Average Error: 14.8 → 0.3
Time: 18.1s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}\]
r \cdot \frac{\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 r26699 = r;
        double r26700 = b;
        double r26701 = sin(r26700);
        double r26702 = a;
        double r26703 = r26702 + r26700;
        double r26704 = cos(r26703);
        double r26705 = r26701 / r26704;
        double r26706 = r26699 * r26705;
        return r26706;
}
↓
double f(double r, double a, double b) {
        double r26707 = r;
        double r26708 = b;
        double r26709 = sin(r26708);
        double r26710 = r26707 * r26709;
        double r26711 = a;
        double r26712 = cos(r26711);
        double r26713 = cos(r26708);
        double r26714 = r26712 * r26713;
        double r26715 = sin(r26711);
        double r26716 = r26715 * r26709;
        double r26717 = r26714 - r26716;
        double r26718 = r26710 / r26717;
        return r26718;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.8

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

  2. Simplified14.8

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

  4. Applied cos-sum0.3

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

  5. Simplified0.3

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

  6. Simplified0.3

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

  8. Applied associate-*r/0.3

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

  9. Simplified0.3

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

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

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

  12. Applied times-frac0.3

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

  13. Simplified0.3

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

  15. Applied associate-*r/0.3

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

  16. Simplified0.3

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

  17. Final simplification0.3

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

Reproduce

herbie shell --seed 2019195 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))