Falkner and Boettcher, Appendix A

Average Error: 2.1 → 2.1

Time: 4.9s

Precision: 64

Math

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

↓

\[\frac{a \cdot {k}^{m}}{\left(\sqrt[3]{1 + 10 \cdot k} \cdot \sqrt[3]{1 + 10 \cdot k}\right) \cdot \sqrt[3]{1 + 10 \cdot k} + k \cdot k}\]

C

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 ((a * pow(k, m)) / (((cbrt((1.0 + (10.0 * k))) * cbrt((1.0 + (10.0 * k)))) * cbrt((1.0 + (10.0 * k)))) + (k * k)));
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

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 add-cube-cbrt 2.1

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

4. Final simplification 2.1

   \[\leadsto \frac{a \cdot {k}^{m}}{\left(\sqrt[3]{1 + 10 \cdot k} \cdot \sqrt[3]{1 + 10 \cdot k}\right) \cdot \sqrt[3]{1 + 10 \cdot k} + k \cdot k}\]

Reproduce

herbie shell --seed 2020103 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))