Average Error: 15.1 → 0.3
Time: 21.0s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, \left(-\sin b\right) \cdot \sin a\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, \left(-\sin b\right) \cdot \sin a\right)}
double f(double r, double a, double b) {
        double r1120136 = r;
        double r1120137 = b;
        double r1120138 = sin(r1120137);
        double r1120139 = a;
        double r1120140 = r1120139 + r1120137;
        double r1120141 = cos(r1120140);
        double r1120142 = r1120138 / r1120141;
        double r1120143 = r1120136 * r1120142;
        return r1120143;
}
↓
double f(double r, double a, double b) {
        double r1120144 = r;
        double r1120145 = b;
        double r1120146 = sin(r1120145);
        double r1120147 = r1120144 * r1120146;
        double r1120148 = a;
        double r1120149 = cos(r1120148);
        double r1120150 = cos(r1120145);
        double r1120151 = -r1120146;
        double r1120152 = sin(r1120148);
        double r1120153 = r1120151 * r1120152;
        double r1120154 = fma(r1120149, r1120150, r1120153);
        double r1120155 = r1120147 / r1120154;
        return r1120155;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.1

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))