Average Error: 14.5 → 0.3
Time: 17.8s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, \left(-\sin b\right) \cdot \sin a\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, \left(-\sin b\right) \cdot \sin a\right)}
double f(double r, double a, double b) {
        double r25957 = r;
        double r25958 = b;
        double r25959 = sin(r25958);
        double r25960 = r25957 * r25959;
        double r25961 = a;
        double r25962 = r25961 + r25958;
        double r25963 = cos(r25962);
        double r25964 = r25960 / r25963;
        return r25964;
}
↓
double f(double r, double a, double b) {
        double r25965 = r;
        double r25966 = b;
        double r25967 = sin(r25966);
        double r25968 = a;
        double r25969 = cos(r25968);
        double r25970 = cos(r25966);
        double r25971 = -r25967;
        double r25972 = sin(r25968);
        double r25973 = r25971 * r25972;
        double r25974 = fma(r25969, r25970, r25973);
        double r25975 = r25967 / r25974;
        double r25976 = r25965 * r25975;
        return r25976;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.5

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]

  2. Simplified14.5

    \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}}\]
  3. Using strategy rm

  4. Applied cos-sum0.3

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

  5. Simplified0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a \cdot \cos b} - \sin b \cdot \sin a}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity0.3

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

  8. Applied times-frac0.3

    \[\leadsto \color{blue}{\frac{r}{1} \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a}}\]

  9. Simplified0.3

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

  10. Simplified0.3

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

  11. Taylor expanded around inf 0.3

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))