Average Error: 2.1 → 1.9
Time: 6.1s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{1}{\frac{\mathsf{fma}\left(k, \frac{k}{a}, \mathsf{fma}\left(1, \frac{1}{a}, 10 \cdot \frac{k}{a}\right)\right)}{{k}^{m}}}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\frac{1}{\frac{\mathsf{fma}\left(k, \frac{k}{a}, \mathsf{fma}\left(1, \frac{1}{a}, 10 \cdot \frac{k}{a}\right)\right)}{{k}^{m}}}
double code(double a, double k, double m) {
	return ((a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k)));
}
↓
double code(double a, double k, double m) {
	return (1.0 / (fma(k, (k / a), fma(1.0, (1.0 / a), (10.0 * (k / a)))) / pow(k, m)));
}

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

    \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)}{a}}{{k}^{m}}}}\]

  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. Simplified3.8

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

  7. Taylor expanded around 0 3.8

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

  8. Simplified1.9

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

  9. Final simplification1.9

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

Reproduce

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