Average Error: 14.9 → 0.4
Time: 7.1s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\frac{\mathsf{fma}\left(\cos a, \cos b, \sin a \cdot \sin b\right) \cdot \left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)}{\mathsf{fma}\left(\cos a, \cos b, \sin a \cdot \sin b\right)}}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\frac{\mathsf{fma}\left(\cos a, \cos b, \sin a \cdot \sin b\right) \cdot \left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)}{\mathsf{fma}\left(\cos a, \cos b, \sin a \cdot \sin b\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(a), cos(b), (sin(a) * sin(b))) * ((cos(a) * cos(b)) - (sin(a) * sin(b)))) / fma(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 14.9

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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. Simplified0.4

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

  7. Simplified0.4

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

  8. Final simplification0.4

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

Reproduce

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