Average Error: 15.0 → 0.4
Time: 7.2s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\mathsf{fma}\left(\cos b, \cos a, \sin a \cdot \sin b\right)} \cdot \frac{\sin b \cdot \mathsf{fma}\left(\cos a, \cos b, \sin a \cdot \sin b\right)}{\cos b \cdot \cos a - \sin a \cdot \sin b}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\mathsf{fma}\left(\cos b, \cos a, \sin a \cdot \sin b\right)} \cdot \frac{\sin b \cdot \mathsf{fma}\left(\cos a, \cos b, \sin a \cdot \sin b\right)}{\cos b \cdot \cos a - \sin a \cdot \sin b}
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(b), cos(a), (sin(a) * sin(b)))) * ((sin(b) * fma(cos(a), cos(b), (sin(a) * sin(b)))) / ((cos(b) * cos(a)) - (sin(a) * 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 15.0

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 flip--0.4

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

  6. Applied associate-/r/0.4

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

  7. Simplified0.5

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

  9. Applied associate-*l*0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))