Falkner and Boettcher, Appendix A

Average Error: 3.46% → 0.13%

Time: 14.2s

Precision: binary64

Cost: 13508

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 2.65 \cdot 10^{+34}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{m \cdot \log k}}{k} \cdot \frac{a}{k}\\ \end{array} \]

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 2.65e+34)
   (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k)))
   (* (/ (exp (* m (log k))) 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 <= 2.65e+34) {
		tmp = (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = (exp((m * log(k))) / k) * (a / k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = (a * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
end function

↓

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 2.65d+34) then
        tmp = (a * (k ** m)) / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = (exp((m * log(k))) / k) * (a / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	return (a * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}

↓

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.65e+34) {
		tmp = (a * Math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = (Math.exp((m * Math.log(k))) / k) * (a / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	return (a * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))

↓

def code(a, k, m):
	tmp = 0
	if k <= 2.65e+34:
		tmp = (a * math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = (math.exp((m * math.log(k))) / 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 <= 2.65e+34)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64(Float64(exp(Float64(m * log(k))) / k) * Float64(a / k));
	end
	return tmp
end

MATLAB

function tmp = code(a, k, m)
	tmp = (a * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
end

↓

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 2.65e+34)
		tmp = (a * (k ^ m)) / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = (exp((m * log(k))) / k) * (a / k);
	end
	tmp_2 = 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, 2.65e+34], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(m * N[Log[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.6500000000000002e34

   1. Initial program 0.08

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

   if 2.6500000000000002e34 < k

   1. Initial program 9.95

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

   2. Taylor expanded in k around inf 9.95

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

   3. Simplified 0.22

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

      [Start] 9.95	\[ \frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{{k}^{2}} \]
      *-commutative [=>] 9.95	\[ \frac{\color{blue}{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}}{{k}^{2}} \]
      unpow2 [=>] 9.95	\[ \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{\color{blue}{k \cdot k}} \]
      times-frac [=>] 0.22	\[ \color{blue}{\frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{k} \cdot \frac{a}{k}} \]
      mul-1-neg [=>] 0.22	\[ \frac{e^{\color{blue}{-\log \left(\frac{1}{k}\right) \cdot m}}}{k} \cdot \frac{a}{k} \]
      distribute-rgt-neg-in [=>] 0.22	\[ \frac{e^{\color{blue}{\log \left(\frac{1}{k}\right) \cdot \left(-m\right)}}}{k} \cdot \frac{a}{k} \]
      log-rec [=>] 0.22	\[ \frac{e^{\color{blue}{\left(-\log k\right)} \cdot \left(-m\right)}}{k} \cdot \frac{a}{k} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.13

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.65 \cdot 10^{+34}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{m \cdot \log k}}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.63%
Cost	7300

\[\begin{array}{l} \mathbf{if}\;k \leq 1.32 \cdot 10^{+126}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{elif}\;k \leq 7.8 \cdot 10^{+195}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 2
Error	3.46%
Cost	7296

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

Alternative 3
Error	4.68%
Cost	7040

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

Alternative 4
Error	4.34%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{-11} \lor \neg \left(m \leq 8.8 \cdot 10^{-11}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 5
Error	23.56%
Cost	1240

\[\begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-303}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 3 \cdot 10^{-229}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 10^{-218}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{-159}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 4 \cdot 10^{-123}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 6
Error	23.35%
Cost	1240

\[\begin{array}{l} t_0 := \frac{a}{1 + k \cdot 10}\\ \mathbf{if}\;k \leq 6 \cdot 10^{-304}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 3 \cdot 10^{-229}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-219}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 4.5 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 3.9 \cdot 10^{-125}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 10.2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 7
Error	23.69%
Cost	1112

\[\begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-304}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 3 \cdot 10^{-229}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{-216}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{-158}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 3.9 \cdot 10^{-125}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 8
Error	4.27%
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -1.2:\\ \;\;\;\;0\\ \mathbf{elif}\;m \leq 0.43:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Error	5.48%
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq -1.15:\\ \;\;\;\;0\\ \mathbf{elif}\;m \leq 0.42:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10
Error	26.12%
Cost	328

\[\begin{array}{l} \mathbf{if}\;m \leq -3.3 \cdot 10^{-13}:\\ \;\;\;\;0\\ \mathbf{elif}\;m \leq 1.56 \cdot 10^{-45}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Error	73.68%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023090 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))