Falkner and Boettcher, Appendix A

Average Error: 1.8 → 0.1

Time: 11.9s

Precision: binary64

Cost: 7428

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 3.45 \cdot 10^{+134}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;{k}^{\left(m + -1\right)} \cdot \frac{a}{k}\\ \end{array} \]

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 3.45e+134)
   (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k)))
   (* (pow k (+ m -1.0)) (/ 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 <= 3.45e+134) {
		tmp = (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = pow(k, (m + -1.0)) * (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 <= 3.45d+134) then
        tmp = (a * (k ** m)) / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = (k ** (m + (-1.0d0))) * (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 <= 3.45e+134) {
		tmp = (a * Math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = Math.pow(k, (m + -1.0)) * (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 <= 3.45e+134:
		tmp = (a * math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = math.pow(k, (m + -1.0)) * (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 <= 3.45e+134)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64((k ^ Float64(m + -1.0)) * 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 <= 3.45e+134)
		tmp = (a * (k ^ m)) / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = (k ^ (m + -1.0)) * (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, 3.45e+134], 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, N[(m + -1.0), $MachinePrecision]], $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.4500000000000001e134

   1. Initial program 0.1

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

   if 3.4500000000000001e134 < k

   1. Initial program 8.1

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

   2. Taylor expanded in a around 0 8.1

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

   3. Simplified 8.2

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

      [Start] 8.1	\[ \frac{e^{\log k \cdot m} \cdot a}{1 + \left({k}^{2} + 10 \cdot k\right)} \]
      associate-/l* [=>] 8.2	\[ \color{blue}{\frac{e^{\log k \cdot m}}{\frac{1 + \left({k}^{2} + 10 \cdot k\right)}{a}}} \]
      exp-to-pow [=>] 8.2	\[ \frac{\color{blue}{{k}^{m}}}{\frac{1 + \left({k}^{2} + 10 \cdot k\right)}{a}} \]
      unpow2 [=>] 8.2	\[ \frac{{k}^{m}}{\frac{1 + \left(\color{blue}{k \cdot k} + 10 \cdot k\right)}{a}} \]
      distribute-rgt-in [<=] 8.2	\[ \frac{{k}^{m}}{\frac{1 + \color{blue}{k \cdot \left(k + 10\right)}}{a}} \]

   4. Applied egg-rr 8.2

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

   5. Taylor expanded in k around inf 8.2

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

   6. Simplified 0.5

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

      [Start] 8.2	\[ \frac{{k}^{m}}{\frac{{k}^{2}}{a}} \]
      unpow2 [=>] 8.2	\[ \frac{{k}^{m}}{\frac{\color{blue}{k \cdot k}}{a}} \]
      associate-/l* [=>] 0.5	\[ \frac{{k}^{m}}{\color{blue}{\frac{k}{\frac{a}{k}}}} \]

   7. Applied egg-rr 0.1

      \[\leadsto \color{blue}{{k}^{\left(m - 1\right)} \cdot \frac{a}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.45 \cdot 10^{+134}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;{k}^{\left(m + -1\right)} \cdot \frac{a}{k}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	7300

\[\begin{array}{l} \mathbf{if}\;k \leq 2600000000000:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{\left(m + -1\right)} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 2
Error	1.5
Cost	7044

\[\begin{array}{l} \mathbf{if}\;k \leq 1.65 \cdot 10^{-21}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;{k}^{\left(m + -1\right)} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 3
Error	2.2
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -1.2 \cdot 10^{-9} \lor \neg \left(m \leq 1.4 \cdot 10^{-7}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 4
Error	14.2
Cost	1352

\[\begin{array}{l} \mathbf{if}\;m \leq -1.25:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 5.4 \cdot 10^{+22}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{\left(k + 10\right) + \left(\frac{100}{k} + \frac{1000}{k \cdot k}\right)}\\ \end{array} \]

Alternative 5
Error	19.3
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -0.76 \lor \neg \left(m \leq 4.1 \cdot 10^{+89}\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 6
Error	18.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -0.68 \lor \neg \left(m \leq 1.25 \cdot 10^{+95}\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 7
Error	23.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -1.25:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a - a \cdot \left(k \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 8
Error	23.1
Cost	712

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

Alternative 9
Error	23.9
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 10
Error	23.3
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 11
Error	23.9
Cost	448

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

Alternative 12
Error	46.4
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023018 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))