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

Average Error: 14.7 → 0.3

Time: 18.7s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r452113 = r;
        double r452114 = b;
        double r452115 = sin(r452114);
        double r452116 = a;
        double r452117 = r452116 + r452114;
        double r452118 = cos(r452117);
        double r452119 = r452115 / r452118;
        double r452120 = r452113 * r452119;
        return r452120;
}

↓

double f(double r, double a, double b) {
        double r452121 = r;
        double r452122 = b;
        double r452123 = sin(r452122);
        double r452124 = r452121 * r452123;
        double r452125 = a;
        double r452126 = cos(r452125);
        double r452127 = cos(r452122);
        double r452128 = r452126 * r452127;
        double r452129 = sin(r452125);
        double r452130 = r452123 * r452129;
        double r452131 = r452128 - r452130;
        double r452132 = r452124 / r452131;
        return r452132;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.7

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
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 associate-*r/ 0.3

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

6. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))