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 r281583 = a;
        double r281584 = k;
        double r281585 = m;
        double r281586 = pow(r281584, r281585);
        double r281587 = r281583 * r281586;
        double r281588 = 1.0;
        double r281589 = 10.0;
        double r281590 = r281589 * r281584;
        double r281591 = r281588 + r281590;
        double r281592 = r281584 * r281584;
        double r281593 = r281591 + r281592;
        double r281594 = r281587 / r281593;
        return r281594;
}
↓
double f(double a, double k, double m) {
        double r281595 = 1.0;
        double r281596 = k;
        double r281597 = 10.0;
        double r281598 = r281597 + r281596;
        double r281599 = r281596 * r281598;
        double r281600 = 1.0;
        double r281601 = r281599 + r281600;
        double r281602 = sqrt(r281601);
        double r281603 = r281595 / r281602;
        double r281604 = a;
        double r281605 = m;
        double r281606 = pow(r281596, r281605);
        double r281607 = r281602 / r281606;
        double r281608 = r281604 / r281607;
        double r281609 = r281603 * r281608;
        return r281609;
}

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))))