Average Error: 15.4 → 0.3
Time: 13.4s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\sin b \cdot r}{\cos b \cdot \cos a + \left(-\sin a \cdot \sin b\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{\sin b \cdot r}{\cos b \cdot \cos a + \left(-\sin a \cdot \sin b\right)}
double f(double r, double a, double b) {
        double r24052 = r;
        double r24053 = b;
        double r24054 = sin(r24053);
        double r24055 = r24052 * r24054;
        double r24056 = a;
        double r24057 = r24056 + r24053;
        double r24058 = cos(r24057);
        double r24059 = r24055 / r24058;
        return r24059;
}
↓
double f(double r, double a, double b) {
        double r24060 = b;
        double r24061 = sin(r24060);
        double r24062 = r;
        double r24063 = r24061 * r24062;
        double r24064 = cos(r24060);
        double r24065 = a;
        double r24066 = cos(r24065);
        double r24067 = r24064 * r24066;
        double r24068 = sin(r24065);
        double r24069 = r24068 * r24061;
        double r24070 = -r24069;
        double r24071 = r24067 + r24070;
        double r24072 = r24063 / r24071;
        return r24072;
}

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

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 add-cbrt-cube0.4

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

  11. Applied add-cbrt-cube0.4

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

  12. Applied cbrt-unprod0.4

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

  13. Simplified0.4

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019308 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))