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

Average Error: 15.1 → 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 r459780 = r;
        double r459781 = b;
        double r459782 = sin(r459781);
        double r459783 = r459780 * r459782;
        double r459784 = a;
        double r459785 = r459784 + r459781;
        double r459786 = cos(r459785);
        double r459787 = r459783 / r459786;
        return r459787;
}

↓

double f(double r, double a, double b) {
        double r459788 = r;
        double r459789 = b;
        double r459790 = cos(r459789);
        double r459791 = a;
        double r459792 = cos(r459791);
        double r459793 = r459790 * r459792;
        double r459794 = sin(r459789);
        double r459795 = sin(r459791);
        double r459796 = r459794 * r459795;
        double r459797 = r459793 - r459796;
        double r459798 = r459797 / r459794;
        double r459799 = r459788 / r459798;
        return r459799;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.1

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 2019153 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))