Average Error: 14.8 → 0.3
Time: 25.4s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, -\sin a \cdot \sin b\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, -\sin a \cdot \sin b\right)}
double f(double r, double a, double b) {
        double r1026818 = r;
        double r1026819 = b;
        double r1026820 = sin(r1026819);
        double r1026821 = r1026818 * r1026820;
        double r1026822 = a;
        double r1026823 = r1026822 + r1026819;
        double r1026824 = cos(r1026823);
        double r1026825 = r1026821 / r1026824;
        return r1026825;
}
↓
double f(double r, double a, double b) {
        double r1026826 = r;
        double r1026827 = b;
        double r1026828 = sin(r1026827);
        double r1026829 = cos(r1026827);
        double r1026830 = a;
        double r1026831 = cos(r1026830);
        double r1026832 = sin(r1026830);
        double r1026833 = r1026832 * r1026828;
        double r1026834 = -r1026833;
        double r1026835 = fma(r1026829, r1026831, r1026834);
        double r1026836 = r1026828 / r1026835;
        double r1026837 = r1026826 * r1026836;
        return r1026837;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.8

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 div-inv0.4

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

  7. Applied associate-*l*0.4

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

  8. Simplified0.3

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

  10. Applied fma-neg0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))