Falkner and Boettcher, Appendix A

Average Error: 2.1 → 0.1

Time: 6.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 4.1034082139038655 \cdot 10^{+147}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{k}^{\left(-m\right)}}{a} + \left(k + 10\right) \cdot \frac{\frac{k}{{k}^{m}}}{a}}\\ \end{array}\]

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 4.1034082139038655e+147)
   (/ (* a (pow k m)) (+ 1.0 (* k (+ k 10.0))))
   (/ 1.0 (+ (/ (pow k (- m)) a) (* (+ k 10.0) (/ (/ k (pow k m)) a))))))

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) {
	double tmp;
	if (k <= 4.1034082139038655e+147) {
		tmp = (a * pow(k, m)) / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = 1.0 / ((pow(k, -m) / a) + ((k + 10.0) * ((k / pow(k, m)) / a)));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

1. Split input into 2 regimes

2. if k < 4.1034082139038655e147

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   if 4.1034082139038655e147 < k

   1. Initial program 9.9

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

   3. Applied clear-num_binary64_2123 9.9

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

   4. Simplified 9.9

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

   5. Taylor expanded around inf 9.9

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

   6. Simplified 0.4

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

   8. Applied div-inv_binary64_2121 0.4

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

   9. Applied div-inv_binary64_2121 0.4

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

   10. Applied distribute-rgt-out_binary64_2077 0.4

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

   11. Applied associate-*r*_binary64_2064 0.4

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

   12. Simplified 0.4

       \[\leadsto \frac{1}{\frac{{k}^{\left(-m\right)}}{a} + \color{blue}{\frac{\frac{k}{{k}^{m}}}{a}} \cdot \left(k + 10\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.1034082139038655 \cdot 10^{+147}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{k}^{\left(-m\right)}}{a} + \left(k + 10\right) \cdot \frac{\frac{k}{{k}^{m}}}{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020346 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))