Average Error: 15.1 → 0.4
Time: 6.5s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sqrt[3]{{\left(\sin a \cdot \sin b\right)}^{3}}}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sqrt[3]{{\left(\sin a \cdot \sin b\right)}^{3}}}
double f(double r, double a, double b) {
        double r16494 = r;
        double r16495 = b;
        double r16496 = sin(r16495);
        double r16497 = r16494 * r16496;
        double r16498 = a;
        double r16499 = r16498 + r16495;
        double r16500 = cos(r16499);
        double r16501 = r16497 / r16500;
        return r16501;
}
↓
double f(double r, double a, double b) {
        double r16502 = r;
        double r16503 = b;
        double r16504 = sin(r16503);
        double r16505 = cos(r16503);
        double r16506 = a;
        double r16507 = cos(r16506);
        double r16508 = r16505 * r16507;
        double r16509 = sin(r16506);
        double r16510 = r16509 * r16504;
        double r16511 = 3.0;
        double r16512 = pow(r16510, r16511);
        double r16513 = cbrt(r16512);
        double r16514 = r16508 - r16513;
        double r16515 = r16504 / r16514;
        double r16516 = r16502 * r16515;
        return r16516;
}

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. Simplified0.3

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

  10. Applied add-cbrt-cube0.4

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

  11. Applied add-cbrt-cube0.4

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

  12. Applied cbrt-unprod0.4

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))