Average Error: 14.9 → 0.4
Time: 5.4s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\mathsf{fma}\left(\frac{\cos a}{\sin b}, \cos b, \frac{-\sin a}{1}\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\mathsf{fma}\left(\frac{\cos a}{\sin b}, \cos b, \frac{-\sin a}{1}\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) / 1.0)));
}

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

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 *-commutative0.3

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

  6. Applied fma-neg0.3

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

  8. Applied associate-/l*0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))