Average Error: 14.8 → 0.3
Time: 17.7s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\frac{r \cdot \sin b}{\frac{\cos a \cdot \cos b - \sin b \cdot \sin a}{\sin b \cdot \sin a - \cos a \cdot \cos b} \cdot \frac{\sin b \cdot \sin a + \cos a \cdot \cos b}{\sin b \cdot \sin a + \cos a \cdot \cos b}}}{\sin b \cdot \sin a - \cos a \cdot \cos b}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{\frac{r \cdot \sin b}{\frac{\cos a \cdot \cos b - \sin b \cdot \sin a}{\sin b \cdot \sin a - \cos a \cdot \cos b} \cdot \frac{\sin b \cdot \sin a + \cos a \cdot \cos b}{\sin b \cdot \sin a + \cos a \cdot \cos b}}}{\sin b \cdot \sin a - \cos a \cdot \cos b}
double f(double r, double a, double b) {
        double r28671 = r;
        double r28672 = b;
        double r28673 = sin(r28672);
        double r28674 = r28671 * r28673;
        double r28675 = a;
        double r28676 = r28675 + r28672;
        double r28677 = cos(r28676);
        double r28678 = r28674 / r28677;
        return r28678;
}
↓
double f(double r, double a, double b) {
        double r28679 = r;
        double r28680 = b;
        double r28681 = sin(r28680);
        double r28682 = r28679 * r28681;
        double r28683 = a;
        double r28684 = cos(r28683);
        double r28685 = cos(r28680);
        double r28686 = r28684 * r28685;
        double r28687 = sin(r28683);
        double r28688 = r28681 * r28687;
        double r28689 = r28686 - r28688;
        double r28690 = r28688 - r28686;
        double r28691 = r28689 / r28690;
        double r28692 = r28688 + r28686;
        double r28693 = r28692 / r28692;
        double r28694 = r28691 * r28693;
        double r28695 = r28682 / r28694;
        double r28696 = r28695 / r28690;
        return r28696;
}

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. Simplified14.8

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

  4. Applied cos-sum0.3

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

  5. Simplified0.3

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

  7. Applied flip--0.4

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  11. Applied flip-+0.4

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

  12. Applied associate-/r/0.4

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

  13. Applied associate-/r*0.4

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2019195 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))