Average Error: 14.7 → 0.4
Time: 22.5s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\mathsf{fma}\left(\cos b \cdot \cos a, \frac{1}{\sin b}, -\sin a\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\mathsf{fma}\left(\cos b \cdot \cos a, \frac{1}{\sin b}, -\sin a\right)}
double f(double r, double a, double b) {
        double r24491 = r;
        double r24492 = b;
        double r24493 = sin(r24492);
        double r24494 = r24491 * r24493;
        double r24495 = a;
        double r24496 = r24495 + r24492;
        double r24497 = cos(r24496);
        double r24498 = r24494 / r24497;
        return r24498;
}
↓
double f(double r, double a, double b) {
        double r24499 = r;
        double r24500 = b;
        double r24501 = cos(r24500);
        double r24502 = a;
        double r24503 = cos(r24502);
        double r24504 = r24501 * r24503;
        double r24505 = 1.0;
        double r24506 = sin(r24500);
        double r24507 = r24505 / r24506;
        double r24508 = sin(r24502);
        double r24509 = -r24508;
        double r24510 = fma(r24504, r24507, r24509);
        double r24511 = r24499 / r24510;
        return r24511;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.7

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 associate-/l*0.4

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

  6. Simplified0.4

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

  8. Applied div-inv0.4

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

  9. Applied fma-neg0.4

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

  10. Final simplification0.4

    \[\leadsto \frac{r}{\mathsf{fma}\left(\cos b \cdot \cos a, \frac{1}{\sin b}, -\sin a\right)}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))