Average Error: 15.2 → 0.3
Time: 24.2s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot \sin b\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot \sin b
double f(double r, double a, double b) {
        double r795322 = r;
        double r795323 = b;
        double r795324 = sin(r795323);
        double r795325 = a;
        double r795326 = r795325 + r795323;
        double r795327 = cos(r795326);
        double r795328 = r795324 / r795327;
        double r795329 = r795322 * r795328;
        return r795329;
}
↓
double f(double r, double a, double b) {
        double r795330 = r;
        double r795331 = a;
        double r795332 = cos(r795331);
        double r795333 = b;
        double r795334 = cos(r795333);
        double r795335 = r795332 * r795334;
        double r795336 = sin(r795333);
        double r795337 = sin(r795331);
        double r795338 = r795336 * r795337;
        double r795339 = r795335 - r795338;
        double r795340 = r795330 / r795339;
        double r795341 = r795340 * r795336;
        return r795341;
}

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 15.2

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

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

  6. Applied *-un-lft-identity0.3

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

  7. Applied times-frac0.3

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))