Falkner and Boettcher, Appendix A

Average Accuracy: 96.9% → 99.9%

Time: 16.3s

Precision: binary64

Cost: 13572

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+37}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{k}}{\frac{k}{a}}\\ \end{array} \]

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 2e+37)
   (* a (/ (pow k m) (fma k (+ k 10.0) 1.0)))
   (/ (/ (pow k m) k) (/ k a))))

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 <= 2e+37) {
		tmp = a * (pow(k, m) / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = (pow(k, m) / k) / (k / a);
	}
	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 <= 2e+37)
		tmp = Float64(a * Float64((k ^ m) / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = Float64(Float64((k ^ m) / k) / Float64(k / a));
	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, 2e+37], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision] / N[(k / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.99999999999999991e37

   1. Initial program 99.9%

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

   2. Simplified 99.9%

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

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

   if 1.99999999999999991e37 < k

   1. Initial program 91.1%

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

   2. Simplified 91.1%

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

      [Start] 91.1	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-*r/ [<=] 91.1	\[ \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      associate-+l+ [=>] 91.1	\[ a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
      +-commutative [=>] 91.1	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}} \]
      distribute-rgt-out [=>] 91.1	\[ a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]
      fma-def [=>] 91.1	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
      +-commutative [=>] 91.1	\[ a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

   3. Taylor expanded in k around inf 91.1%

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

   4. Simplified 91.1%

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

      [Start] 91.1	\[ a \cdot \frac{{k}^{m}}{{k}^{2}} \]
      unpow2 [=>] 91.1	\[ a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot k}} \]

   5. Applied egg-rr 99.8%

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

      [Start] 91.1	\[ a \cdot \frac{{k}^{m}}{k \cdot k} \]
      *-commutative [=>] 91.1	\[ \color{blue}{\frac{{k}^{m}}{k \cdot k} \cdot a} \]
      associate-/r* [=>] 91.6	\[ \color{blue}{\frac{\frac{{k}^{m}}{k}}{k}} \cdot a \]
      associate-*l/ [=>] 99.8	\[ \color{blue}{\frac{\frac{{k}^{m}}{k} \cdot a}{k}} \]
      associate-/l* [=>] 99.8	\[ \color{blue}{\frac{\frac{{k}^{m}}{k}}{\frac{k}{a}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+37}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{k}}{\frac{k}{a}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot k + k \cdot 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{k}}{\frac{k}{a}}\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{k}}{\frac{k}{a}}\\ \end{array} \]

Alternative 3
Accuracy	95.7%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{-17}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 5.4 \cdot 10^{+146}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{k \cdot k}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+205}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 4
Accuracy	95.7%
Cost	7176

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 5.4 \cdot 10^{+146}:\\ \;\;\;\;\frac{t_0}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+205}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 5
Accuracy	98.1%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{-17}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{k}}{\frac{k}{a}}\\ \end{array} \]

Alternative 6
Accuracy	96.1%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -0.25 \lor \neg \left(m \leq 2.1 \cdot 10^{-6}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 7
Accuracy	95.8%
Cost	6920

\[\begin{array}{l} \mathbf{if}\;m \leq -4.8 \cdot 10^{-20}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \mathbf{elif}\;m \leq 5.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 8
Accuracy	69.9%
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -1.5 \lor \neg \left(m \leq 4 \cdot 10^{+19}\right):\\ \;\;\;\;\left(1 + a \cdot \frac{0.1}{k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 9
Accuracy	72.9%
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -0.4:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 4 \cdot 10^{+19}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + a \cdot \frac{0.1}{k}\right) + -1\\ \end{array} \]

Alternative 10
Accuracy	74.0%
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -0.52:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 4 \cdot 10^{+19}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + a \cdot \frac{0.1}{k}\right) + -1\\ \end{array} \]

Alternative 11
Accuracy	63.0%
Cost	716

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 5.2 \cdot 10^{+206}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Accuracy	63.3%
Cost	716

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -0.43:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 5.2 \cdot 10^{+206}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Accuracy	63.2%
Cost	716

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -0.43:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{+205}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Accuracy	62.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;k \leq -0.062 \lor \neg \left(k \leq 0.075\right):\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 15
Accuracy	63.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{elif}\;k \leq 5.2 \cdot 10^{+206}:\\ \;\;\;\;\frac{\frac{a}{k}}{k + 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 16
Accuracy	39.1%
Cost	585

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

Alternative 17
Accuracy	62.3%
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 18
Accuracy	63.0%
Cost	584

\[\begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+37}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{+206}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 19
Accuracy	26.7%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023135 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))