Average Error: 2.1 → 1.9
Time: 5.7s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{1}{\frac{\frac{k}{a} \cdot k + \left(1 \cdot \frac{1}{a} + 10 \cdot \frac{k}{a}\right)}{{k}^{m}}}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\frac{1}{\frac{\frac{k}{a} \cdot k + \left(1 \cdot \frac{1}{a} + 10 \cdot \frac{k}{a}\right)}{{k}^{m}}}
double f(double a, double k, double m) {
        double r220085 = a;
        double r220086 = k;
        double r220087 = m;
        double r220088 = pow(r220086, r220087);
        double r220089 = r220085 * r220088;
        double r220090 = 1.0;
        double r220091 = 10.0;
        double r220092 = r220091 * r220086;
        double r220093 = r220090 + r220092;
        double r220094 = r220086 * r220086;
        double r220095 = r220093 + r220094;
        double r220096 = r220089 / r220095;
        return r220096;
}
↓
double f(double a, double k, double m) {
        double r220097 = 1.0;
        double r220098 = k;
        double r220099 = a;
        double r220100 = r220098 / r220099;
        double r220101 = r220100 * r220098;
        double r220102 = 1.0;
        double r220103 = r220097 / r220099;
        double r220104 = r220102 * r220103;
        double r220105 = 10.0;
        double r220106 = r220105 * r220100;
        double r220107 = r220104 + r220106;
        double r220108 = r220101 + r220107;
        double r220109 = m;
        double r220110 = pow(r220098, r220109);
        double r220111 = r220108 / r220110;
        double r220112 = r220097 / r220111;
        return r220112;
}

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 2.1

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

  3. Applied clear-num2.2

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

  4. Simplified2.2

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

  5. Taylor expanded around 0 3.8

    \[\leadsto \frac{1}{\frac{\color{blue}{\frac{{k}^{2}}{a} + \left(1 \cdot \frac{1}{a} + 10 \cdot \frac{k}{a}\right)}}{{k}^{m}}}\]

  6. Taylor expanded around 0 3.8

    \[\leadsto \frac{1}{\frac{\color{blue}{\frac{{k}^{2}}{a}} + \left(1 \cdot \frac{1}{a} + 10 \cdot \frac{k}{a}\right)}{{k}^{m}}}\]

  7. Simplified1.9

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

  8. Final simplification1.9

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

Reproduce

herbie shell --seed 2020049 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))