Average Error: 14.7 → 0.3
Time: 24.8s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[-\sin b \cdot \frac{r}{\mathsf{fma}\left(\sin a, \sin b, -\cos b \cdot \cos a\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
-\sin b \cdot \frac{r}{\mathsf{fma}\left(\sin a, \sin b, -\cos b \cdot \cos a\right)}
double f(double r, double a, double b) {
        double r24515 = r;
        double r24516 = b;
        double r24517 = sin(r24516);
        double r24518 = r24515 * r24517;
        double r24519 = a;
        double r24520 = r24519 + r24516;
        double r24521 = cos(r24520);
        double r24522 = r24518 / r24521;
        return r24522;
}
↓
double f(double r, double a, double b) {
        double r24523 = b;
        double r24524 = sin(r24523);
        double r24525 = r;
        double r24526 = a;
        double r24527 = sin(r24526);
        double r24528 = cos(r24523);
        double r24529 = cos(r24526);
        double r24530 = r24528 * r24529;
        double r24531 = -r24530;
        double r24532 = fma(r24527, r24524, r24531);
        double r24533 = r24525 / r24532;
        double r24534 = r24524 * r24533;
        double r24535 = -r24534;
        return r24535;
}

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 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. Using strategy rm

  7. Applied frac-2neg0.3

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

  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 distribute-frac-neg0.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

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))))