Average Error: 14.8 → 0.5
Time: 6.0s
Precision: binary64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\left(r \cdot \sin b\right) \cdot \left(\frac{\cos a \cdot \cos b}{{\left(\cos b\right)}^{2} \cdot {\left(\cos a\right)}^{2} - {\left(\sin a\right)}^{2} \cdot {\left(\sin b\right)}^{2}} + \frac{\sin a \cdot \sin b}{{\left(\cos b\right)}^{2} \cdot {\left(\cos a\right)}^{2} - {\left(\sin a\right)}^{2} \cdot {\left(\sin b\right)}^{2}}\right)\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\left(r \cdot \sin b\right) \cdot \left(\frac{\cos a \cdot \cos b}{{\left(\cos b\right)}^{2} \cdot {\left(\cos a\right)}^{2} - {\left(\sin a\right)}^{2} \cdot {\left(\sin b\right)}^{2}} + \frac{\sin a \cdot \sin b}{{\left(\cos b\right)}^{2} \cdot {\left(\cos a\right)}^{2} - {\left(\sin a\right)}^{2} \cdot {\left(\sin b\right)}^{2}}\right)
double code(double r, double a, double b) {
	return ((double) (((double) (r * ((double) sin(b)))) / ((double) cos(((double) (a + b))))));
}
↓
double code(double r, double a, double b) {
	return ((double) (((double) (r * ((double) sin(b)))) * ((double) (((double) (((double) (((double) cos(a)) * ((double) cos(b)))) / ((double) (((double) (((double) pow(((double) cos(b)), 2.0)) * ((double) pow(((double) cos(a)), 2.0)))) - ((double) (((double) pow(((double) sin(a)), 2.0)) * ((double) pow(((double) sin(b)), 2.0)))))))) + ((double) (((double) (((double) sin(a)) * ((double) sin(b)))) / ((double) (((double) (((double) pow(((double) cos(b)), 2.0)) * ((double) pow(((double) cos(a)), 2.0)))) - ((double) (((double) pow(((double) sin(a)), 2.0)) * ((double) pow(((double) sin(b)), 2.0))))))))))));
}

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 14.8

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 div-inv0.4

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

  7. Applied flip--0.4

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

  8. Applied associate-/r/0.5

    \[\leadsto \left(r \cdot \sin b\right) \cdot \color{blue}{\left(\frac{1}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)} \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)\right)}\]
  9. Using strategy rm

  10. Applied distribute-lft-in0.5

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

  11. Simplified0.5

    \[\leadsto \left(r \cdot \sin b\right) \cdot \left(\color{blue}{\frac{\cos a \cdot \cos b}{{\left(\cos b\right)}^{2} \cdot {\left(\cos a\right)}^{2} - {\left(\sin a\right)}^{2} \cdot {\left(\sin b\right)}^{2}}} + \frac{1}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)} \cdot \left(\sin a \cdot \sin b\right)\right)\]

  12. Simplified0.5

    \[\leadsto \left(r \cdot \sin b\right) \cdot \left(\frac{\cos a \cdot \cos b}{{\left(\cos b\right)}^{2} \cdot {\left(\cos a\right)}^{2} - {\left(\sin a\right)}^{2} \cdot {\left(\sin b\right)}^{2}} + \color{blue}{\frac{\sin a \cdot \sin b}{{\left(\cos b\right)}^{2} \cdot {\left(\cos a\right)}^{2} - {\left(\sin a\right)}^{2} \cdot {\left(\sin b\right)}^{2}}}\right)\]

  13. Final simplification0.5

    \[\leadsto \left(r \cdot \sin b\right) \cdot \left(\frac{\cos a \cdot \cos b}{{\left(\cos b\right)}^{2} \cdot {\left(\cos a\right)}^{2} - {\left(\sin a\right)}^{2} \cdot {\left(\sin b\right)}^{2}} + \frac{\sin a \cdot \sin b}{{\left(\cos b\right)}^{2} \cdot {\left(\cos a\right)}^{2} - {\left(\sin a\right)}^{2} \cdot {\left(\sin b\right)}^{2}}\right)\]

Reproduce

herbie shell --seed 2020162 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))