Average Error: 15.3 → 0.4
Time: 20.5s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\sin b \cdot \frac{r}{\mathsf{fma}\left(\cos a, \cos b, \mathsf{log1p}\left(\mathsf{expm1}\left(\left(-\sin b\right) \cdot \sin a\right)\right)\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\sin b \cdot \frac{r}{\mathsf{fma}\left(\cos a, \cos b, \mathsf{log1p}\left(\mathsf{expm1}\left(\left(-\sin b\right) \cdot \sin a\right)\right)\right)}
double f(double r, double a, double b) {
        double r27839 = r;
        double r27840 = b;
        double r27841 = sin(r27840);
        double r27842 = a;
        double r27843 = r27842 + r27840;
        double r27844 = cos(r27843);
        double r27845 = r27841 / r27844;
        double r27846 = r27839 * r27845;
        return r27846;
}
↓
double f(double r, double a, double b) {
        double r27847 = b;
        double r27848 = sin(r27847);
        double r27849 = r;
        double r27850 = a;
        double r27851 = cos(r27850);
        double r27852 = cos(r27847);
        double r27853 = -r27848;
        double r27854 = sin(r27850);
        double r27855 = r27853 * r27854;
        double r27856 = expm1(r27855);
        double r27857 = log1p(r27856);
        double r27858 = fma(r27851, r27852, r27857);
        double r27859 = r27849 / r27858;
        double r27860 = r27848 * r27859;
        return r27860;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.3

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

  2. Simplified15.3

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

  4. Applied cos-sum0.3

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

  5. Simplified0.3

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

  6. Simplified0.3

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

  8. Applied fma-neg0.3

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

  9. Simplified0.3

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

  11. Applied log1p-expm1-u0.4

    \[\leadsto \sin b \cdot \frac{r}{\mathsf{fma}\left(\cos a, \cos b, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin a \cdot \left(-\sin b\right)\right)\right)}\right)}\]

  12. Simplified0.4

    \[\leadsto \sin b \cdot \frac{r}{\mathsf{fma}\left(\cos a, \cos b, \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin b \cdot \left(-\sin a\right)\right)}\right)\right)}\]

  13. Final simplification0.4

    \[\leadsto \sin b \cdot \frac{r}{\mathsf{fma}\left(\cos a, \cos b, \mathsf{log1p}\left(\mathsf{expm1}\left(\left(-\sin b\right) \cdot \sin a\right)\right)\right)}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))