r*sin(b)/cos(a+b), B

Average Error: 15.4 → 0.3

Time: 20.6s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r27944 = r;
        double r27945 = b;
        double r27946 = sin(r27945);
        double r27947 = a;
        double r27948 = r27947 + r27945;
        double r27949 = cos(r27948);
        double r27950 = r27946 / r27949;
        double r27951 = r27944 * r27950;
        return r27951;
}

↓

double f(double r, double a, double b) {
        double r27952 = r;
        double r27953 = b;
        double r27954 = sin(r27953);
        double r27955 = r27952 * r27954;
        double r27956 = cos(r27953);
        double r27957 = a;
        double r27958 = cos(r27957);
        double r27959 = sin(r27957);
        double r27960 = -r27959;
        double r27961 = r27960 * r27954;
        double r27962 = fma(r27956, r27958, r27961);
        double r27963 = r27955 / r27962;
        return r27963;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.4

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

2. Simplified 15.4

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

4. Applied cos-sum 0.3

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

5. Simplified 0.3

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

6. Simplified 0.3

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

8. Applied fma-neg 0.3

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

9. Simplified 0.3

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

11. Applied pow1 0.3

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

12. Applied pow1 0.3

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

13. Applied pow-prod-down 0.3

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

14. Simplified 0.3

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

15. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))