Average Error: 14.7 → 0.4
Time: 23.7s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\mathsf{fma}\left(\cos b, \frac{\cos a}{\sin b}, -\sin a\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\mathsf{fma}\left(\cos b, \frac{\cos a}{\sin b}, -\sin a\right)}
double f(double r, double a, double b) {
        double r27548 = r;
        double r27549 = b;
        double r27550 = sin(r27549);
        double r27551 = a;
        double r27552 = r27551 + r27549;
        double r27553 = cos(r27552);
        double r27554 = r27550 / r27553;
        double r27555 = r27548 * r27554;
        return r27555;
}
↓
double f(double r, double a, double b) {
        double r27556 = r;
        double r27557 = b;
        double r27558 = cos(r27557);
        double r27559 = a;
        double r27560 = cos(r27559);
        double r27561 = sin(r27557);
        double r27562 = r27560 / r27561;
        double r27563 = sin(r27559);
        double r27564 = -r27563;
        double r27565 = fma(r27558, r27562, r27564);
        double r27566 = r27556 / r27565;
        return r27566;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.7

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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. Taylor expanded around inf 0.3

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

  5. Simplified0.4

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

  6. Taylor expanded around inf 0.4

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

  7. Simplified0.4

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

  8. Taylor expanded around inf 0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))