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

Average Error: 15.5 → 0.4

Time: 27.2s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r1212325 = r;
        double r1212326 = b;
        double r1212327 = sin(r1212326);
        double r1212328 = a;
        double r1212329 = r1212328 + r1212326;
        double r1212330 = cos(r1212329);
        double r1212331 = r1212327 / r1212330;
        double r1212332 = r1212325 * r1212331;
        return r1212332;
}

↓

double f(double r, double a, double b) {
        double r1212333 = r;
        double r1212334 = b;
        double r1212335 = sin(r1212334);
        double r1212336 = r1212333 * r1212335;
        double r1212337 = a;
        double r1212338 = cos(r1212337);
        double r1212339 = cos(r1212334);
        double r1212340 = r1212338 * r1212339;
        double r1212341 = r1212340 * r1212340;
        double r1212342 = r1212340 * r1212341;
        double r1212343 = sin(r1212337);
        double r1212344 = r1212335 * r1212343;
        double r1212345 = r1212344 + r1212340;
        double r1212346 = r1212340 * r1212345;
        double r1212347 = r1212344 * r1212344;
        double r1212348 = r1212346 + r1212347;
        double r1212349 = r1212342 / r1212348;
        double r1212350 = r1212344 * r1212347;
        double r1212351 = r1212350 / r1212348;
        double r1212352 = r1212349 - r1212351;
        double r1212353 = r1212336 / r1212352;
        return r1212353;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.5

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

7. Applied flip3-- 0.4

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\frac{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}}}\]

8. Simplified 0.4

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

9. Simplified 0.4

   \[\leadsto \frac{r \cdot \sin b}{\frac{\left(\cos a \cdot \cos b\right) \cdot \left(\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right)\right) - \left(\left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)\right) \cdot \left(\sin b \cdot \sin a\right)}{\color{blue}{\left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin b \cdot \sin a + \cos a \cdot \cos b\right)}}}\]
10. Using strategy rm

11. Applied div-sub 0.4

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

12. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019170 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))