Average Error: 2.0 → 2.1
Time: 27.8s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{1}{\sqrt{k \cdot \left(10 + k\right) + 1}} \cdot \frac{a}{\frac{\sqrt{k \cdot \left(10 + k\right) + 1}}{{k}^{m}}}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\frac{1}{\sqrt{k \cdot \left(10 + k\right) + 1}} \cdot \frac{a}{\frac{\sqrt{k \cdot \left(10 + k\right) + 1}}{{k}^{m}}}
double f(double a, double k, double m) {
        double r184916 = a;
        double r184917 = k;
        double r184918 = m;
        double r184919 = pow(r184917, r184918);
        double r184920 = r184916 * r184919;
        double r184921 = 1.0;
        double r184922 = 10.0;
        double r184923 = r184922 * r184917;
        double r184924 = r184921 + r184923;
        double r184925 = r184917 * r184917;
        double r184926 = r184924 + r184925;
        double r184927 = r184920 / r184926;
        return r184927;
}
↓
double f(double a, double k, double m) {
        double r184928 = 1.0;
        double r184929 = k;
        double r184930 = 10.0;
        double r184931 = r184930 + r184929;
        double r184932 = r184929 * r184931;
        double r184933 = 1.0;
        double r184934 = r184932 + r184933;
        double r184935 = sqrt(r184934);
        double r184936 = r184928 / r184935;
        double r184937 = a;
        double r184938 = m;
        double r184939 = pow(r184929, r184938);
        double r184940 = r184935 / r184939;
        double r184941 = r184937 / r184940;
        double r184942 = r184936 * r184941;
        return r184942;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 2.0

    \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]

  2. Simplified2.0

    \[\leadsto \color{blue}{\frac{a}{\frac{k \cdot \left(10 + k\right) + 1}{{k}^{m}}}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity2.0

    \[\leadsto \frac{a}{\frac{k \cdot \left(10 + k\right) + 1}{{\color{blue}{\left(1 \cdot k\right)}}^{m}}}\]

  5. Applied unpow-prod-down2.0

    \[\leadsto \frac{a}{\frac{k \cdot \left(10 + k\right) + 1}{\color{blue}{{1}^{m} \cdot {k}^{m}}}}\]

  6. Applied add-sqr-sqrt2.1

    \[\leadsto \frac{a}{\frac{\color{blue}{\sqrt{k \cdot \left(10 + k\right) + 1} \cdot \sqrt{k \cdot \left(10 + k\right) + 1}}}{{1}^{m} \cdot {k}^{m}}}\]

  7. Applied times-frac2.1

    \[\leadsto \frac{a}{\color{blue}{\frac{\sqrt{k \cdot \left(10 + k\right) + 1}}{{1}^{m}} \cdot \frac{\sqrt{k \cdot \left(10 + k\right) + 1}}{{k}^{m}}}}\]

  8. Applied *-un-lft-identity2.1

    \[\leadsto \frac{\color{blue}{1 \cdot a}}{\frac{\sqrt{k \cdot \left(10 + k\right) + 1}}{{1}^{m}} \cdot \frac{\sqrt{k \cdot \left(10 + k\right) + 1}}{{k}^{m}}}\]

  9. Applied times-frac2.1

    \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{k \cdot \left(10 + k\right) + 1}}{{1}^{m}}} \cdot \frac{a}{\frac{\sqrt{k \cdot \left(10 + k\right) + 1}}{{k}^{m}}}}\]

  10. Simplified2.1

    \[\leadsto \color{blue}{\frac{1}{\sqrt{k \cdot \left(10 + k\right) + 1}}} \cdot \frac{a}{\frac{\sqrt{k \cdot \left(10 + k\right) + 1}}{{k}^{m}}}\]

  11. Final simplification2.1

    \[\leadsto \frac{1}{\sqrt{k \cdot \left(10 + k\right) + 1}} \cdot \frac{a}{\frac{\sqrt{k \cdot \left(10 + k\right) + 1}}{{k}^{m}}}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))