Average Error: 15.1 → 0.4
Time: 32.8s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\left(\frac{1}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot \sin b\right) \cdot r\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\left(\frac{1}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot \sin b\right) \cdot r
double f(double r, double a, double b) {
        double r749231 = r;
        double r749232 = b;
        double r749233 = sin(r749232);
        double r749234 = r749231 * r749233;
        double r749235 = a;
        double r749236 = r749235 + r749232;
        double r749237 = cos(r749236);
        double r749238 = r749234 / r749237;
        return r749238;
}
↓
double f(double r, double a, double b) {
        double r749239 = 1.0;
        double r749240 = a;
        double r749241 = cos(r749240);
        double r749242 = b;
        double r749243 = cos(r749242);
        double r749244 = r749241 * r749243;
        double r749245 = sin(r749242);
        double r749246 = sin(r749240);
        double r749247 = r749245 * r749246;
        double r749248 = r749244 - r749247;
        double r749249 = r749239 / r749248;
        double r749250 = r749249 * r749245;
        double r749251 = r;
        double r749252 = r749250 * r749251;
        return r749252;
}

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

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 *-un-lft-identity0.3

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

  6. Applied times-frac0.3

    \[\leadsto \color{blue}{\frac{r}{1} \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]

  7. Simplified0.3

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

  9. Applied div-inv0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))