Average Error: 15.1 → 0.3
Time: 7.1s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\cos a \cdot \cos b - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)}
double f(double r, double a, double b) {
        double r17461 = r;
        double r17462 = b;
        double r17463 = sin(r17462);
        double r17464 = a;
        double r17465 = r17464 + r17462;
        double r17466 = cos(r17465);
        double r17467 = r17463 / r17466;
        double r17468 = r17461 * r17467;
        return r17468;
}
↓
double f(double r, double a, double b) {
        double r17469 = r;
        double r17470 = b;
        double r17471 = sin(r17470);
        double r17472 = r17469 * r17471;
        double r17473 = a;
        double r17474 = cos(r17473);
        double r17475 = cos(r17470);
        double r17476 = r17474 * r17475;
        double r17477 = sin(r17473);
        double r17478 = r17477 * r17471;
        double r17479 = log1p(r17478);
        double r17480 = expm1(r17479);
        double r17481 = r17476 - r17480;
        double r17482 = r17472 / r17481;
        return r17482;
}

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 15.1

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 associate-*r/0.3

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

  7. Applied expm1-log1p-u0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))