Average Error: 14.9 → 0.4
Time: 8.0s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\frac{\mathsf{fma}\left(\cos a \cdot 1, \cos b, -\sin b \cdot \sin a\right)}{\sin b}}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\frac{\mathsf{fma}\left(\cos a \cdot 1, \cos b, -\sin b \cdot \sin a\right)}{\sin b}}
double code(double r, double a, double b) {
	return ((double) (r * ((double) (((double) sin(b)) / ((double) cos(((double) (a + b))))))));
}
↓
double code(double r, double a, double b) {
	return ((double) (r / ((double) (((double) fma(((double) (((double) cos(a)) * 1.0)), ((double) cos(b)), ((double) -(((double) (((double) sin(b)) * ((double) sin(a)))))))) / ((double) sin(b))))));
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.9

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

  3. Applied cos-sum0.3

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

  5. Applied clear-num0.4

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

  6. Applied un-div-inv0.4

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

  8. Applied *-un-lft-identity0.4

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

  9. Applied associate-*r*0.4

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

  10. Applied fma-neg0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))