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

Average Error: 14.8 → 0.3

Time: 17.1s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r27557 = r;
        double r27558 = b;
        double r27559 = sin(r27558);
        double r27560 = r27557 * r27559;
        double r27561 = a;
        double r27562 = r27561 + r27558;
        double r27563 = cos(r27562);
        double r27564 = r27560 / r27563;
        return r27564;
}

↓

double f(double r, double a, double b) {
        double r27565 = r;
        double r27566 = a;
        double r27567 = cos(r27566);
        double r27568 = b;
        double r27569 = cos(r27568);
        double r27570 = r27567 * r27569;
        double r27571 = sin(r27568);
        double r27572 = sin(r27566);
        double r27573 = r27571 * r27572;
        double r27574 = r27570 - r27573;
        double r27575 = r27565 / r27574;
        double r27576 = r27575 * r27571;
        return r27576;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.8

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 flip-- 0.4

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

6. Simplified 0.4

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

8. Applied *-un-lft-identity 0.4

   \[\leadsto \frac{r \cdot \sin b}{\frac{\left(\cos a \cdot \cos b - \sin b \cdot \sin a\right) \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)}{\color{blue}{1 \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)}}}\]

9. Applied times-frac 0.3

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\frac{\cos a \cdot \cos b - \sin b \cdot \sin a}{1} \cdot \frac{\cos a \cdot \cos b + \sin a \cdot \sin b}{\cos a \cdot \cos b + \sin a \cdot \sin b}}}\]

10. Applied times-frac 0.3

    \[\leadsto \color{blue}{\frac{r}{\frac{\cos a \cdot \cos b - \sin b \cdot \sin a}{1}} \cdot \frac{\sin b}{\frac{\cos a \cdot \cos b + \sin a \cdot \sin b}{\cos a \cdot \cos b + \sin a \cdot \sin b}}}\]

11. Simplified 0.3

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

12. Simplified 0.3

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

13. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019195 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))