Average Error: 15.0 → 0.3
Time: 6.6s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\cos b \cdot \cos a - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin a \cdot \sin b\right)\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\cos b \cdot \cos a - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin a \cdot \sin b\right)\right)}
double f(double r, double a, double b) {
        double r17404 = r;
        double r17405 = b;
        double r17406 = sin(r17405);
        double r17407 = r17404 * r17406;
        double r17408 = a;
        double r17409 = r17408 + r17405;
        double r17410 = cos(r17409);
        double r17411 = r17407 / r17410;
        return r17411;
}
↓
double f(double r, double a, double b) {
        double r17412 = r;
        double r17413 = b;
        double r17414 = sin(r17413);
        double r17415 = cos(r17413);
        double r17416 = a;
        double r17417 = cos(r17416);
        double r17418 = r17415 * r17417;
        double r17419 = sin(r17416);
        double r17420 = r17419 * r17414;
        double r17421 = expm1(r17420);
        double r17422 = log1p(r17421);
        double r17423 = r17418 - r17422;
        double r17424 = r17414 / r17423;
        double r17425 = r17412 * r17424;
        return r17425;
}

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 15.0

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 *-un-lft-identity0.3

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

  6. Applied times-frac0.3

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

  7. Simplified0.3

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

  8. Simplified0.3

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

  10. Applied log1p-expm1-u0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))