Average Error: 14.4 → 0.3
Time: 21.1s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\sin b}{\mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right) + \mathsf{fma}\left(\cos a, \cos b, \left(-\sin b\right) \cdot \sin a\right)} \cdot r\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{\sin b}{\mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right) + \mathsf{fma}\left(\cos a, \cos b, \left(-\sin b\right) \cdot \sin a\right)} \cdot r
double f(double r, double a, double b) {
        double r966635 = r;
        double r966636 = b;
        double r966637 = sin(r966636);
        double r966638 = r966635 * r966637;
        double r966639 = a;
        double r966640 = r966639 + r966636;
        double r966641 = cos(r966640);
        double r966642 = r966638 / r966641;
        return r966642;
}
↓
double f(double r, double a, double b) {
        double r966643 = b;
        double r966644 = sin(r966643);
        double r966645 = -r966644;
        double r966646 = a;
        double r966647 = sin(r966646);
        double r966648 = r966647 * r966644;
        double r966649 = fma(r966645, r966647, r966648);
        double r966650 = cos(r966646);
        double r966651 = cos(r966643);
        double r966652 = r966645 * r966647;
        double r966653 = fma(r966650, r966651, r966652);
        double r966654 = r966649 + r966653;
        double r966655 = r966644 / r966654;
        double r966656 = r;
        double r966657 = r966655 * r966656;
        return r966657;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.4

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 *-un-lft-identity0.3

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

  6. Applied times-frac0.3

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

  7. Simplified0.3

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

  9. Applied prod-diff0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))