Average Error: 1.9 → 2.0
Time: 5.3s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{a}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot \frac{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}}{{k}^{m}}}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\frac{a}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot \frac{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}}{{k}^{m}}}
double f(double a, double k, double m) {
        double r140308 = a;
        double r140309 = k;
        double r140310 = m;
        double r140311 = pow(r140309, r140310);
        double r140312 = r140308 * r140311;
        double r140313 = 1.0;
        double r140314 = 10.0;
        double r140315 = r140314 * r140309;
        double r140316 = r140313 + r140315;
        double r140317 = r140309 * r140309;
        double r140318 = r140316 + r140317;
        double r140319 = r140312 / r140318;
        return r140319;
}
↓
double f(double a, double k, double m) {
        double r140320 = a;
        double r140321 = 1.0;
        double r140322 = 10.0;
        double r140323 = k;
        double r140324 = r140322 * r140323;
        double r140325 = r140321 + r140324;
        double r140326 = r140323 * r140323;
        double r140327 = r140325 + r140326;
        double r140328 = sqrt(r140327);
        double r140329 = m;
        double r140330 = pow(r140323, r140329);
        double r140331 = r140328 / r140330;
        double r140332 = r140328 * r140331;
        double r140333 = r140320 / r140332;
        return r140333;
}

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 1.9

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

  3. Applied associate-/l*1.9

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

  5. Applied *-un-lft-identity1.9

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

  6. Applied unpow-prod-down1.9

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

  7. Applied add-sqr-sqrt2.0

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

  8. Applied times-frac2.0

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

  9. Simplified2.0

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

  10. Final simplification2.0

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

Reproduce

herbie shell --seed 2020035 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))