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

Average Error: 14.8 → 0.4

Time: 24.8s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r760221 = r;
        double r760222 = b;
        double r760223 = sin(r760222);
        double r760224 = a;
        double r760225 = r760224 + r760222;
        double r760226 = cos(r760225);
        double r760227 = r760223 / r760226;
        double r760228 = r760221 * r760227;
        return r760228;
}

↓

double f(double r, double a, double b) {
        double r760229 = b;
        double r760230 = sin(r760229);
        double r760231 = a;
        double r760232 = cos(r760231);
        double r760233 = cos(r760229);
        double r760234 = r760232 * r760233;
        double r760235 = sin(r760231);
        double r760236 = r760235 * r760230;
        double r760237 = r760234 - r760236;
        double r760238 = r;
        double r760239 = r760237 / r760238;
        double r760240 = r760230 / r760239;
        return r760240;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.8

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
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. Taylor expanded around inf 0.3

   \[\leadsto \color{blue}{\frac{\sin b \cdot r}{\cos a \cdot \cos b - \sin b \cdot \sin a}}\]
5. Using strategy rm

6. Applied associate-/l* 0.4

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

7. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019135 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))