Average Error: 15.4 → 0.4
Time: 17.5s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\left(\frac{r}{\mathsf{fma}\left(\sin a, -\sin b, \cos a \cdot \cos b\right)} \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, \sin a, \cos a \cdot \cos b\right)}\right) \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\left(\frac{r}{\mathsf{fma}\left(\sin a, -\sin b, \cos a \cdot \cos b\right)} \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, \sin a, \cos a \cdot \cos b\right)}\right) \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)
double f(double r, double a, double b) {
        double r26835 = r;
        double r26836 = b;
        double r26837 = sin(r26836);
        double r26838 = r26835 * r26837;
        double r26839 = a;
        double r26840 = r26839 + r26836;
        double r26841 = cos(r26840);
        double r26842 = r26838 / r26841;
        return r26842;
}
↓
double f(double r, double a, double b) {
        double r26843 = r;
        double r26844 = a;
        double r26845 = sin(r26844);
        double r26846 = b;
        double r26847 = sin(r26846);
        double r26848 = -r26847;
        double r26849 = cos(r26844);
        double r26850 = cos(r26846);
        double r26851 = r26849 * r26850;
        double r26852 = fma(r26845, r26848, r26851);
        double r26853 = r26843 / r26852;
        double r26854 = fma(r26847, r26845, r26851);
        double r26855 = r26847 / r26854;
        double r26856 = r26853 * r26855;
        double r26857 = r26845 * r26847;
        double r26858 = r26851 + r26857;
        double r26859 = r26856 * r26858;
        return r26859;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.4

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

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

  6. Applied flip--0.4

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

  7. Applied associate-/r/0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))