Average Error: 14.7 → 0.4
Time: 25.6s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\cos a \cdot \frac{\cos b}{\sin b} - \sin a}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\cos a \cdot \frac{\cos b}{\sin b} - \sin a}
double f(double r, double a, double b) {
        double r24850 = r;
        double r24851 = b;
        double r24852 = sin(r24851);
        double r24853 = r24850 * r24852;
        double r24854 = a;
        double r24855 = r24854 + r24851;
        double r24856 = cos(r24855);
        double r24857 = r24853 / r24856;
        return r24857;
}
↓
double f(double r, double a, double b) {
        double r24858 = r;
        double r24859 = a;
        double r24860 = cos(r24859);
        double r24861 = b;
        double r24862 = cos(r24861);
        double r24863 = sin(r24861);
        double r24864 = r24862 / r24863;
        double r24865 = r24860 * r24864;
        double r24866 = sin(r24859);
        double r24867 = r24865 - r24866;
        double r24868 = r24858 / r24867;
        return r24868;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.7

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

  3. Applied cos-sum0.3

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

  5. Applied *-un-lft-identity0.3

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

  6. Applied times-frac0.3

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

  7. Simplified0.3

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

  9. Applied *-un-lft-identity0.3

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

  10. Applied associate-*l*0.3

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

  11. Simplified0.4

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

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

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

  14. Applied times-frac0.4

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2019304 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))