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

Average Error: 14.9 → 0.4

Time: 6.4s

Precision: 64

Math

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

↓

\[r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sqrt[3]{{\left(\sin a \cdot \sin b\right)}^{3}}}\]

C

double f(double r, double a, double b) {
        double r16441 = r;
        double r16442 = b;
        double r16443 = sin(r16442);
        double r16444 = r16441 * r16443;
        double r16445 = a;
        double r16446 = r16445 + r16442;
        double r16447 = cos(r16446);
        double r16448 = r16444 / r16447;
        return r16448;
}

↓

double f(double r, double a, double b) {
        double r16449 = r;
        double r16450 = b;
        double r16451 = sin(r16450);
        double r16452 = cos(r16450);
        double r16453 = a;
        double r16454 = cos(r16453);
        double r16455 = r16452 * r16454;
        double r16456 = sin(r16453);
        double r16457 = r16456 * r16451;
        double r16458 = 3.0;
        double r16459 = pow(r16457, r16458);
        double r16460 = cbrt(r16459);
        double r16461 = r16455 - r16460;
        double r16462 = r16451 / r16461;
        double r16463 = r16449 * r16462;
        return r16463;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.9

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

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

6. Applied add-cbrt-cube 0.4

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

7. Applied cbrt-unprod 0.4

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

8. Simplified 0.4

   \[\leadsto \frac{r \cdot \sin b}{\cos a \cdot \cos b - \sqrt[3]{\color{blue}{{\left(\sin a \cdot \sin b\right)}^{3}}}}\]
9. Using strategy rm

10. Applied *-un-lft-identity 0.4

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{1 \cdot \left(\cos a \cdot \cos b - \sqrt[3]{{\left(\sin a \cdot \sin b\right)}^{3}}\right)}}\]

11. Applied times-frac 0.4

    \[\leadsto \color{blue}{\frac{r}{1} \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin a \cdot \sin b\right)}^{3}}}}\]

12. Simplified 0.4

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

13. Simplified 0.3

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

15. Applied add-cbrt-cube 0.4

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

16. Applied add-cbrt-cube 0.4

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

17. Applied cbrt-unprod 0.4

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

18. Simplified 0.4

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

19. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020081 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))