Falkner and Boettcher, Appendix A

Average Error: 1.9 → 0.1

Time: 18.4s

Precision: binary64

Cost: 19972

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{+144}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{\frac{1}{\frac{1}{{k}^{m}}}}{k}\\ \end{array} \]

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 1e+144)
   (/ (* a (pow k m)) (fma k k (fma k 10.0 1.0)))
   (* (/ a k) (/ (/ 1.0 (/ 1.0 (pow k m))) 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 <= 1e+144) {
		tmp = (a * pow(k, m)) / fma(k, k, fma(k, 10.0, 1.0));
	} else {
		tmp = (a / k) * ((1.0 / (1.0 / pow(k, m))) / 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 <= 1e+144)
		tmp = Float64(Float64(a * (k ^ m)) / fma(k, k, fma(k, 10.0, 1.0)));
	else
		tmp = Float64(Float64(a / k) * Float64(Float64(1.0 / Float64(1.0 / (k ^ m))) / 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, 1e+144], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(k * k + N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] * N[(N[(1.0 / N[(1.0 / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000002e144

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      +-commutative [=>] 0.1	\[ \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k + \left(1 + 10 \cdot k\right)}} \]
      fma-def [=>] 0.1	\[ \frac{a \cdot {k}^{m}}{\color{blue}{\mathsf{fma}\left(k, k, 1 + 10 \cdot k\right)}} \]
      +-commutative [=>] 0.1	\[ \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k, \color{blue}{10 \cdot k + 1}\right)} \]
      *-commutative [=>] 0.1	\[ \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k, \color{blue}{k \cdot 10} + 1\right)} \]
      fma-def [=>] 0.1	\[ \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k, \color{blue}{\mathsf{fma}\left(k, 10, 1\right)}\right)} \]

   if 1.00000000000000002e144 < k

   1. Initial program 9.2

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

   2. Simplified 9.2

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

      [Start] 9.2	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      +-commutative [=>] 9.2	\[ \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k + \left(1 + 10 \cdot k\right)}} \]
      fma-def [=>] 9.2	\[ \frac{a \cdot {k}^{m}}{\color{blue}{\mathsf{fma}\left(k, k, 1 + 10 \cdot k\right)}} \]
      +-commutative [=>] 9.2	\[ \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k, \color{blue}{10 \cdot k + 1}\right)} \]
      *-commutative [=>] 9.2	\[ \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k, \color{blue}{k \cdot 10} + 1\right)} \]
      fma-def [=>] 9.2	\[ \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k, \color{blue}{\mathsf{fma}\left(k, 10, 1\right)}\right)} \]

   3. Taylor expanded in k around inf 9.2

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

   4. Simplified 0.1

      \[\leadsto \color{blue}{\frac{a}{k} \cdot \frac{\frac{1}{\frac{1}{{k}^{m}}}}{k}} \]
      Proof

      [Start] 9.2	\[ \frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{{k}^{2}} \]
      unpow2 [=>] 9.2	\[ \frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{\color{blue}{k \cdot k}} \]
      times-frac [=>] 0.1	\[ \color{blue}{\frac{a}{k} \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{k}} \]
      mul-1-neg [=>] 0.1	\[ \frac{a}{k} \cdot \frac{e^{\color{blue}{-\log \left(\frac{1}{k}\right) \cdot m}}}{k} \]
      exp-neg [=>] 0.1	\[ \frac{a}{k} \cdot \frac{\color{blue}{\frac{1}{e^{\log \left(\frac{1}{k}\right) \cdot m}}}}{k} \]
      log-rec [=>] 0.1	\[ \frac{a}{k} \cdot \frac{\frac{1}{e^{\color{blue}{\left(-\log k\right)} \cdot m}}}{k} \]
      distribute-lft-neg-out [=>] 0.1	\[ \frac{a}{k} \cdot \frac{\frac{1}{e^{\color{blue}{-\log k \cdot m}}}}{k} \]
      rec-exp [<=] 0.1	\[ \frac{a}{k} \cdot \frac{\frac{1}{\color{blue}{\frac{1}{e^{\log k \cdot m}}}}}{k} \]
      exp-to-pow [=>] 0.1	\[ \frac{a}{k} \cdot \frac{\frac{1}{\frac{1}{\color{blue}{{k}^{m}}}}}{k} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{+144}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{\frac{1}{\frac{1}{{k}^{m}}}}{k}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{+145}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{\frac{1}{\frac{1}{{k}^{m}}}}{k}\\ \end{array} \]

Alternative 2
Error	0.7
Cost	7300

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

Alternative 3
Error	5.2
Cost	7180

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 1.35 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1180000000000:\\ \;\;\;\;\frac{a}{1 + \frac{k}{\frac{1}{k + 10}}}\\ \mathbf{elif}\;k \leq 1.52 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot \frac{k}{a}\right)}^{-1}\\ \end{array} \]

Alternative 4
Error	2.3
Cost	7176

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\frac{t_0}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot \frac{k}{a}\right)}^{-1}\\ \end{array} \]

Alternative 5
Error	0.9
Cost	7172

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

Alternative 6
Error	5.2
Cost	7052

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 3.5 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 118000000000:\\ \;\;\;\;\frac{a}{1 + \frac{k}{\frac{1}{k + 10}}}\\ \mathbf{elif}\;k \leq 82000000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 7
Error	1.1
Cost	7044

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

Alternative 8
Error	19.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -17500000000000 \lor \neg \left(m \leq 4.7 \cdot 10^{+79}\right):\\ \;\;\;\;-1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 9
Error	18.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -17500000000000 \lor \neg \left(m \leq 4 \cdot 10^{+79}\right):\\ \;\;\;\;-1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 10
Error	23.0
Cost	712

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

Alternative 11
Error	23.9
Cost	585

\[\begin{array}{l} \mathbf{if}\;k \leq -1 \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 12
Error	23.2
Cost	584

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

Alternative 13
Error	23.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{+108}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 14
Error	46.6
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023038 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))