Average Error: 15.3 → 0.4
Time: 6.4s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\frac{\cos b \cdot \cos a - \sin a \cdot \sin b}{\sin b}}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\frac{\cos b \cdot \cos a - \sin a \cdot \sin b}{\sin b}}
double f(double r, double a, double b) {
        double r17475 = r;
        double r17476 = b;
        double r17477 = sin(r17476);
        double r17478 = a;
        double r17479 = r17478 + r17476;
        double r17480 = cos(r17479);
        double r17481 = r17477 / r17480;
        double r17482 = r17475 * r17481;
        return r17482;
}
↓
double f(double r, double a, double b) {
        double r17483 = r;
        double r17484 = b;
        double r17485 = cos(r17484);
        double r17486 = a;
        double r17487 = cos(r17486);
        double r17488 = r17485 * r17487;
        double r17489 = sin(r17486);
        double r17490 = sin(r17484);
        double r17491 = r17489 * r17490;
        double r17492 = r17488 - r17491;
        double r17493 = r17492 / r17490;
        double r17494 = r17483 / r17493;
        return r17494;
}

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 15.3

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 add-cbrt-cube0.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \color{blue}{\sqrt[3]{\left(\sin b \cdot \sin b\right) \cdot \sin b}}}\]

  6. Applied add-cbrt-cube0.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \color{blue}{\sqrt[3]{\left(\sin a \cdot \sin a\right) \cdot \sin a}} \cdot \sqrt[3]{\left(\sin b \cdot \sin b\right) \cdot \sin b}}\]

  7. Applied cbrt-unprod0.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \color{blue}{\sqrt[3]{\left(\left(\sin a \cdot \sin a\right) \cdot \sin a\right) \cdot \left(\left(\sin b \cdot \sin b\right) \cdot \sin b\right)}}}\]

  8. Simplified0.4

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

  10. Applied *-un-lft-identity0.4

    \[\leadsto \color{blue}{\left(1 \cdot r\right)} \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin a \cdot \sin b\right)}^{3}}}\]

  11. Applied associate-*l*0.4

    \[\leadsto \color{blue}{1 \cdot \left(r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin a \cdot \sin b\right)}^{3}}}\right)}\]

  12. Simplified0.3

    \[\leadsto 1 \cdot \color{blue}{\frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin a \cdot \sin b}}\]
  13. Using strategy rm

  14. Applied associate-/l*0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020056 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))