Average Error: 14.9 → 0.4
Time: 28.7s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\mathsf{fma}\left(\left(\cos a\right), \left(\cos b\right), \left(\sin a \cdot \left(-\sin b\right)\right)\right)\right)\right)\right)\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\mathsf{fma}\left(\left(\cos a\right), \left(\cos b\right), \left(\sin a \cdot \left(-\sin b\right)\right)\right)\right)\right)\right)\right)}
double f(double r, double a, double b) {
        double r1020221 = r;
        double r1020222 = b;
        double r1020223 = sin(r1020222);
        double r1020224 = r1020221 * r1020223;
        double r1020225 = a;
        double r1020226 = r1020225 + r1020222;
        double r1020227 = cos(r1020226);
        double r1020228 = r1020224 / r1020227;
        return r1020228;
}
↓
double f(double r, double a, double b) {
        double r1020229 = r;
        double r1020230 = b;
        double r1020231 = sin(r1020230);
        double r1020232 = r1020229 * r1020231;
        double r1020233 = a;
        double r1020234 = cos(r1020233);
        double r1020235 = cos(r1020230);
        double r1020236 = sin(r1020233);
        double r1020237 = -r1020231;
        double r1020238 = r1020236 * r1020237;
        double r1020239 = fma(r1020234, r1020235, r1020238);
        double r1020240 = expm1(r1020239);
        double r1020241 = log1p(r1020240);
        double r1020242 = r1020232 / r1020241;
        return r1020242;
}

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. Using strategy rm

  5. Applied fma-neg0.3

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

  7. Applied log1p-expm1-u0.4

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\mathsf{fma}\left(\left(\cos a\right), \left(\cos b\right), \left(-\sin a \cdot \sin b\right)\right)\right)\right)\right)\right)}}\]

  8. Final simplification0.4

    \[\leadsto \frac{r \cdot \sin b}{\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\mathsf{fma}\left(\left(\cos a\right), \left(\cos b\right), \left(\sin a \cdot \left(-\sin b\right)\right)\right)\right)\right)\right)\right)}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))