Average Error: 15.4 → 0.4
Time: 21.4s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\frac{r}{\sin b \cdot \sin a + \cos a \cdot \cos b} \cdot \frac{\left(\cos a \cdot \cos b - \sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a + \cos a \cdot \cos b\right)}{\cos a \cdot \cos b - \sin b \cdot \sin a}}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot \sin b\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{\frac{r}{\sin b \cdot \sin a + \cos a \cdot \cos b} \cdot \frac{\left(\cos a \cdot \cos b - \sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a + \cos a \cdot \cos b\right)}{\cos a \cdot \cos b - \sin b \cdot \sin a}}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot \sin b
double f(double r, double a, double b) {
        double r31047 = r;
        double r31048 = b;
        double r31049 = sin(r31048);
        double r31050 = a;
        double r31051 = r31050 + r31048;
        double r31052 = cos(r31051);
        double r31053 = r31049 / r31052;
        double r31054 = r31047 * r31053;
        return r31054;
}
↓
double f(double r, double a, double b) {
        double r31055 = r;
        double r31056 = b;
        double r31057 = sin(r31056);
        double r31058 = a;
        double r31059 = sin(r31058);
        double r31060 = r31057 * r31059;
        double r31061 = cos(r31058);
        double r31062 = cos(r31056);
        double r31063 = r31061 * r31062;
        double r31064 = r31060 + r31063;
        double r31065 = r31055 / r31064;
        double r31066 = r31063 - r31060;
        double r31067 = r31066 * r31064;
        double r31068 = r31067 / r31066;
        double r31069 = r31065 * r31068;
        double r31070 = r31069 / r31066;
        double r31071 = r31070 * r31057;
        return r31071;
}

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 15.4

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

  2. Simplified15.4

    \[\leadsto \color{blue}{\sin b \cdot \frac{r}{\cos \left(b + a\right)}}\]
  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 flip--0.4

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

  9. Applied associate-/r/0.5

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

  10. Simplified0.4

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

  12. Applied flip-+0.5

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

  13. Applied associate-*r/0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019174 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))