Average Error: 15.4 → 0.4
Time: 20.1s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\sin b \cdot \frac{r}{\cos a \cdot \cos b - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin b \cdot \sin a\right)\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\sin b \cdot \frac{r}{\cos a \cdot \cos b - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin b \cdot \sin a\right)\right)}
double f(double r, double a, double b) {
        double r29421 = r;
        double r29422 = b;
        double r29423 = sin(r29422);
        double r29424 = a;
        double r29425 = r29424 + r29422;
        double r29426 = cos(r29425);
        double r29427 = r29423 / r29426;
        double r29428 = r29421 * r29427;
        return r29428;
}
↓
double f(double r, double a, double b) {
        double r29429 = b;
        double r29430 = sin(r29429);
        double r29431 = r;
        double r29432 = a;
        double r29433 = cos(r29432);
        double r29434 = cos(r29429);
        double r29435 = r29433 * r29434;
        double r29436 = sin(r29432);
        double r29437 = r29430 * r29436;
        double r29438 = log1p(r29437);
        double r29439 = expm1(r29438);
        double r29440 = r29435 - r29439;
        double r29441 = r29431 / r29440;
        double r29442 = r29430 * r29441;
        return r29442;
}

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 15.4

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

  2. Simplified15.4

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

  4. Applied cos-sum0.3

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

  5. Simplified0.3

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

  6. Simplified0.3

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

  8. Applied expm1-log1p-u0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))