Average Error: 15.2 → 0.3
Time: 23.3s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, -\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, -\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)\right)}
double f(double r, double a, double b) {
        double r1088953 = r;
        double r1088954 = b;
        double r1088955 = sin(r1088954);
        double r1088956 = r1088953 * r1088955;
        double r1088957 = a;
        double r1088958 = r1088957 + r1088954;
        double r1088959 = cos(r1088958);
        double r1088960 = r1088956 / r1088959;
        return r1088960;
}
↓
double f(double r, double a, double b) {
        double r1088961 = r;
        double r1088962 = b;
        double r1088963 = sin(r1088962);
        double r1088964 = r1088961 * r1088963;
        double r1088965 = a;
        double r1088966 = cos(r1088965);
        double r1088967 = cos(r1088962);
        double r1088968 = sin(r1088965);
        double r1088969 = r1088963 * r1088968;
        double r1088970 = expm1(r1088969);
        double r1088971 = log1p(r1088970);
        double r1088972 = -r1088971;
        double r1088973 = fma(r1088966, r1088967, r1088972);
        double r1088974 = r1088964 / r1088973;
        return r1088974;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.2

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 *-un-lft-identity0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{1 \cdot \mathsf{fma}\left(\cos a, \cos b, -\sin a \cdot \sin b\right)}}\]

  8. Applied associate-/r*0.3

    \[\leadsto \color{blue}{\frac{\frac{r \cdot \sin b}{1}}{\mathsf{fma}\left(\cos a, \cos b, -\sin a \cdot \sin b\right)}}\]

  9. Simplified0.3

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

  11. Applied log1p-expm1-u0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))