Average Error: 15.4 → 0.4
Time: 13.1s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \left(\frac{\sin b}{\cos b \cdot \left({\left(\cos a\right)}^{2} \cdot \cos b\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)} \cdot \left(\cos b \cdot \cos a + \sin a \cdot \sin b\right)\right)\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \left(\frac{\sin b}{\cos b \cdot \left({\left(\cos a\right)}^{2} \cdot \cos b\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)} \cdot \left(\cos b \cdot \cos a + \sin a \cdot \sin b\right)\right)
double f(double r, double a, double b) {
        double r78 = r;
        double r79 = b;
        double r80 = sin(r79);
        double r81 = r78 * r80;
        double r82 = a;
        double r83 = r82 + r79;
        double r84 = cos(r83);
        double r85 = r81 / r84;
        return r85;
}
↓
double f(double r, double a, double b) {
        double r86 = r;
        double r87 = b;
        double r88 = sin(r87);
        double r89 = cos(r87);
        double r90 = a;
        double r91 = cos(r90);
        double r92 = 2.0;
        double r93 = pow(r91, r92);
        double r94 = r93 * r89;
        double r95 = r89 * r94;
        double r96 = sin(r90);
        double r97 = r96 * r88;
        double r98 = r97 * r97;
        double r99 = r95 - r98;
        double r100 = r88 / r99;
        double r101 = r89 * r91;
        double r102 = r101 + r97;
        double r103 = r100 * r102;
        double r104 = r86 * r103;
        return r104;
}

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 15.4

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 flip--0.4

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

  11. Applied associate-/r/0.4

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

  13. Applied associate-*l*0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020025 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))