Average Error: 15.1 → 0.4
Time: 6.6s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\mathsf{fma}\left(\frac{\cos a}{\sin b}, \cos b, -\sin a\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\mathsf{fma}\left(\frac{\cos a}{\sin b}, \cos b, -\sin a\right)}
double code(double r, double a, double b) {
	return (r * (sin(b) / cos((a + b))));
}
↓
double code(double r, double a, double b) {
	return (r / fma((cos(a) / sin(b)), cos(b), -sin(a)));
}

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 15.1

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 associate-*r/0.3

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

  7. Applied associate-/l*0.4

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

  8. Taylor expanded around inf 0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))