Falkner and Boettcher, Appendix A

Average Error: 2.1 → 0.2

Time: 14.3s

Precision: binary64

Cost: 13572

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{+140}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

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 1.9e+140)
   (* a (/ (pow k m) (fma k (+ k 10.0) 1.0)))
   (/ (pow k m) (/ k (/ a 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) {
	double tmp;
	if (k <= 1.9e+140) {
		tmp = a * (pow(k, m) / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = pow(k, m) / (k / (a / k));
	}
	return tmp;
}

Julia

function code(a, k, m)
	return Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
end

↓

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.9e+140)
		tmp = Float64(a * Float64((k ^ m) / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = Float64((k ^ m) / Float64(k / Float64(a / k)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, k_, m_] := If[LessEqual[k, 1.9e+140], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] / N[(k / N[(a / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.9e140

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-*r/ [<=] 0.2	\[ \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      associate-+l+ [=>] 0.2	\[ a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
      +-commutative [=>] 0.2	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}} \]
      distribute-rgt-out [=>] 0.1	\[ a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]
      fma-def [=>] 0.1	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
      +-commutative [=>] 0.1	\[ a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

   if 1.9e140 < k

   1. Initial program 10.3

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

   2. Taylor expanded in a around 0 10.3

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

   3. Simplified 10.3

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

      [Start] 10.3	\[ \frac{e^{\log k \cdot m} \cdot a}{1 + \left({k}^{2} + 10 \cdot k\right)} \]
      associate-/l* [=>] 10.3	\[ \color{blue}{\frac{e^{\log k \cdot m}}{\frac{1 + \left({k}^{2} + 10 \cdot k\right)}{a}}} \]
      exp-to-pow [=>] 10.3	\[ \frac{\color{blue}{{k}^{m}}}{\frac{1 + \left({k}^{2} + 10 \cdot k\right)}{a}} \]
      *-commutative [=>] 10.3	\[ \frac{{k}^{m}}{\frac{1 + \left({k}^{2} + \color{blue}{k \cdot 10}\right)}{a}} \]
      unpow2 [=>] 10.3	\[ \frac{{k}^{m}}{\frac{1 + \left(\color{blue}{k \cdot k} + k \cdot 10\right)}{a}} \]
      distribute-lft-in [<=] 10.3	\[ \frac{{k}^{m}}{\frac{1 + \color{blue}{k \cdot \left(k + 10\right)}}{a}} \]

   4. Taylor expanded in k around inf 10.3

      \[\leadsto \frac{{k}^{m}}{\color{blue}{\frac{{k}^{2}}{a}}} \]

   5. Simplified 0.4

      \[\leadsto \frac{{k}^{m}}{\color{blue}{\frac{k}{\frac{a}{k}}}} \]
      Proof

      [Start] 10.3	\[ \frac{{k}^{m}}{\frac{{k}^{2}}{a}} \]
      unpow2 [=>] 10.3	\[ \frac{{k}^{m}}{\frac{\color{blue}{k \cdot k}}{a}} \]
      associate-/l* [=>] 0.4	\[ \frac{{k}^{m}}{\color{blue}{\frac{k}{\frac{a}{k}}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{+140}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{+74}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	7300

\[\begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{+31}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

Alternative 3
Error	1.8
Cost	7176

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{+162}:\\ \;\;\;\;\frac{t_0}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k}}{\frac{k}{a}}\\ \end{array} \]

Alternative 4
Error	0.6
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 10:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

Alternative 5
Error	0.9
Cost	7044

\[\begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

Alternative 6
Error	2.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;m \leq -0.31:\\ \;\;\;\;0\\ \mathbf{elif}\;m \leq 0.00027:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	2.7
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -0.24:\\ \;\;\;\;0\\ \mathbf{elif}\;m \leq 0.00027:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Error	13.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq 2.25 \cdot 10^{-305}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 9
Error	12.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq 2.25 \cdot 10^{-305}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 10
Error	3.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq -0.32:\\ \;\;\;\;0\\ \mathbf{elif}\;m \leq 0.00027:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Error	14.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;k \leq 2.25 \cdot 10^{-305}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 12
Error	12.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;k \leq 2.25 \cdot 10^{-305}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 13
Error	16.5
Cost	328

\[\begin{array}{l} \mathbf{if}\;k \leq 2.25 \cdot 10^{-305}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 20:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14
Error	46.5
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023059 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))