Falkner and Boettcher, Appendix A

Average Accuracy: 97.1% → 99.7%

Time: 22.1s

Precision: binary64

Cost: 13572

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+115}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

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 1.1e+115)
   (* (pow k m) (/ a (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 <= 1.1e+115) {
		tmp = pow(k, m) * (a / 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 <= 1.1e+115)
		tmp = Float64((k ^ m) * Float64(a / 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, 1.1e+115], N[(N[Power[k, m], $MachinePrecision] * N[(a / 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 < 1.1e115

   1. Initial program 99.9%

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

   2. Simplified 99.9%

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

      [Start] 99.9	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      *-commutative [=>] 99.9	\[ \frac{\color{blue}{{k}^{m} \cdot a}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-*r/ [<=] 99.9	\[ \color{blue}{{k}^{m} \cdot \frac{a}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      associate-+l+ [=>] 99.9	\[ {k}^{m} \cdot \frac{a}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
      +-commutative [=>] 99.9	\[ {k}^{m} \cdot \frac{a}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}} \]
      distribute-rgt-out [=>] 99.9	\[ {k}^{m} \cdot \frac{a}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]
      fma-def [=>] 99.9	\[ {k}^{m} \cdot \frac{a}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
      +-commutative [=>] 99.9	\[ {k}^{m} \cdot \frac{a}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

   if 1.1e115 < k

   1. Initial program 87.9%

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

   2. Taylor expanded in k around inf 87.9%

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

   3. Simplified 87.9%

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

      [Start] 87.9	\[ \frac{a \cdot {k}^{m}}{{k}^{2}} \]
      unpow2 [=>] 87.9	\[ \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]

   4. Applied egg-rr 99.9%

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

      [Start] 87.9	\[ \frac{a \cdot {k}^{m}}{k \cdot k} \]
      *-commutative [=>] 87.9	\[ \frac{\color{blue}{{k}^{m} \cdot a}}{k \cdot k} \]
      times-frac [=>] 99.9	\[ \color{blue}{\frac{{k}^{m}}{k} \cdot \frac{a}{k}} \]

   5. Applied egg-rr 99.0%

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

      [Start] 99.9	\[ \frac{{k}^{m}}{k} \cdot \frac{a}{k} \]
      associate-*l/ [=>] 99.9	\[ \color{blue}{\frac{{k}^{m} \cdot \frac{a}{k}}{k}} \]
      associate-/l* [=>] 99.0	\[ \color{blue}{\frac{{k}^{m}}{\frac{k}{\frac{a}{k}}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+115}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	7300

\[\begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+115}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

Alternative 2
Accuracy	97.2%
Cost	7180

\[\begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-9}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{elif}\;k \leq 360000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{+161}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 3
Accuracy	98.8%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-10}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{elif}\;k \leq 820000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

Alternative 4
Accuracy	98.5%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{-9}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{elif}\;k \leq 175000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

Alternative 5
Accuracy	98.5%
Cost	7172

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

Alternative 6
Accuracy	98.6%
Cost	7172

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

Alternative 7
Accuracy	92.0%
Cost	7052

\[\begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;k \leq 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{+62}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 8
Accuracy	65.6%
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 9
Accuracy	65.6%
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 10
Accuracy	65.7%
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 11
Accuracy	69.6%
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -1.2 \cdot 10^{+65}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 3.7 \cdot 10^{-245}:\\ \;\;\;\;\left(1 + -10 \cdot \left(k \cdot a\right)\right) + -1\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 12
Accuracy	71.6%
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;k \leq 3.7 \cdot 10^{-245}:\\ \;\;\;\;\left(1 + -10 \cdot \left(k \cdot a\right)\right) + -1\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 13
Accuracy	85.7%
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+24}:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 1.25:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -10 \cdot \left(k \cdot a\right)\right) + -1\\ \end{array} \]

Alternative 14
Accuracy	42.3%
Cost	716

\[\begin{array}{l} t_0 := \frac{a}{k} \cdot 0.1\\ \mathbf{if}\;k \leq -0.1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Accuracy	64.4%
Cost	716

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Accuracy	65.4%
Cost	716

\[\begin{array}{l} \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 17
Accuracy	33.7%
Cost	452

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

Alternative 18
Accuracy	27.7%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023143 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))