Average Error: 15.0 → 0.4
Time: 7.1s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \left(\frac{\frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin a \cdot \sin b\right)}}{\cos b \cdot \cos a - \sin a \cdot \sin b} \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)\right)\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
r \cdot \left(\frac{\frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin a \cdot \sin b\right)}}{\cos b \cdot \cos a - \sin a \cdot \sin b} \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)\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 * (((sin(b) / fma(cos(b), cos(a), (sin(a) * sin(b)))) / ((cos(b) * cos(a)) - (sin(a) * sin(b)))) * ((cos(a) * cos(b)) + (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

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

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

  6. Applied associate-/r/0.4

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

  7. Simplified0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2020092 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))