Average Error: 14.9 → 0.3
Time: 39.3s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\left(\cos a\right), \left(\cos b\right), \left(\left(-\sin b\right) \cdot \sin a\right)\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\mathsf{fma}\left(\left(\cos a\right), \left(\cos b\right), \left(\left(-\sin b\right) \cdot \sin a\right)\right)}
double f(double r, double a, double b) {
        double r1342191 = r;
        double r1342192 = b;
        double r1342193 = sin(r1342192);
        double r1342194 = a;
        double r1342195 = r1342194 + r1342192;
        double r1342196 = cos(r1342195);
        double r1342197 = r1342193 / r1342196;
        double r1342198 = r1342191 * r1342197;
        return r1342198;
}
↓
double f(double r, double a, double b) {
        double r1342199 = r;
        double r1342200 = b;
        double r1342201 = sin(r1342200);
        double r1342202 = a;
        double r1342203 = cos(r1342202);
        double r1342204 = cos(r1342200);
        double r1342205 = -r1342201;
        double r1342206 = sin(r1342202);
        double r1342207 = r1342205 * r1342206;
        double r1342208 = fma(r1342203, r1342204, r1342207);
        double r1342209 = r1342201 / r1342208;
        double r1342210 = r1342199 * r1342209;
        return r1342210;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.9

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

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))