Falkner and Boettcher, Appendix A

Average Error: 1.9 → 0.2

Time: 6.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le 7.43032575854467656 \cdot 10^{150}:\\ \;\;\;\;\frac{a}{\sqrt{1 + k \cdot \left(k + 10\right)}} \cdot \frac{{k}^{m}}{\sqrt{1 + k \cdot \left(k + 10\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{k} \cdot \frac{a}{k} + \left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{k} \cdot \frac{a}{k}\right) \cdot \left(\frac{99}{k \cdot k} - \frac{10}{k}\right)\\ \end{array}\]

C

double code(double a, double k, double m) {
	return ((double) (((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 <= 7.430325758544677e+150)) {
		VAR = ((double) (((double) (a / ((double) sqrt(((double) (1.0 + ((double) (k * ((double) (k + 10.0)))))))))) * ((double) (((double) pow(k, m)) / ((double) sqrt(((double) (1.0 + ((double) (k * ((double) (k + 10.0))))))))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) pow(((double) exp(m)), ((double) log(k)))) / k)) * ((double) (a / k)))) + ((double) (((double) (((double) (((double) pow(((double) exp(m)), ((double) log(k)))) / k)) * ((double) (a / k)))) * ((double) (((double) (99.0 / ((double) (k * k)))) - ((double) (10.0 / k))))))));
	}
	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 < 7.43032575854467656e150

   1. Initial program 0.1

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

   3. Applied add-sqr-sqrt 0.2

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

   4. Applied times-frac 0.2

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

   5. Simplified 0.2

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

   6. Simplified 0.2

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

   if 7.43032575854467656e150 < k

   1. Initial program 9.5

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

   3. Applied add-sqr-sqrt 9.5

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

   4. Applied times-frac 9.5

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

   5. Simplified 9.5

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

   6. Simplified 9.5

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

   7. Taylor expanded around inf 9.5

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

   8. Simplified 0.1

      \[\leadsto \color{blue}{\frac{{\left(e^{m}\right)}^{\left(0 + \log k\right)}}{k} \cdot \frac{a}{k} + \left(\frac{{\left(e^{m}\right)}^{\left(0 + \log k\right)}}{k} \cdot \frac{a}{k}\right) \cdot \left(\frac{99}{k \cdot k} - \frac{10}{k}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 7.43032575854467656 \cdot 10^{150}:\\ \;\;\;\;\frac{a}{\sqrt{1 + k \cdot \left(k + 10\right)}} \cdot \frac{{k}^{m}}{\sqrt{1 + k \cdot \left(k + 10\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{k} \cdot \frac{a}{k} + \left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{k} \cdot \frac{a}{k}\right) \cdot \left(\frac{99}{k \cdot k} - \frac{10}{k}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))