Falkner and Boettcher, Appendix A

Average Error: 2.3 → 0.1

Time: 14.8s

Precision: binary64

Cost: 13572

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 100000000:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{-a}{k}}{-10 - k}\\ \end{array} \]

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 100000000.0)
   (* a (/ (pow k m) (fma k (+ k 10.0) 1.0)))
   (* (pow k m) (/ (/ (- a) k) (- -10.0 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 <= 100000000.0) {
		tmp = a * (pow(k, m) / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = pow(k, m) * ((-a / k) / (-10.0 - 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 <= 100000000.0)
		tmp = Float64(a * Float64((k ^ m) / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = Float64((k ^ m) * Float64(Float64(Float64(-a) / k) / Float64(-10.0 - 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, 100000000.0], 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[(N[((-a) / k), $MachinePrecision] / N[(-10.0 - k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1e8

   1. Initial program 0.0

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

   2. Simplified 0.0

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

      [Start] 0.0	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-*r/ [<=] 0.0	\[ \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      associate-+l+ [=>] 0.0	\[ a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
      +-commutative [=>] 0.0	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}} \]
      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 1e8 < k

   1. Initial program 5.9

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

   2. Taylor expanded in k around inf 5.9

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{{k}^{2} + 10 \cdot k}} \]

   3. Simplified 5.9

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

      [Start] 5.9	\[ \frac{a \cdot {k}^{m}}{{k}^{2} + 10 \cdot k} \]
      unpow2 [=>] 5.9	\[ \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k} + 10 \cdot k} \]
      distribute-rgt-in [<=] 5.9	\[ \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot \left(k + 10\right)}} \]

   4. Applied egg-rr 5.9

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

   5. Simplified 0.1

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

      [Start] 5.9	\[ \left(a \cdot \left(-{k}^{m}\right)\right) \cdot \frac{1}{k \cdot \left(-\left(k + 10\right)\right)} \]
      associate-*r/ [=>] 5.9	\[ \color{blue}{\frac{\left(a \cdot \left(-{k}^{m}\right)\right) \cdot 1}{k \cdot \left(-\left(k + 10\right)\right)}} \]
      *-rgt-identity [=>] 5.9	\[ \frac{\color{blue}{a \cdot \left(-{k}^{m}\right)}}{k \cdot \left(-\left(k + 10\right)\right)} \]
      associate-*l/ [<=] 5.9	\[ \color{blue}{\frac{a}{k \cdot \left(-\left(k + 10\right)\right)} \cdot \left(-{k}^{m}\right)} \]
      *-commutative [=>] 5.9	\[ \color{blue}{\left(-{k}^{m}\right) \cdot \frac{a}{k \cdot \left(-\left(k + 10\right)\right)}} \]
      associate-/r* [=>] 0.1	\[ \left(-{k}^{m}\right) \cdot \color{blue}{\frac{\frac{a}{k}}{-\left(k + 10\right)}} \]
      neg-sub0 [=>] 0.1	\[ \left(-{k}^{m}\right) \cdot \frac{\frac{a}{k}}{\color{blue}{0 - \left(k + 10\right)}} \]
      +-commutative [=>] 0.1	\[ \left(-{k}^{m}\right) \cdot \frac{\frac{a}{k}}{0 - \color{blue}{\left(10 + k\right)}} \]
      associate--r+ [=>] 0.1	\[ \left(-{k}^{m}\right) \cdot \frac{\frac{a}{k}}{\color{blue}{\left(0 - 10\right) - k}} \]
      metadata-eval [=>] 0.1	\[ \left(-{k}^{m}\right) \cdot \frac{\frac{a}{k}}{\color{blue}{-10} - k} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 100000000:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{-a}{k}}{-10 - k}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 2200000000:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot k + k \cdot 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{-a}{k}}{-10 - k}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	7300

\[\begin{array}{l} \mathbf{if}\;k \leq 200000000000:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{-a}{k}}{-10 - k}\\ \end{array} \]

Alternative 3
Error	0.5
Cost	7236

\[\begin{array}{l} \mathbf{if}\;k \leq 0.085:\\ \;\;\;\;\left(a \cdot {k}^{m}\right) \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{-a}{k}}{-10 - k}\\ \end{array} \]

Alternative 4
Error	0.7
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;\left(a \cdot {k}^{m}\right) \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 5
Error	0.5
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 0.075:\\ \;\;\;\;\left(a \cdot {k}^{m}\right) \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k + 10}\\ \end{array} \]

Alternative 6
Error	0.5
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 0.075:\\ \;\;\;\;\left(a \cdot {k}^{m}\right) \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{a}{k}}{k + 10}\\ \end{array} \]

Alternative 7
Error	0.9
Cost	7044

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

Alternative 8
Error	2.8
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -1.1 \cdot 10^{-6} \lor \neg \left(m \leq 1.35 \cdot 10^{-11}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 9
Error	2.8
Cost	6920

\[\begin{array}{l} \mathbf{if}\;m \leq -1.04 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{elif}\;m \leq 1.42 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 10
Error	18.9
Cost	969

\[\begin{array}{l} \mathbf{if}\;m \leq -1200000 \lor \neg \left(m \leq 5 \cdot 10^{+17}\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k + \left(1 + k \cdot 10\right)}\\ \end{array} \]

Alternative 11
Error	19.7
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -1200000 \lor \neg \left(m \leq 7.6 \cdot 10^{+17}\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 12
Error	18.9
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -1200000 \lor \neg \left(m \leq 3.2 \cdot 10^{+17}\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 13
Error	24.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;k \leq -0.44 \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \]

Alternative 14
Error	23.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -0.44:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 15
Error	23.5
Cost	712

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

Alternative 16
Error	24.8
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 17
Error	23.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{+156}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 18
Error	23.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k + 10}\\ \end{array} \]

Alternative 19
Error	47.3
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023054 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))