, double b) {
        double r1068123 = r;
        double r1068124 = b;
        double r1068125 = cos(r1068124);
        double r1068126 = a;
        double r1068127 = cos(r1068126);
        double r1068128 = r1068125 * r1068127;
        double r1068129 = sin(r1068124);
        double r1068130 = sin(r1068126);
        double r1068131 = r1068129 * r1068130;
        double r1068132 = r1068128 - r1068131;
        double r1068133 = r1068132 / r1068129;
        double r1068134 = r1068123 / r1068133;
        return r1068134;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.2

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 2019172 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))