Average Error: 14.5 → 0.4
Time: 21.2s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\cos b \cdot \cos a - \sqrt[3]{{\left(\sin a \cdot \sin b\right)}^{3}}} \cdot \sin b\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\cos b \cdot \cos a - \sqrt[3]{{\left(\sin a \cdot \sin b\right)}^{3}}} \cdot \sin b
double f(double r, double a, double b) {
        double r27701 = r;
        double r27702 = b;
        double r27703 = sin(r27702);
        double r27704 = a;
        double r27705 = r27704 + r27702;
        double r27706 = cos(r27705);
        double r27707 = r27703 / r27706;
        double r27708 = r27701 * r27707;
        return r27708;
}
↓
double f(double r, double a, double b) {
        double r27709 = r;
        double r27710 = b;
        double r27711 = cos(r27710);
        double r27712 = a;
        double r27713 = cos(r27712);
        double r27714 = r27711 * r27713;
        double r27715 = sin(r27712);
        double r27716 = sin(r27710);
        double r27717 = r27715 * r27716;
        double r27718 = 3.0;
        double r27719 = pow(r27717, r27718);
        double r27720 = cbrt(r27719);
        double r27721 = r27714 - r27720;
        double r27722 = r27709 / r27721;
        double r27723 = r27722 * r27716;
        return r27723;
}

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 14.5

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

  2. Simplified14.5

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

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

  9. Applied add-cbrt-cube0.4

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

  10. Applied cbrt-unprod0.4

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

  11. Simplified0.4

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

  13. Applied *-commutative0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))