Average Error: 14.7 → 0.3
Time: 24.7s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot \sin b\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot \sin b
double f(double r, double a, double b) {
        double r810553 = r;
        double r810554 = b;
        double r810555 = sin(r810554);
        double r810556 = r810553 * r810555;
        double r810557 = a;
        double r810558 = r810557 + r810554;
        double r810559 = cos(r810558);
        double r810560 = r810556 / r810559;
        return r810560;
}
↓
double f(double r, double a, double b) {
        double r810561 = r;
        double r810562 = a;
        double r810563 = cos(r810562);
        double r810564 = b;
        double r810565 = cos(r810564);
        double r810566 = r810563 * r810565;
        double r810567 = sin(r810564);
        double r810568 = sin(r810562);
        double r810569 = r810567 * r810568;
        double r810570 = r810566 - r810569;
        double r810571 = r810561 / r810570;
        double r810572 = r810571 * r810567;
        return r810572;
}

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 14.7

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

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

  10. Applied *-un-lft-identity0.3

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

  11. Applied times-frac0.3

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019132 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))