Average Error: 14.8 → 0.3
Time: 23.7s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, -\mathsf{log1p}\left(\mathsf{expm1}\left(\sin a \cdot \sin b\right)\right)\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, -\mathsf{log1p}\left(\mathsf{expm1}\left(\sin a \cdot \sin b\right)\right)\right)}
double f(double r, double a, double b) {
        double r25775 = r;
        double r25776 = b;
        double r25777 = sin(r25776);
        double r25778 = a;
        double r25779 = r25778 + r25776;
        double r25780 = cos(r25779);
        double r25781 = r25777 / r25780;
        double r25782 = r25775 * r25781;
        return r25782;
}
↓
double f(double r, double a, double b) {
        double r25783 = r;
        double r25784 = b;
        double r25785 = sin(r25784);
        double r25786 = r25783 * r25785;
        double r25787 = a;
        double r25788 = cos(r25787);
        double r25789 = cos(r25784);
        double r25790 = sin(r25787);
        double r25791 = r25790 * r25785;
        double r25792 = expm1(r25791);
        double r25793 = log1p(r25792);
        double r25794 = -r25793;
        double r25795 = fma(r25788, r25789, r25794);
        double r25796 = r25786 / r25795;
        return r25796;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.8

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

  3. Applied cos-sum0.3

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

  5. Applied fma-neg0.3

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

  7. Applied associate-*r/0.3

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

  9. Applied log1p-expm1-u0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))