Average Error: 14.7 → 0.3
Time: 24.2s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\sin b \cdot \frac{r}{\mathsf{fma}\left(-\sin a, \sin b, \cos a \cdot \cos b\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\sin b \cdot \frac{r}{\mathsf{fma}\left(-\sin a, \sin b, \cos a \cdot \cos b\right)}
double f(double r, double a, double b) {
        double r25031 = r;
        double r25032 = b;
        double r25033 = sin(r25032);
        double r25034 = r25031 * r25033;
        double r25035 = a;
        double r25036 = r25035 + r25032;
        double r25037 = cos(r25036);
        double r25038 = r25034 / r25037;
        return r25038;
}
↓
double f(double r, double a, double b) {
        double r25039 = b;
        double r25040 = sin(r25039);
        double r25041 = r;
        double r25042 = a;
        double r25043 = sin(r25042);
        double r25044 = -r25043;
        double r25045 = cos(r25042);
        double r25046 = cos(r25039);
        double r25047 = r25045 * r25046;
        double r25048 = fma(r25044, r25040, r25047);
        double r25049 = r25041 / r25048;
        double r25050 = r25040 * r25049;
        return r25050;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.7

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

  5. Applied prod-diff0.3

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

  6. Simplified0.3

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

  7. Simplified0.3

    \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(-\sin a, \sin b, \cos a \cdot \cos b\right) + \color{blue}{\left(0 \cdot \sin a\right) \cdot \sin b}}\]

  8. Taylor expanded around inf 0.3

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

  9. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))