Average Error: 15.5 → 0.4
Time: 24.5s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\left(r \cdot \sin b\right) \cdot \frac{-1}{\mathsf{fma}\left(-\cos b, \cos a, \sin a \cdot \sin b\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\left(r \cdot \sin b\right) \cdot \frac{-1}{\mathsf{fma}\left(-\cos b, \cos a, \sin a \cdot \sin b\right)}
double f(double r, double a, double b) {
        double r25533 = r;
        double r25534 = b;
        double r25535 = sin(r25534);
        double r25536 = r25533 * r25535;
        double r25537 = a;
        double r25538 = r25537 + r25534;
        double r25539 = cos(r25538);
        double r25540 = r25536 / r25539;
        return r25540;
}
↓
double f(double r, double a, double b) {
        double r25541 = r;
        double r25542 = b;
        double r25543 = sin(r25542);
        double r25544 = r25541 * r25543;
        double r25545 = -1.0;
        double r25546 = cos(r25542);
        double r25547 = -r25546;
        double r25548 = a;
        double r25549 = cos(r25548);
        double r25550 = sin(r25548);
        double r25551 = r25550 * r25543;
        double r25552 = fma(r25547, r25549, r25551);
        double r25553 = r25545 / r25552;
        double r25554 = r25544 * r25553;
        return r25554;
}

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 *-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 frac-2neg0.3

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

  10. Simplified0.3

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

  12. Applied div-inv0.4

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

  13. Applied associate-*r*0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))