Average Error: 14.7 → 0.4
Time: 34.0s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)\right)}
double f(double r, double a, double b) {
        double r24684 = r;
        double r24685 = b;
        double r24686 = sin(r24685);
        double r24687 = a;
        double r24688 = r24687 + r24685;
        double r24689 = cos(r24688);
        double r24690 = r24686 / r24689;
        double r24691 = r24684 * r24690;
        return r24691;
}
↓
double f(double r, double a, double b) {
        double r24692 = r;
        double r24693 = b;
        double r24694 = sin(r24693);
        double r24695 = a;
        double r24696 = cos(r24695);
        double r24697 = cos(r24693);
        double r24698 = r24696 * r24697;
        double r24699 = sin(r24695);
        double r24700 = r24699 * r24694;
        double r24701 = r24698 - r24700;
        double r24702 = log1p(r24701);
        double r24703 = expm1(r24702);
        double r24704 = r24694 / r24703;
        double r24705 = r24692 * r24704;
        return r24705;
}

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 14.7

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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-u0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))