Average Error: 15.5 → 0.3
Time: 21.6s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}
double f(double r, double a, double b) {
        double r1030387 = r;
        double r1030388 = b;
        double r1030389 = sin(r1030388);
        double r1030390 = r1030387 * r1030389;
        double r1030391 = a;
        double r1030392 = r1030391 + r1030388;
        double r1030393 = cos(r1030392);
        double r1030394 = r1030390 / r1030393;
        return r1030394;
}
↓
double f(double r, double a, double b) {
        double r1030395 = r;
        double r1030396 = b;
        double r1030397 = sin(r1030396);
        double r1030398 = r1030395 * r1030397;
        double r1030399 = a;
        double r1030400 = cos(r1030399);
        double r1030401 = cos(r1030396);
        double r1030402 = r1030400 * r1030401;
        double r1030403 = sin(r1030399);
        double r1030404 = r1030403 * r1030397;
        double r1030405 = r1030402 - r1030404;
        double r1030406 = r1030398 / r1030405;
        return r1030406;
}

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 15.5

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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. Taylor expanded around inf 0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))