Average Error: 1.9 → 2.0
Time: 20.7s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{1}{\sqrt{\frac{k \cdot \left(10 + k\right) + 1}{{k}^{m}}}} \cdot \frac{a}{\sqrt{\frac{k \cdot \left(10 + k\right) + 1}{{k}^{m}}}}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\frac{1}{\sqrt{\frac{k \cdot \left(10 + k\right) + 1}{{k}^{m}}}} \cdot \frac{a}{\sqrt{\frac{k \cdot \left(10 + k\right) + 1}{{k}^{m}}}}
double f(double a, double k, double m) {
        double r251021 = a;
        double r251022 = k;
        double r251023 = m;
        double r251024 = pow(r251022, r251023);
        double r251025 = r251021 * r251024;
        double r251026 = 1.0;
        double r251027 = 10.0;
        double r251028 = r251027 * r251022;
        double r251029 = r251026 + r251028;
        double r251030 = r251022 * r251022;
        double r251031 = r251029 + r251030;
        double r251032 = r251025 / r251031;
        return r251032;
}
↓
double f(double a, double k, double m) {
        double r251033 = 1.0;
        double r251034 = k;
        double r251035 = 10.0;
        double r251036 = r251035 + r251034;
        double r251037 = r251034 * r251036;
        double r251038 = 1.0;
        double r251039 = r251037 + r251038;
        double r251040 = m;
        double r251041 = pow(r251034, r251040);
        double r251042 = r251039 / r251041;
        double r251043 = sqrt(r251042);
        double r251044 = r251033 / r251043;
        double r251045 = a;
        double r251046 = r251045 / r251043;
        double r251047 = r251044 * r251046;
        return r251047;
}

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. Simplified1.9

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

  4. Applied add-sqr-sqrt2.0

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

  5. Applied *-un-lft-identity2.0

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

  6. Applied times-frac2.0

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

  7. Final simplification2.0

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

Reproduce

herbie shell --seed 2019305 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))