Falkner and Boettcher, Appendix A

Average Error: 97.0% → 99.9%

Time: 12.2s

Precision: binary64

Cost: 7428.00

Math

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

↓

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

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 1e+14)
   (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k)))
   (* (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 <= 1e+14) {
		tmp = (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = pow(k, m) * ((a / k) / 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 <= 1d+14) then
        tmp = (a * (k ** m)) / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = (k ** m) * ((a / k) / 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 <= 1e+14) {
		tmp = (a * Math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = Math.pow(k, m) * ((a / k) / 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 <= 1e+14:
		tmp = (a * math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = math.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 <= 1e+14)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64((k ^ m) * Float64(Float64(a / k) / 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 <= 1e+14)
		tmp = (a * (k ^ m)) / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = (k ^ m) * ((a / k) / 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, 1e+14], 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[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1e14

   1. Initial program 99.9

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

   if 1e14 < k

   1. Initial program 92.0

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

   2. Taylor expanded in k around 0 91.9

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

   3. Taylor expanded in k around inf 91.9

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

   4. Simplified 99.7

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

      [Start] 91.9	\[ \frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{{k}^{2}} \]
      associate-/l* [=>] 91.9	\[ \color{blue}{\frac{a}{\frac{{k}^{2}}{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}}} \]
      associate-/r/ [=>] 91.9	\[ \color{blue}{\frac{a}{{k}^{2}} \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}} \]
      *-commutative [=>] 91.9	\[ \color{blue}{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot \frac{a}{{k}^{2}}} \]
      mul-1-neg [=>] 91.9	\[ e^{\color{blue}{-\log \left(\frac{1}{k}\right) \cdot m}} \cdot \frac{a}{{k}^{2}} \]
      log-rec [=>] 91.9	\[ e^{-\color{blue}{\left(-\log k\right)} \cdot m} \cdot \frac{a}{{k}^{2}} \]
      distribute-lft-neg-out [=>] 91.9	\[ e^{-\color{blue}{\left(-\log k \cdot m\right)}} \cdot \frac{a}{{k}^{2}} \]
      remove-double-neg [=>] 91.9	\[ e^{\color{blue}{\log k \cdot m}} \cdot \frac{a}{{k}^{2}} \]
      exp-to-pow [=>] 91.9	\[ \color{blue}{{k}^{m}} \cdot \frac{a}{{k}^{2}} \]
      unpow2 [=>] 91.9	\[ {k}^{m} \cdot \frac{a}{\color{blue}{k \cdot k}} \]
      associate-/r* [=>] 99.7	\[ {k}^{m} \cdot \color{blue}{\frac{\frac{a}{k}}{k}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9

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

Alternatives

Alternative 1
Error	99.8%
Cost	7300.00

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

Alternative 2
Error	98.8%
Cost	7172.00

\[\begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-5}:\\ \;\;\;\;\left(a \cdot {k}^{m}\right) \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 3
Error	98.6%
Cost	7044.00

\[\begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-5}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 4
Error	96.1%
Cost	6921.00

\[\begin{array}{l} \mathbf{if}\;m \leq -0.95 \lor \neg \left(m \leq 0.0052\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 5
Error	69.6%
Cost	841.00

\[\begin{array}{l} \mathbf{if}\;m \leq -0.68 \lor \neg \left(m \leq 5.2 \cdot 10^{+49}\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 6
Error	70.7%
Cost	841.00

\[\begin{array}{l} \mathbf{if}\;m \leq -1.62 \lor \neg \left(m \leq 4 \cdot 10^{+54}\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 7
Error	63.0%
Cost	712.00

\[\begin{array}{l} \mathbf{if}\;k \leq -0.44:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.098:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 8
Error	63.1%
Cost	712.00

\[\begin{array}{l} \mathbf{if}\;k \leq -9.8:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 10.2:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 9
Error	61.8%
Cost	585.00

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

Alternative 10
Error	62.8%
Cost	584.00

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

Alternative 11
Error	62.8%
Cost	580.00

\[\begin{array}{l} \mathbf{if}\;k \leq 300000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 12
Error	27.4%
Cost	64.00

\[a \]

Reproduce

herbie shell --seed 2023121 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))