Average Error: 15.2 → 0.3
Time: 23.7s
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 r1088955 = r;
        double r1088956 = b;
        double r1088957 = sin(r1088956);
        double r1088958 = r1088955 * r1088957;
        double r1088959 = a;
        double r1088960 = r1088959 + r1088956;
        double r1088961 = cos(r1088960);
        double r1088962 = r1088958 / r1088961;
        return r1088962;
}
↓
double f(double r, double a, double b) {
        double r1088963 = r;
        double r1088964 = b;
        double r1088965 = sin(r1088964);
        double r1088966 = r1088963 * r1088965;
        double r1088967 = a;
        double r1088968 = cos(r1088967);
        double r1088969 = cos(r1088964);
        double r1088970 = sin(r1088967);
        double r1088971 = r1088965 * r1088970;
        double r1088972 = expm1(r1088971);
        double r1088973 = log1p(r1088972);
        double r1088974 = -r1088973;
        double r1088975 = fma(r1088968, r1088969, r1088974);
        double r1088976 = r1088966 / r1088975;
        return r1088976;
}

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))))