Average Error: 14.9 → 0.3
Time: 23.2s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(-\sin a, \sin b, \cos b \cdot \cos a\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\mathsf{fma}\left(-\sin a, \sin b, \cos b \cdot \cos a\right)}
double f(double r, double a, double b) {
        double r24042 = r;
        double r24043 = b;
        double r24044 = sin(r24043);
        double r24045 = r24042 * r24044;
        double r24046 = a;
        double r24047 = r24046 + r24043;
        double r24048 = cos(r24047);
        double r24049 = r24045 / r24048;
        return r24049;
}
↓
double f(double r, double a, double b) {
        double r24050 = r;
        double r24051 = b;
        double r24052 = sin(r24051);
        double r24053 = r24050 * r24052;
        double r24054 = a;
        double r24055 = sin(r24054);
        double r24056 = -r24055;
        double r24057 = cos(r24051);
        double r24058 = cos(r24054);
        double r24059 = r24057 * r24058;
        double r24060 = fma(r24056, r24052, r24059);
        double r24061 = r24053 / r24060;
        return r24061;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.9

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

  3. Applied cos-sum0.3

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

  4. Taylor expanded around inf 0.3

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

  5. Simplified0.3

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

  6. Taylor expanded around inf 0.3

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

  7. Simplified0.3

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

  8. Taylor expanded around inf 0.3

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

  9. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))