Falkner and Boettcher, Appendix A

Average Error: 2.3 → 0.1

Time: 14.3s

Precision: binary64

Cost: 13572

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{+17}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{-a}{k}}{-k}\\ \end{array} \]

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 2.8e+17)
   (* a (/ (pow k m) (fma k (+ k 10.0) 1.0)))
   (/ (* (pow k m) (/ (- a) k)) (- 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 <= 2.8e+17) {
		tmp = a * (pow(k, m) / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = (pow(k, m) * (-a / k)) / -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 <= 2.8e+17)
		tmp = Float64(a * Float64((k ^ m) / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = Float64(Float64((k ^ m) * Float64(Float64(-a) / k)) / Float64(-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, 2.8e+17], 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] * N[((-a) / k), $MachinePrecision]), $MachinePrecision] / (-k)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.8e17

   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 2.8e17 < k

   1. Initial program 6.1

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

   2. Taylor expanded in k around inf 6.1

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

   3. Simplified 6.1

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

      [Start] 6.1	\[ \frac{a \cdot {k}^{m}}{{k}^{2}} \]
      unpow2 [=>] 6.1	\[ \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]

   4. Applied egg-rr 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{+17}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{-a}{k}}{-k}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{+31}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{-a}{k}}{-k}\\ \end{array} \]

Alternative 2
Error	0.9
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 0.0022:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{-a}{k}}{-k}\\ \end{array} \]

Alternative 3
Error	0.7
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 0.0022:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{-a}{k}}{-k}\\ \end{array} \]

Alternative 4
Error	2.2
Cost	7048

\[\begin{array}{l} \mathbf{if}\;k \leq 0.0022:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 8 \cdot 10^{+141}:\\ \;\;\;\;\frac{a}{{k}^{\left(2 - m\right)}}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{+283}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 5
Error	0.9
Cost	7044

\[\begin{array}{l} \mathbf{if}\;k \leq 0.0022:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

Alternative 6
Error	2.9
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -100000 \lor \neg \left(m \leq 0.088\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 7
Error	10.6
Cost	1224

\[\begin{array}{l} \mathbf{if}\;m \leq -100000:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 1.45:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + \left(-100 + \left(\frac{-1000}{k} - \frac{10000}{k \cdot k}\right)\right)}\\ \end{array} \]

Alternative 8
Error	11.6
Cost	1224

\[\begin{array}{l} \mathbf{if}\;m \leq -100000:\\ \;\;\;\;\frac{a}{1 + \left(0.1 + k \cdot 0.01\right) \cdot \left(k \cdot \left(100 - k \cdot k\right)\right)}\\ \mathbf{elif}\;m \leq 1.5:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + \left(-100 + \left(\frac{-1000}{k} - \frac{10000}{k \cdot k}\right)\right)}\\ \end{array} \]

Alternative 9
Error	21.2
Cost	972

\[\begin{array}{l} t_0 := \left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{if}\;k \leq -1.02 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq 8 \cdot 10^{+126}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 10
Error	16.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -215000:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 1.6:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + \left(-100 - \frac{1000}{k}\right)}\\ \end{array} \]

Alternative 11
Error	22.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -0.43:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.15:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 12
Error	22.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -0.43:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;k \leq 0.15:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 13
Error	22.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -10:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;k \leq 10.5:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 14
Error	20.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq -100000:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 15
Error	24.2
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 16
Error	23.1
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 17
Error	23.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{+31}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{+283}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 18
Error	46.9
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023046 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))