Average Error: 2.0 → 2.1
Time: 25.5s
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 r141062 = a;
        double r141063 = k;
        double r141064 = m;
        double r141065 = pow(r141063, r141064);
        double r141066 = r141062 * r141065;
        double r141067 = 1.0;
        double r141068 = 10.0;
        double r141069 = r141068 * r141063;
        double r141070 = r141067 + r141069;
        double r141071 = r141063 * r141063;
        double r141072 = r141070 + r141071;
        double r141073 = r141066 / r141072;
        return r141073;
}
↓
double f(double a, double k, double m) {
        double r141074 = 1.0;
        double r141075 = k;
        double r141076 = 10.0;
        double r141077 = r141076 + r141075;
        double r141078 = r141075 * r141077;
        double r141079 = 1.0;
        double r141080 = r141078 + r141079;
        double r141081 = sqrt(r141080);
        double r141082 = r141074 / r141081;
        double r141083 = a;
        double r141084 = m;
        double r141085 = pow(r141075, r141084);
        double r141086 = r141081 / r141085;
        double r141087 = r141083 / r141086;
        double r141088 = r141082 * r141087;
        return r141088;
}

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))))