Average Error: 15.1 → 0.5
Time: 6.7s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\left(\sin a \cdot \sin b + \cos a \cdot \cos b\right) \cdot \left(r \cdot \frac{\sin b}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)}\right)\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\left(\sin a \cdot \sin b + \cos a \cdot \cos b\right) \cdot \left(r \cdot \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)}\right)
double f(double r, double a, double b) {
        double r16204 = r;
        double r16205 = b;
        double r16206 = sin(r16205);
        double r16207 = a;
        double r16208 = r16207 + r16205;
        double r16209 = cos(r16208);
        double r16210 = r16206 / r16209;
        double r16211 = r16204 * r16210;
        return r16211;
}
↓
double f(double r, double a, double b) {
        double r16212 = a;
        double r16213 = sin(r16212);
        double r16214 = b;
        double r16215 = sin(r16214);
        double r16216 = r16213 * r16215;
        double r16217 = cos(r16212);
        double r16218 = cos(r16214);
        double r16219 = r16217 * r16218;
        double r16220 = r16216 + r16219;
        double r16221 = r;
        double r16222 = r16219 * r16219;
        double r16223 = r16216 * r16216;
        double r16224 = r16222 - r16223;
        double r16225 = r16215 / r16224;
        double r16226 = r16221 * r16225;
        double r16227 = r16220 * r16226;
        return r16227;
}

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 15.1

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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. Applied associate-*r*0.5

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

  9. Applied *-un-lft-identity0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2020003 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))