Average Error: 15.1 → 0.4
Time: 23.2s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\sin b}{\frac{\cos b \cdot \cos a - \sin b \cdot \sin a}{\cos b \cdot \cos a + \sin b \cdot \sin a}} \cdot \frac{r}{\cos b \cdot \cos a + \sin b \cdot \sin a}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{\sin b}{\frac{\cos b \cdot \cos a - \sin b \cdot \sin a}{\cos b \cdot \cos a + \sin b \cdot \sin a}} \cdot \frac{r}{\cos b \cdot \cos a + \sin b \cdot \sin a}
double f(double r, double a, double b) {
        double r545953 = r;
        double r545954 = b;
        double r545955 = sin(r545954);
        double r545956 = r545953 * r545955;
        double r545957 = a;
        double r545958 = r545957 + r545954;
        double r545959 = cos(r545958);
        double r545960 = r545956 / r545959;
        return r545960;
}
↓
double f(double r, double a, double b) {
        double r545961 = b;
        double r545962 = sin(r545961);
        double r545963 = cos(r545961);
        double r545964 = a;
        double r545965 = cos(r545964);
        double r545966 = r545963 * r545965;
        double r545967 = sin(r545964);
        double r545968 = r545962 * r545967;
        double r545969 = r545966 - r545968;
        double r545970 = r545966 + r545968;
        double r545971 = r545969 / r545970;
        double r545972 = r545962 / r545971;
        double r545973 = r;
        double r545974 = r545973 / r545970;
        double r545975 = r545972 * r545974;
        return r545975;
}

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 15.1

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

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

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

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

  8. Applied difference-of-squares0.4

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

  9. Applied times-frac0.3

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

  10. Applied times-frac0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))