Average Error: 14.8 → 0.3
Time: 24.0s
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 r25787 = r;
        double r25788 = b;
        double r25789 = sin(r25788);
        double r25790 = a;
        double r25791 = r25790 + r25788;
        double r25792 = cos(r25791);
        double r25793 = r25789 / r25792;
        double r25794 = r25787 * r25793;
        return r25794;
}
↓
double f(double r, double a, double b) {
        double r25795 = r;
        double r25796 = b;
        double r25797 = sin(r25796);
        double r25798 = r25795 * r25797;
        double r25799 = a;
        double r25800 = cos(r25799);
        double r25801 = cos(r25796);
        double r25802 = sin(r25799);
        double r25803 = r25802 * r25797;
        double r25804 = expm1(r25803);
        double r25805 = log1p(r25804);
        double r25806 = -r25805;
        double r25807 = fma(r25800, r25801, r25806);
        double r25808 = r25798 / r25807;
        return r25808;
}

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)))))