Average Error: 15.3 → 0.3
Time: 21.1s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\sin b \cdot r}{\cos b \cdot \cos a - \sin b \cdot \sin a}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{\sin b \cdot r}{\cos b \cdot \cos a - \sin b \cdot \sin a}
double f(double r, double a, double b) {
        double r28873 = r;
        double r28874 = b;
        double r28875 = sin(r28874);
        double r28876 = a;
        double r28877 = r28876 + r28874;
        double r28878 = cos(r28877);
        double r28879 = r28875 / r28878;
        double r28880 = r28873 * r28879;
        return r28880;
}
↓
double f(double r, double a, double b) {
        double r28881 = b;
        double r28882 = sin(r28881);
        double r28883 = r;
        double r28884 = r28882 * r28883;
        double r28885 = cos(r28881);
        double r28886 = a;
        double r28887 = cos(r28886);
        double r28888 = r28885 * r28887;
        double r28889 = sin(r28886);
        double r28890 = r28882 * r28889;
        double r28891 = r28888 - r28890;
        double r28892 = r28884 / r28891;
        return r28892;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.3

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

  2. Simplified15.3

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

  4. Applied cos-sum0.3

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

  5. Simplified0.3

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

  6. Simplified0.3

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

  8. Applied add-cbrt-cube0.4

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

  9. Applied add-cbrt-cube0.4

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

  10. Applied cbrt-unprod0.4

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

  11. Simplified0.4

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

  13. Applied *-un-lft-identity0.4

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

  14. Applied associate-*l*0.4

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

  15. Simplified0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2019179 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))