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

Average Error: 15.0 → 0.4

Time: 25.0s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r25508 = r;
        double r25509 = b;
        double r25510 = sin(r25509);
        double r25511 = r25508 * r25510;
        double r25512 = a;
        double r25513 = r25512 + r25509;
        double r25514 = cos(r25513);
        double r25515 = r25511 / r25514;
        return r25515;
}

↓

double f(double r, double a, double b) {
        double r25516 = r;
        double r25517 = a;
        double r25518 = cos(r25517);
        double r25519 = b;
        double r25520 = cos(r25519);
        double r25521 = r25518 * r25520;
        double r25522 = sin(r25519);
        double r25523 = r25521 / r25522;
        double r25524 = sin(r25517);
        double r25525 = r25523 - r25524;
        double r25526 = r25516 / r25525;
        return r25526;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.0

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

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

6. Applied times-frac 0.3

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

7. Simplified 0.3

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

9. Applied pow1 0.3

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

10. Applied pow1 0.3

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

11. Applied pow-prod-down 0.3

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

12. Simplified 0.4

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

13. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))