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

Average Error: 15.2 → 0.4

Time: 6.9s

Precision: 64

Math

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

↓

\[\frac{r}{\cos b \cdot \cos a - \sin a \cdot \sin b} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\sin b}}\]

C

double f(double r, double a, double b) {
        double r17739 = r;
        double r17740 = b;
        double r17741 = sin(r17740);
        double r17742 = a;
        double r17743 = r17742 + r17740;
        double r17744 = cos(r17743);
        double r17745 = r17741 / r17744;
        double r17746 = r17739 * r17745;
        return r17746;
}

↓

double f(double r, double a, double b) {
        double r17747 = r;
        double r17748 = b;
        double r17749 = cos(r17748);
        double r17750 = a;
        double r17751 = cos(r17750);
        double r17752 = r17749 * r17751;
        double r17753 = sin(r17750);
        double r17754 = sin(r17748);
        double r17755 = r17753 * r17754;
        double r17756 = r17752 - r17755;
        double r17757 = r17747 / r17756;
        double r17758 = 1.0;
        double r17759 = cbrt(r17758);
        double r17760 = r17758 / r17754;
        double r17761 = r17759 / r17760;
        double r17762 = r17757 * r17761;
        return r17762;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.2

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

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

6. Applied add-cbrt-cube 0.4

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

7. Applied cbrt-unprod 0.4

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

8. Simplified 0.4

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

10. Applied clear-num 0.4

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

11. Simplified 0.4

    \[\leadsto r \cdot \frac{1}{\color{blue}{\frac{\cos b \cdot \cos a - \sin a \cdot \sin b}{\sin b}}}\]
12. Using strategy rm

13. Applied div-inv 0.4

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

14. Applied add-cube-cbrt 0.4

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

15. Applied times-frac 0.4

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

16. Applied associate-*r* 0.5

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

17. Simplified 0.4

    \[\leadsto \color{blue}{\frac{r}{\cos b \cdot \cos a - \sin a \cdot \sin b}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\sin b}}\]

18. Final simplification 0.4

    \[\leadsto \frac{r}{\cos b \cdot \cos a - \sin a \cdot \sin b} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\sin b}}\]

Reproduce

herbie shell --seed 2020064 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))