Falkner and Boettcher, Appendix A

Average Error: 2.0 → 0.1

Time: 15.0s

Precision: binary64

Cost: 7428

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 32500000000000:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{\frac{k + 10}{{k}^{m}}}\\ \end{array} \]

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 32500000000000.0)
   (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k)))
   (/ (/ a k) (/ (+ k 10.0) (pow k m)))))

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 <= 32500000000000.0) {
		tmp = (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = (a / k) / ((k + 10.0) / pow(k, m));
	}
	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 <= 32500000000000.0d0) then
        tmp = (a * (k ** m)) / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = (a / k) / ((k + 10.0d0) / (k ** m))
    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 <= 32500000000000.0) {
		tmp = (a * Math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = (a / k) / ((k + 10.0) / Math.pow(k, m));
	}
	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 <= 32500000000000.0:
		tmp = (a * math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = (a / k) / ((k + 10.0) / math.pow(k, m))
	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 <= 32500000000000.0)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64(Float64(a / k) / Float64(Float64(k + 10.0) / (k ^ m)));
	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 <= 32500000000000.0)
		tmp = (a * (k ^ m)) / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = (a / k) / ((k + 10.0) / (k ^ m));
	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, 32500000000000.0], 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[(a / k), $MachinePrecision] / N[(N[(k + 10.0), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.25e13

   1. Initial program 0.1

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

   if 3.25e13 < k

   1. Initial program 5.3

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

   2. Taylor expanded in k around inf 5.3

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{10 \cdot k} + k \cdot k} \]

   3. Simplified 5.3

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot 10} + k \cdot k} \]
      Proof

      [Start] 5.3	\[ \frac{a \cdot {k}^{m}}{10 \cdot k + k \cdot k} \]
      *-commutative [=>] 5.3	\[ \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot 10} + k \cdot k} \]

   4. Taylor expanded in a around 0 5.3

      \[\leadsto \color{blue}{\frac{e^{\log k \cdot m} \cdot a}{{k}^{2} + 10 \cdot k}} \]

   5. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\frac{a}{k}}{\frac{k + 10}{{k}^{m}}}} \]
      Proof

      [Start] 5.3	\[ \frac{e^{\log k \cdot m} \cdot a}{{k}^{2} + 10 \cdot k} \]
      *-commutative [=>] 5.3	\[ \frac{\color{blue}{a \cdot e^{\log k \cdot m}}}{{k}^{2} + 10 \cdot k} \]
      exp-to-pow [=>] 5.3	\[ \frac{a \cdot \color{blue}{{k}^{m}}}{{k}^{2} + 10 \cdot k} \]
      unpow2 [=>] 5.3	\[ \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k} + 10 \cdot k} \]
      distribute-rgt-in [<=] 5.3	\[ \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot \left(k + 10\right)}} \]
      associate-/r* [=>] 0.1	\[ \color{blue}{\frac{\frac{a \cdot {k}^{m}}{k}}{k + 10}} \]
      associate-*l/ [<=] 0.1	\[ \frac{\color{blue}{\frac{a}{k} \cdot {k}^{m}}}{k + 10} \]
      associate-/l* [=>] 0.1	\[ \color{blue}{\frac{\frac{a}{k}}{\frac{k + 10}{{k}^{m}}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 32500000000000:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{\frac{k + 10}{{k}^{m}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.9
Cost	7176

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{t_0}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 2
Error	1.8
Cost	7176

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;t_0 \cdot \left(1 + k \cdot -10\right)\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{t_0}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 3
Error	2.0
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{+148}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 4
Error	0.5
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 0.076:\\ \;\;\;\;\left(a \cdot {k}^{m}\right) \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{\frac{k + 10}{{k}^{m}}}\\ \end{array} \]

Alternative 5
Error	2.6
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -1.2 \cdot 10^{-16} \lor \neg \left(m \leq 2.5 \cdot 10^{-6}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\left(2 + k \cdot \left(k + 10\right)\right) + -1}\\ \end{array} \]

Alternative 6
Error	18.6
Cost	969

\[\begin{array}{l} \mathbf{if}\;m \leq -220000000 \lor \neg \left(m \leq 2.75 \cdot 10^{+88}\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\left(2 + k \cdot \left(k + 10\right)\right) + -1}\\ \end{array} \]

Alternative 7
Error	19.3
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -220000000 \lor \neg \left(m \leq 4.5 \cdot 10^{+89}\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 8
Error	18.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -220000000 \lor \neg \left(m \leq 2.2 \cdot 10^{+88}\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 9
Error	22.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -0.44:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 10
Error	22.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -9.6:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 11
Error	23.7
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 12
Error	23.0
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 13
Error	23.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 14
Error	46.3
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023041 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))