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

Average Error: 15.3 → 0.4

Time: 23.1s

Precision: 64

Math

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

↓

\[\frac{r \cdot \sin b}{\left(\left(\sin a \cdot \sin b\right) \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right) + {\left(\cos a \cdot \cos b\right)}^{2}\right) \cdot \frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\left(\sin a \cdot \sin b\right) \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right) + {\left(\cos a \cdot \cos b\right)}^{2}}}\]

C

double f(double r, double a, double b) {
        double r27916 = r;
        double r27917 = b;
        double r27918 = sin(r27917);
        double r27919 = r27916 * r27918;
        double r27920 = a;
        double r27921 = r27920 + r27917;
        double r27922 = cos(r27921);
        double r27923 = r27919 / r27922;
        return r27923;
}

↓

double f(double r, double a, double b) {
        double r27924 = r;
        double r27925 = b;
        double r27926 = sin(r27925);
        double r27927 = r27924 * r27926;
        double r27928 = a;
        double r27929 = sin(r27928);
        double r27930 = r27929 * r27926;
        double r27931 = cos(r27928);
        double r27932 = cos(r27925);
        double r27933 = r27931 * r27932;
        double r27934 = r27933 + r27930;
        double r27935 = r27930 * r27934;
        double r27936 = 2.0;
        double r27937 = pow(r27933, r27936);
        double r27938 = r27935 + r27937;
        double r27939 = r27933 - r27930;
        double r27940 = r27939 / r27938;
        double r27941 = r27938 * r27940;
        double r27942 = r27927 / r27941;
        return r27942;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.3

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 flip3-- 0.4

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

6. Simplified 0.4

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

7. Simplified 0.4

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

9. Applied *-un-lft-identity 0.4

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

10. Applied difference-cubes 0.4

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

11. Applied times-frac 0.4

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

12. Simplified 0.4

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\left(\left(\sin a \cdot \sin b\right) \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right) + {\left(\cos a \cdot \cos b\right)}^{2}\right)} \cdot \frac{\cos a \cdot \cos b - \sin b \cdot \sin a}{\left(\sin b \cdot \sin a\right) \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right)}}\]

13. Simplified 0.4

    \[\leadsto \frac{r \cdot \sin b}{\left(\left(\sin a \cdot \sin b\right) \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right) + {\left(\cos a \cdot \cos b\right)}^{2}\right) \cdot \color{blue}{\frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\left(\sin a \cdot \sin b\right) \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right) + {\left(\cos a \cdot \cos b\right)}^{2}}}}\]

14. Final simplification 0.4

    \[\leadsto \frac{r \cdot \sin b}{\left(\left(\sin a \cdot \sin b\right) \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right) + {\left(\cos a \cdot \cos b\right)}^{2}\right) \cdot \frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\left(\sin a \cdot \sin b\right) \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right) + {\left(\cos a \cdot \cos b\right)}^{2}}}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))