Average Error: 14.8 → 0.3
Time: 20.3s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(-\sin b, \sin a, \cos a \cdot \cos b\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\mathsf{fma}\left(-\sin b, \sin a, \cos a \cdot \cos b\right)}
double f(double r, double a, double b) {
        double r27129 = r;
        double r27130 = b;
        double r27131 = sin(r27130);
        double r27132 = a;
        double r27133 = r27132 + r27130;
        double r27134 = cos(r27133);
        double r27135 = r27131 / r27134;
        double r27136 = r27129 * r27135;
        return r27136;
}
↓
double f(double r, double a, double b) {
        double r27137 = r;
        double r27138 = b;
        double r27139 = sin(r27138);
        double r27140 = r27137 * r27139;
        double r27141 = -r27139;
        double r27142 = a;
        double r27143 = sin(r27142);
        double r27144 = cos(r27142);
        double r27145 = cos(r27138);
        double r27146 = r27144 * r27145;
        double r27147 = fma(r27141, r27143, r27146);
        double r27148 = r27140 / r27147;
        return r27148;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.8

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

  2. Simplified14.8

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

  4. Applied cos-sum0.3

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

  5. Simplified0.3

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

  6. Simplified0.3

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

  7. Taylor expanded around inf 0.3

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))