Falkner and Boettcher, Appendix A

Average Error: 2.1 → 0.1

Time: 5.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 5.2465810226145275 \cdot 10^{+23}:\\ \;\;\;\;a \cdot \left({\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m} \cdot \frac{{\left(\sqrt[3]{k}\right)}^{m}}{1 + k \cdot \left(k + 10\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;99 \cdot \left(a \cdot \frac{{k}^{m}}{{k}^{4}}\right) + \left(\frac{{k}^{m}}{k} \cdot \frac{a}{k} - 10 \cdot \left(a \cdot \frac{{k}^{m}}{{k}^{3}}\right)\right)\\ \end{array}\]

FPCore

(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))

↓

(FPCore (a k m)
 :precision binary64
 (if (<= k 5.2465810226145275e+23)
   (*
    a
    (*
     (pow (* (cbrt k) (cbrt k)) m)
     (/ (pow (cbrt k) m) (+ 1.0 (* k (+ k 10.0))))))
   (+
    (* 99.0 (* a (/ (pow k m) (pow k 4.0))))
    (- (* (/ (pow k m) k) (/ a k)) (* 10.0 (* a (/ (pow k m) (pow k 3.0))))))))

C

double code(double a, double k, double m) {
	return (((double) (a * ((double) pow(k, m)))) / ((double) (((double) (1.0 + ((double) (10.0 * k)))) + ((double) (k * k)))));
}

↓

double code(double a, double k, double m) {
	double VAR;
	if ((k <= 5.2465810226145275e+23)) {
		VAR = ((double) (a * ((double) (((double) pow(((double) (((double) cbrt(k)) * ((double) cbrt(k)))), m)) * (((double) pow(((double) cbrt(k)), m)) / ((double) (1.0 + ((double) (k * ((double) (k + 10.0)))))))))));
	} else {
		VAR = ((double) (((double) (99.0 * ((double) (a * (((double) pow(k, m)) / ((double) pow(k, 4.0))))))) + ((double) (((double) ((((double) pow(k, m)) / k) * (a / k))) - ((double) (10.0 * ((double) (a * (((double) pow(k, m)) / ((double) pow(k, 3.0)))))))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

1. Split input into 2 regimes

2. if k < 5.24658102261452752e23

   1. Initial program 0.0

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

   2. Simplified 0.0

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

   4. Applied *-un-lft-identity 0.0

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

   5. Applied add-cube-cbrt 0.0

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

   6. Applied unpow-prod-down 0.0

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

   7. Applied times-frac 0.0

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

   8. Simplified 0.0

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

   if 5.24658102261452752e23 < k

   1. Initial program 5.7

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

   2. Simplified 5.8

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

   4. Applied add-sqr-sqrt 5.8

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

   5. Applied *-un-lft-identity 5.8

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

   6. Applied unpow-prod-down 5.8

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

   7. Applied times-frac 5.8

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

   8. Applied associate-*r* 5.8

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

   9. Simplified 5.8

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

   10. Taylor expanded around inf 5.7

       \[\leadsto \color{blue}{\left(99 \cdot \frac{e^{m \cdot \left(\log 1 - \log \left(\frac{1}{k}\right)\right)} \cdot a}{{k}^{4}} + \frac{e^{m \cdot \left(\log 1 - \log \left(\frac{1}{k}\right)\right)} \cdot a}{{k}^{2}}\right) - 10 \cdot \frac{e^{m \cdot \left(\log 1 - \log \left(\frac{1}{k}\right)\right)} \cdot a}{{k}^{3}}}\]

   11. Simplified 0.1

       \[\leadsto \color{blue}{99 \cdot \left(\frac{{k}^{m}}{{k}^{4}} \cdot a\right) + \left(\frac{{k}^{m}}{k} \cdot \frac{a}{k} - 10 \cdot \left(\frac{{k}^{m}}{{k}^{3}} \cdot a\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2465810226145275 \cdot 10^{+23}:\\ \;\;\;\;a \cdot \left({\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m} \cdot \frac{{\left(\sqrt[3]{k}\right)}^{m}}{1 + k \cdot \left(k + 10\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;99 \cdot \left(a \cdot \frac{{k}^{m}}{{k}^{4}}\right) + \left(\frac{{k}^{m}}{k} \cdot \frac{a}{k} - 10 \cdot \left(a \cdot \frac{{k}^{m}}{{k}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))