Average Error: 14.8 → 0.4
Time: 22.6s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\frac{\cos a}{\frac{\sin b}{\cos b}} - \sin a}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\frac{\cos a}{\frac{\sin b}{\cos b}} - \sin a}
double f(double r, double a, double b) {
        double r961065 = r;
        double r961066 = b;
        double r961067 = sin(r961066);
        double r961068 = a;
        double r961069 = r961068 + r961066;
        double r961070 = cos(r961069);
        double r961071 = r961067 / r961070;
        double r961072 = r961065 * r961071;
        return r961072;
}
↓
double f(double r, double a, double b) {
        double r961073 = r;
        double r961074 = a;
        double r961075 = cos(r961074);
        double r961076 = b;
        double r961077 = sin(r961076);
        double r961078 = cos(r961076);
        double r961079 = r961077 / r961078;
        double r961080 = r961075 / r961079;
        double r961081 = sin(r961074);
        double r961082 = r961080 - r961081;
        double r961083 = r961073 / r961082;
        return r961083;
}

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

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

  3. Applied cos-sum0.3

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

  4. Taylor expanded around inf 0.3

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

  5. Simplified0.4

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

  7. Applied associate-/l*0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))