Average Error: 15.1 → 0.4
Time: 6.4s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, -\log \left(e^{\sin a \cdot \sin b}\right)\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, -\log \left(e^{\sin a \cdot \sin b}\right)\right)}
double f(double r, double a, double b) {
        double r17257 = r;
        double r17258 = b;
        double r17259 = sin(r17258);
        double r17260 = r17257 * r17259;
        double r17261 = a;
        double r17262 = r17261 + r17258;
        double r17263 = cos(r17262);
        double r17264 = r17260 / r17263;
        return r17264;
}
↓
double f(double r, double a, double b) {
        double r17265 = r;
        double r17266 = b;
        double r17267 = sin(r17266);
        double r17268 = r17265 * r17267;
        double r17269 = a;
        double r17270 = cos(r17269);
        double r17271 = cos(r17266);
        double r17272 = sin(r17269);
        double r17273 = r17272 * r17267;
        double r17274 = exp(r17273);
        double r17275 = log(r17274);
        double r17276 = -r17275;
        double r17277 = fma(r17270, r17271, r17276);
        double r17278 = r17268 / r17277;
        return r17278;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.1

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 fma-neg0.3

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

  7. Applied add-log-exp0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2020003 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))