Average Error: 15.0 → 0.3
Time: 14.2s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, -\sin b \cdot \sin a\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, -\sin b \cdot \sin a\right)}
double f(double r, double a, double b) {
        double r17258 = r;
        double r17259 = b;
        double r17260 = sin(r17259);
        double r17261 = a;
        double r17262 = r17261 + r17259;
        double r17263 = cos(r17262);
        double r17264 = r17260 / r17263;
        double r17265 = r17258 * r17264;
        return r17265;
}
↓
double f(double r, double a, double b) {
        double r17266 = r;
        double r17267 = b;
        double r17268 = sin(r17267);
        double r17269 = r17266 * r17268;
        double r17270 = cos(r17267);
        double r17271 = a;
        double r17272 = cos(r17271);
        double r17273 = sin(r17271);
        double r17274 = r17268 * r17273;
        double r17275 = -r17274;
        double r17276 = fma(r17270, r17272, r17275);
        double r17277 = r17269 / r17276;
        return r17277;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.0

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 associate-*r/0.3

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

  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

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))