Average Error: 15.1 → 0.3
Time: 26.5s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(\left(\cos a\right), \left(\cos b\right), \left(\sin b \cdot \left(-\sin a\right)\right)\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\mathsf{fma}\left(\left(\cos a\right), \left(\cos b\right), \left(\sin b \cdot \left(-\sin a\right)\right)\right)}
double f(double r, double a, double b) {
        double r812036 = r;
        double r812037 = b;
        double r812038 = sin(r812037);
        double r812039 = a;
        double r812040 = r812039 + r812037;
        double r812041 = cos(r812040);
        double r812042 = r812038 / r812041;
        double r812043 = r812036 * r812042;
        return r812043;
}
↓
double f(double r, double a, double b) {
        double r812044 = r;
        double r812045 = b;
        double r812046 = sin(r812045);
        double r812047 = r812044 * r812046;
        double r812048 = a;
        double r812049 = cos(r812048);
        double r812050 = cos(r812045);
        double r812051 = sin(r812048);
        double r812052 = -r812051;
        double r812053 = r812046 * r812052;
        double r812054 = fma(r812049, r812050, r812053);
        double r812055 = r812047 / r812054;
        return r812055;
}

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 add-log-exp0.4

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

  6. Taylor expanded around inf 0.4

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

  8. Applied fma-neg0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))