Average Error: 15.2 → 0.4
Time: 5.8s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\cos b \cdot \cos a - \log \left(e^{\sin a \cdot \sin b}\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\cos b \cdot \cos a - \log \left(e^{\sin a \cdot \sin b}\right)}
double f(double r, double a, double b) {
        double r15711 = r;
        double r15712 = b;
        double r15713 = sin(r15712);
        double r15714 = r15711 * r15713;
        double r15715 = a;
        double r15716 = r15715 + r15712;
        double r15717 = cos(r15716);
        double r15718 = r15714 / r15717;
        return r15718;
}
↓
double f(double r, double a, double b) {
        double r15719 = r;
        double r15720 = b;
        double r15721 = sin(r15720);
        double r15722 = cos(r15720);
        double r15723 = a;
        double r15724 = cos(r15723);
        double r15725 = r15722 * r15724;
        double r15726 = sin(r15723);
        double r15727 = r15726 * r15721;
        double r15728 = exp(r15727);
        double r15729 = log(r15728);
        double r15730 = r15725 - r15729;
        double r15731 = r15721 / r15730;
        double r15732 = r15719 * r15731;
        return r15732;
}

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 15.2

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

  15. Applied add-log-exp0.4

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

  16. Simplified0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2020064 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))