Falkner and Boettcher, Appendix A

Average Error: 2.1 → 0.2

Time: 11.9s

Precision: binary64

Cost: 7428

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k \cdot \frac{k}{a}}\\ \end{array} \]

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 1.8e+86)
   (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k)))
   (/ (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 <= 1.8e+86) {
		tmp = (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = pow(k, m) / (k * (k / a));
	}
	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 <= 1.8d+86) then
        tmp = (a * (k ** m)) / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = (k ** m) / (k * (k / a))
    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 <= 1.8e+86) {
		tmp = (a * Math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = Math.pow(k, m) / (k * (k / a));
	}
	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 <= 1.8e+86:
		tmp = (a * math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = math.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 <= 1.8e+86)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64((k ^ m) / Float64(k * Float64(k / a)));
	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 <= 1.8e+86)
		tmp = (a * (k ^ m)) / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = (k ^ m) / (k * (k / a));
	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, 1.8e+86], 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[Power[k, m], $MachinePrecision] / N[(k * N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.80000000000000003e86

   1. Initial program 0.1

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

   if 1.80000000000000003e86 < k

   1. Initial program 7.4

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

   2. Taylor expanded in a around 0 7.4

      \[\leadsto \color{blue}{\frac{e^{\log k \cdot m} \cdot a}{1 + \left({k}^{2} + 10 \cdot k\right)}} \]

   3. Simplified 7.5

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

      [Start] 7.4	\[ \frac{e^{\log k \cdot m} \cdot a}{1 + \left({k}^{2} + 10 \cdot k\right)} \]
      associate-/l* [=>] 7.5	\[ \color{blue}{\frac{e^{\log k \cdot m}}{\frac{1 + \left({k}^{2} + 10 \cdot k\right)}{a}}} \]
      exp-to-pow [=>] 7.5	\[ \frac{\color{blue}{{k}^{m}}}{\frac{1 + \left({k}^{2} + 10 \cdot k\right)}{a}} \]
      *-commutative [=>] 7.5	\[ \frac{{k}^{m}}{\frac{1 + \left({k}^{2} + \color{blue}{k \cdot 10}\right)}{a}} \]
      unpow2 [=>] 7.5	\[ \frac{{k}^{m}}{\frac{1 + \left(\color{blue}{k \cdot k} + k \cdot 10\right)}{a}} \]
      distribute-lft-in [<=] 7.5	\[ \frac{{k}^{m}}{\frac{1 + \color{blue}{k \cdot \left(k + 10\right)}}{a}} \]

   4. Taylor expanded in k around inf 7.5

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

   5. Simplified 0.5

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

      [Start] 7.5	\[ \frac{{k}^{m}}{\frac{{k}^{2}}{a}} \]
      unpow2 [=>] 7.5	\[ \frac{{k}^{m}}{\frac{\color{blue}{k \cdot k}}{a}} \]
      *-rgt-identity [<=] 7.5	\[ \frac{{k}^{m}}{\frac{\color{blue}{\left(k \cdot k\right) \cdot 1}}{a}} \]
      associate-*r/ [<=] 7.5	\[ \frac{{k}^{m}}{\color{blue}{\left(k \cdot k\right) \cdot \frac{1}{a}}} \]
      associate-*l* [=>] 0.5	\[ \frac{{k}^{m}}{\color{blue}{k \cdot \left(k \cdot \frac{1}{a}\right)}} \]
      associate-*r/ [=>] 0.5	\[ \frac{{k}^{m}}{k \cdot \color{blue}{\frac{k \cdot 1}{a}}} \]
      *-rgt-identity [=>] 0.5	\[ \frac{{k}^{m}}{k \cdot \frac{\color{blue}{k}}{a}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	7300

\[\begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+26}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 2
Error	1.8
Cost	7048

\[\begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{a}{{k}^{\left(2 - m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 3
Error	0.7
Cost	7044

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

Alternative 4
Error	0.9
Cost	7044

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

Alternative 5
Error	2.7
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -1.02 \cdot 10^{-14} \lor \neg \left(m \leq 0.00036\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 6
Error	19.9
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -1.02 \lor \neg \left(m \leq 5 \cdot 10^{+21}\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 7
Error	19.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -0.52 \lor \neg \left(m \leq 3.5 \cdot 10^{+21}\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 8
Error	23.6
Cost	712

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

Alternative 9
Error	23.4
Cost	712

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

Alternative 10
Error	24.5
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 11
Error	23.5
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 12
Error	23.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 13
Error	47.1
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023012 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))