Average Error: 2.0 → 2.1
Time: 27.5s
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 r182989 = a;
        double r182990 = k;
        double r182991 = m;
        double r182992 = pow(r182990, r182991);
        double r182993 = r182989 * r182992;
        double r182994 = 1.0;
        double r182995 = 10.0;
        double r182996 = r182995 * r182990;
        double r182997 = r182994 + r182996;
        double r182998 = r182990 * r182990;
        double r182999 = r182997 + r182998;
        double r183000 = r182993 / r182999;
        return r183000;
}
↓
double f(double a, double k, double m) {
        double r183001 = 1.0;
        double r183002 = k;
        double r183003 = 10.0;
        double r183004 = r183003 + r183002;
        double r183005 = r183002 * r183004;
        double r183006 = 1.0;
        double r183007 = r183005 + r183006;
        double r183008 = sqrt(r183007);
        double r183009 = r183001 / r183008;
        double r183010 = a;
        double r183011 = m;
        double r183012 = pow(r183002, r183011);
        double r183013 = r183008 / r183012;
        double r183014 = r183010 / r183013;
        double r183015 = r183009 * r183014;
        return r183015;
}

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))))