Average Error: 15.4 → 0.3
Time: 36.2s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\left(\cos a\right), \left(\cos b\right), \left(\left(-\sin b\right) \cdot \sin a\right)\right)}\]
\frac{r \cdot \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 r1195026 = r;
        double r1195027 = b;
        double r1195028 = sin(r1195027);
        double r1195029 = r1195026 * r1195028;
        double r1195030 = a;
        double r1195031 = r1195030 + r1195027;
        double r1195032 = cos(r1195031);
        double r1195033 = r1195029 / r1195032;
        return r1195033;
}
↓
double f(double r, double a, double b) {
        double r1195034 = r;
        double r1195035 = b;
        double r1195036 = sin(r1195035);
        double r1195037 = a;
        double r1195038 = cos(r1195037);
        double r1195039 = cos(r1195035);
        double r1195040 = -r1195036;
        double r1195041 = sin(r1195037);
        double r1195042 = r1195040 * r1195041;
        double r1195043 = fma(r1195038, r1195039, r1195042);
        double r1195044 = r1195036 / r1195043;
        double r1195045 = r1195034 * r1195044;
        return r1195045;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.4

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  2. Using strategy rm

  3. Applied cos-sum0.4

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.4

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{1 \cdot \left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)}}\]

  6. Applied times-frac0.3

    \[\leadsto \color{blue}{\frac{r}{1} \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]

  7. Simplified0.3

    \[\leadsto \color{blue}{r} \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}\]
  8. Using strategy rm

  9. Applied fma-neg0.3

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

  10. 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 2019124 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))