Average Error: 15.1 → 0.3
Time: 23.6s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, \left(-\sin b\right) \cdot \sin a\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, \left(-\sin b\right) \cdot \sin a\right)}
double f(double r, double a, double b) {
        double r733518 = r;
        double r733519 = b;
        double r733520 = sin(r733519);
        double r733521 = r733518 * r733520;
        double r733522 = a;
        double r733523 = r733522 + r733519;
        double r733524 = cos(r733523);
        double r733525 = r733521 / r733524;
        return r733525;
}
↓
double f(double r, double a, double b) {
        double r733526 = r;
        double r733527 = b;
        double r733528 = sin(r733527);
        double r733529 = r733526 * r733528;
        double r733530 = a;
        double r733531 = cos(r733530);
        double r733532 = cos(r733527);
        double r733533 = -r733528;
        double r733534 = sin(r733530);
        double r733535 = r733533 * r733534;
        double r733536 = fma(r733531, r733532, r733535);
        double r733537 = r733529 / r733536;
        return r733537;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.1

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 fma-neg0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))