Average Error: 15.5 → 0.3
Time: 15.7s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \cos a\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos b \cdot \cos a\right)}
double f(double r, double a, double b) {
        double r24475 = r;
        double r24476 = b;
        double r24477 = sin(r24476);
        double r24478 = r24475 * r24477;
        double r24479 = a;
        double r24480 = r24479 + r24476;
        double r24481 = cos(r24480);
        double r24482 = r24478 / r24481;
        return r24482;
}
↓
double f(double r, double a, double b) {
        double r24483 = r;
        double r24484 = b;
        double r24485 = sin(r24484);
        double r24486 = a;
        double r24487 = sin(r24486);
        double r24488 = -r24485;
        double r24489 = cos(r24484);
        double r24490 = cos(r24486);
        double r24491 = r24489 * r24490;
        double r24492 = fma(r24487, r24488, r24491);
        double r24493 = r24485 / r24492;
        double r24494 = r24483 * r24493;
        return r24494;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.5

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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. Simplified0.3

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

  7. Taylor expanded around inf 0.3

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

  8. Simplified0.3

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

  10. Applied *-un-lft-identity0.3

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

  11. Applied times-frac0.3

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))