Average Error: 14.5 → 0.4
Time: 26.7s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{1}{\cos b \cdot \cos a - \sin b \cdot \sin a} \cdot \left(r \cdot \sin b\right)\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{1}{\cos b \cdot \cos a - \sin b \cdot \sin a} \cdot \left(r \cdot \sin b\right)
double f(double r, double a, double b) {
        double r750227 = r;
        double r750228 = b;
        double r750229 = sin(r750228);
        double r750230 = a;
        double r750231 = r750230 + r750228;
        double r750232 = cos(r750231);
        double r750233 = r750229 / r750232;
        double r750234 = r750227 * r750233;
        return r750234;
}
↓
double f(double r, double a, double b) {
        double r750235 = 1.0;
        double r750236 = b;
        double r750237 = cos(r750236);
        double r750238 = a;
        double r750239 = cos(r750238);
        double r750240 = r750237 * r750239;
        double r750241 = sin(r750236);
        double r750242 = sin(r750238);
        double r750243 = r750241 * r750242;
        double r750244 = r750240 - r750243;
        double r750245 = r750235 / r750244;
        double r750246 = r;
        double r750247 = r750246 * r750241;
        double r750248 = r750245 * r750247;
        return r750248;
}

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 14.5

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

  5. Applied fma-neg0.3

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

  7. Applied associate-*r/0.3

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

  9. Applied div-inv0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))