Average Error: 2.8 → 2.8
Time: 9.0s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{\left(a \cdot {\left(\sqrt{k}\right)}^{m}\right) \cdot {\left(\sqrt{k}\right)}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\frac{\left(a \cdot {\left(\sqrt{k}\right)}^{m}\right) \cdot {\left(\sqrt{k}\right)}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
double f(double a, double k, double m) {
        double r331363 = a;
        double r331364 = k;
        double r331365 = m;
        double r331366 = pow(r331364, r331365);
        double r331367 = r331363 * r331366;
        double r331368 = 1.0;
        double r331369 = 10.0;
        double r331370 = r331369 * r331364;
        double r331371 = r331368 + r331370;
        double r331372 = r331364 * r331364;
        double r331373 = r331371 + r331372;
        double r331374 = r331367 / r331373;
        return r331374;
}
↓
double f(double a, double k, double m) {
        double r331375 = a;
        double r331376 = k;
        double r331377 = sqrt(r331376);
        double r331378 = m;
        double r331379 = pow(r331377, r331378);
        double r331380 = r331375 * r331379;
        double r331381 = r331380 * r331379;
        double r331382 = 1.0;
        double r331383 = 10.0;
        double r331384 = r331383 * r331376;
        double r331385 = r331382 + r331384;
        double r331386 = r331376 * r331376;
        double r331387 = r331385 + r331386;
        double r331388 = r331381 / r331387;
        return r331388;
}

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 2.8

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

  3. Applied add-sqr-sqrt2.8

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

  4. Applied unpow-prod-down2.8

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

  5. Applied associate-*r*2.8

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

  6. Final simplification2.8

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

Reproduce

herbie shell --seed 2020046 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))