Average Error: 15.5 → 0.4
Time: 6.3s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\left(r \cdot \sin b\right) \cdot \frac{2}{\left(\cos b \cdot \cos a - \sin a \cdot \sin b\right) \cdot 2}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\left(r \cdot \sin b\right) \cdot \frac{2}{\left(\cos b \cdot \cos a - \sin a \cdot \sin b\right) \cdot 2}
double f(double r, double a, double b) {
        double r16576 = r;
        double r16577 = b;
        double r16578 = sin(r16577);
        double r16579 = r16576 * r16578;
        double r16580 = a;
        double r16581 = r16580 + r16577;
        double r16582 = cos(r16581);
        double r16583 = r16579 / r16582;
        return r16583;
}
↓
double f(double r, double a, double b) {
        double r16584 = r;
        double r16585 = b;
        double r16586 = sin(r16585);
        double r16587 = r16584 * r16586;
        double r16588 = 2.0;
        double r16589 = cos(r16585);
        double r16590 = a;
        double r16591 = cos(r16590);
        double r16592 = r16589 * r16591;
        double r16593 = sin(r16590);
        double r16594 = r16593 * r16586;
        double r16595 = r16592 - r16594;
        double r16596 = r16595 * r16588;
        double r16597 = r16588 / r16596;
        double r16598 = r16587 * r16597;
        return r16598;
}

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 15.5

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

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

  6. Applied add-cbrt-cube0.4

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

  7. Applied cbrt-unprod0.4

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

  8. Simplified0.4

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

  10. Applied div-inv0.4

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

  11. Simplified0.4

    \[\leadsto \left(r \cdot \sin b\right) \cdot \color{blue}{\frac{2}{\left(\cos b \cdot \cos a - \sin a \cdot \sin b\right) \cdot 2}}\]

  12. Final simplification0.4

    \[\leadsto \left(r \cdot \sin b\right) \cdot \frac{2}{\left(\cos b \cdot \cos a - \sin a \cdot \sin b\right) \cdot 2}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))