Average Error: 15.3 → 0.4
Time: 6.1s
Precision: binary64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\sin b \cdot r}{\frac{\cos b \cdot \left(\cos b \cdot \left(\cos a \cdot \cos a\right)\right) - \sin b \cdot \left(\sin b \cdot \left(\sin a \cdot \sin a\right)\right)}{\cos b \cdot \cos a + \sin b \cdot \sin a}}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{\sin b \cdot r}{\frac{\cos b \cdot \left(\cos b \cdot \left(\cos a \cdot \cos a\right)\right) - \sin b \cdot \left(\sin b \cdot \left(\sin a \cdot \sin a\right)\right)}{\cos b \cdot \cos a + \sin b \cdot \sin a}}
double code(double r, double a, double b) {
	return (((double) (r * ((double) sin(b)))) / ((double) cos(((double) (a + b)))));
}
↓
double code(double r, double a, double b) {
	return (((double) (((double) sin(b)) * r)) / (((double) (((double) (((double) cos(b)) * ((double) (((double) cos(b)) * ((double) (((double) cos(a)) * ((double) cos(a)))))))) - ((double) (((double) sin(b)) * ((double) (((double) sin(b)) * ((double) (((double) sin(a)) * ((double) sin(a)))))))))) / ((double) (((double) (((double) cos(b)) * ((double) cos(a)))) + ((double) (((double) sin(b)) * ((double) 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 15.3

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

  2. Simplified15.3

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

  4. Applied cos-sum0.3

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

  6. Applied associate-*r/0.3

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

  7. Simplified0.3

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

  9. Applied flip--0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020182 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))