Falkner and Boettcher, Appendix A

Average Error: 2.1 → 0.2

Time: 14.8s

Precision: binary64

Cost: 13572

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+63}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{k}^{m} \cdot \frac{a}{k}}\right)}^{-1}\\ \end{array} \]

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 1.5e+63)
   (* a (/ (pow k m) (fma k (+ k 10.0) 1.0)))
   (pow (/ k (* (pow k m) (/ a k))) -1.0)))

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 <= 1.5e+63) {
		tmp = a * (pow(k, m) / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = pow((k / (pow(k, m) * (a / k))), -1.0);
	}
	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 <= 1.5e+63)
		tmp = Float64(a * Float64((k ^ m) / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = Float64(k / Float64((k ^ m) * Float64(a / k))) ^ -1.0;
	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, 1.5e+63], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(k / N[(N[Power[k, m], $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.5e63

   1. Initial program 0.1

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

   2. Simplified 0.0

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

      [Start] 0.1	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-*r/ [<=] 0.1	\[ \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      associate-+l+ [=>] 0.1	\[ a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
      +-commutative [=>] 0.1	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}} \]
      distribute-rgt-out [=>] 0.0	\[ a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]
      fma-def [=>] 0.0	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
      +-commutative [=>] 0.0	\[ a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

   if 1.5e63 < k

   1. Initial program 6.8

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

   2. Taylor expanded in a around 0 6.8

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

   3. Simplified 7.1

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

      [Start] 6.8	\[ \frac{e^{\log k \cdot m} \cdot a}{1 + \left({k}^{2} + 10 \cdot k\right)} \]
      associate-/l* [=>] 7.1	\[ \color{blue}{\frac{e^{\log k \cdot m}}{\frac{1 + \left({k}^{2} + 10 \cdot k\right)}{a}}} \]
      exp-to-pow [=>] 7.1	\[ \frac{\color{blue}{{k}^{m}}}{\frac{1 + \left({k}^{2} + 10 \cdot k\right)}{a}} \]
      unpow2 [=>] 7.1	\[ \frac{{k}^{m}}{\frac{1 + \left(\color{blue}{k \cdot k} + 10 \cdot k\right)}{a}} \]
      distribute-rgt-in [<=] 7.1	\[ \frac{{k}^{m}}{\frac{1 + \color{blue}{k \cdot \left(k + 10\right)}}{a}} \]

   4. Taylor expanded in k around inf 7.1

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

   5. Simplified 0.6

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

      [Start] 7.1	\[ \frac{{k}^{m}}{\frac{{k}^{2}}{a}} \]
      unpow2 [=>] 7.1	\[ \frac{{k}^{m}}{\frac{\color{blue}{k \cdot k}}{a}} \]
      *-rgt-identity [<=] 7.1	\[ \frac{{k}^{m}}{\frac{\color{blue}{\left(k \cdot k\right) \cdot 1}}{a}} \]
      associate-*r/ [<=] 7.1	\[ \frac{{k}^{m}}{\color{blue}{\left(k \cdot k\right) \cdot \frac{1}{a}}} \]
      associate-*l* [=>] 0.6	\[ \frac{{k}^{m}}{\color{blue}{k \cdot \left(k \cdot \frac{1}{a}\right)}} \]
      associate-*r/ [=>] 0.6	\[ \frac{{k}^{m}}{k \cdot \color{blue}{\frac{k \cdot 1}{a}}} \]
      *-rgt-identity [=>] 0.6	\[ \frac{{k}^{m}}{k \cdot \frac{\color{blue}{k}}{a}} \]

   6. Applied egg-rr 0.6

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+63}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{k}^{m} \cdot \frac{a}{k}}\right)}^{-1}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	13508

\[\begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+131}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{k}^{m} \cdot \frac{a}{k}}\right)}^{-1}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+109}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 3
Error	0.3
Cost	7300

\[\begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{+109}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 4
Error	2.7
Cost	7044

\[\begin{array}{l} \mathbf{if}\;m \leq -1.7 \cdot 10^{-10}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.155:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 5
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 6
Error	2.7
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -7.6 \lor \neg \left(m \leq 0.104\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 7
Error	23.6
Cost	712

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

Alternative 8
Error	23.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -9.6:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 9
Error	20.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq -7.6:\\ \;\;\;\;-1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 10
Error	20.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq -1000000:\\ \;\;\;\;-1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 11
Error	24.5
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	24.5
Cost	448

\[\frac{a}{1 + k \cdot k} \]

Alternative 13
Error	46.9
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023045 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))