Average Error: 15.0 → 0.3
Time: 7.2s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\cos a \cdot \cos b - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\cos a \cdot \cos b - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)}
double f(double r, double a, double b) {
        double r17681 = r;
        double r17682 = b;
        double r17683 = sin(r17682);
        double r17684 = a;
        double r17685 = r17684 + r17682;
        double r17686 = cos(r17685);
        double r17687 = r17683 / r17686;
        double r17688 = r17681 * r17687;
        return r17688;
}
↓
double f(double r, double a, double b) {
        double r17689 = r;
        double r17690 = b;
        double r17691 = sin(r17690);
        double r17692 = a;
        double r17693 = cos(r17692);
        double r17694 = cos(r17690);
        double r17695 = r17693 * r17694;
        double r17696 = sin(r17692);
        double r17697 = r17696 * r17691;
        double r17698 = log1p(r17697);
        double r17699 = expm1(r17698);
        double r17700 = r17695 - r17699;
        double r17701 = r17691 / r17700;
        double r17702 = r17689 * r17701;
        return r17702;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.0

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 expm1-log1p-u0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))