Average Error: 14.8 → 0.4
Time: 6.2s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \left(\frac{\sin b}{\left(\cos b \cdot \cos a\right) \cdot \left(\cos b \cdot \cos a\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)} \cdot \left(\cos b \cdot \cos a + \sin a \cdot \sin b\right)\right)\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \left(\frac{\sin b}{\left(\cos b \cdot \cos a\right) \cdot \left(\cos b \cdot \cos a\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)} \cdot \left(\cos b \cdot \cos a + \sin a \cdot \sin b\right)\right)
double f(double r, double a, double b) {
        double r17778 = r;
        double r17779 = b;
        double r17780 = sin(r17779);
        double r17781 = r17778 * r17780;
        double r17782 = a;
        double r17783 = r17782 + r17779;
        double r17784 = cos(r17783);
        double r17785 = r17781 / r17784;
        return r17785;
}
↓
double f(double r, double a, double b) {
        double r17786 = r;
        double r17787 = b;
        double r17788 = sin(r17787);
        double r17789 = cos(r17787);
        double r17790 = a;
        double r17791 = cos(r17790);
        double r17792 = r17789 * r17791;
        double r17793 = r17792 * r17792;
        double r17794 = sin(r17790);
        double r17795 = r17794 * r17788;
        double r17796 = r17795 * r17795;
        double r17797 = r17793 - r17796;
        double r17798 = r17788 / r17797;
        double r17799 = r17792 + r17795;
        double r17800 = r17798 * r17799;
        double r17801 = r17786 * r17800;
        return r17801;
}

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 14.8

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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. Simplified0.3

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

  10. Applied flip--0.4

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

  11. Applied associate-/r/0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020046 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))