Average Error: 14.5 → 0.3
Time: 29.9s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\sin b \cdot \frac{r}{\cos a \cdot \cos b - \sin b \cdot \sin a}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\sin b \cdot \frac{r}{\cos a \cdot \cos b - \sin b \cdot \sin a}
double f(double r, double a, double b) {
        double r1169303 = r;
        double r1169304 = b;
        double r1169305 = sin(r1169304);
        double r1169306 = a;
        double r1169307 = r1169306 + r1169304;
        double r1169308 = cos(r1169307);
        double r1169309 = r1169305 / r1169308;
        double r1169310 = r1169303 * r1169309;
        return r1169310;
}
↓
double f(double r, double a, double b) {
        double r1169311 = b;
        double r1169312 = sin(r1169311);
        double r1169313 = r;
        double r1169314 = a;
        double r1169315 = cos(r1169314);
        double r1169316 = cos(r1169311);
        double r1169317 = r1169315 * r1169316;
        double r1169318 = sin(r1169314);
        double r1169319 = r1169312 * r1169318;
        double r1169320 = r1169317 - r1169319;
        double r1169321 = r1169313 / r1169320;
        double r1169322 = r1169312 * r1169321;
        return r1169322;
}

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 +-commutative14.5

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

  4. Applied cos-sum0.3

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

  6. Applied log1p-expm1-u0.3

    \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}}\]
  7. Using strategy rm

  8. Applied clear-num0.4

    \[\leadsto r \cdot \color{blue}{\frac{1}{\frac{\cos b \cdot \cos a - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}{\sin b}}}\]

  9. Applied un-div-inv0.4

    \[\leadsto \color{blue}{\frac{r}{\frac{\cos b \cdot \cos a - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}{\sin b}}}\]
  10. Using strategy rm

  11. Applied associate-/r/0.3

    \[\leadsto \color{blue}{\frac{r}{\cos b \cdot \cos a - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)} \cdot \sin b}\]

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))