Falkner and Boettcher, Appendix A

Average Error: 2.3 → 0.7

Time: 9.7s

Precision: binary64

Cost: 7044

Math

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

↓

\[\begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;k \leq 0.5:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{k}}{k}\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))

↓

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a))) (if (<= k 0.5) t_0 (/ (/ t_0 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 t_0 = pow(k, m) * a;
	double tmp;
	if (k <= 0.5) {
		tmp = t_0;
	} else {
		tmp = (t_0 / 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) :: t_0
    real(8) :: tmp
    t_0 = (k ** m) * a
    if (k <= 0.5d0) then
        tmp = t_0
    else
        tmp = (t_0 / 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 t_0 = Math.pow(k, m) * a;
	double tmp;
	if (k <= 0.5) {
		tmp = t_0;
	} else {
		tmp = (t_0 / 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):
	t_0 = math.pow(k, m) * a
	tmp = 0
	if k <= 0.5:
		tmp = t_0
	else:
		tmp = (t_0 / 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)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (k <= 0.5)
		tmp = t_0;
	else
		tmp = Float64(Float64(t_0 / 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)
	t_0 = (k ^ m) * a;
	tmp = 0.0;
	if (k <= 0.5)
		tmp = t_0;
	else
		tmp = (t_0 / 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_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[k, 0.5], t$95$0, N[(N[(t$95$0 / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.5

   1. Initial program 0.1

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

   2. Taylor expanded in k around 0 25.4

      \[\leadsto \color{blue}{e^{\log k \cdot m} \cdot a} \]

   3. Simplified 0.6

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

   if 0.5 < k

   1. Initial program 5.9

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

   2. Taylor expanded in k around inf 6.7

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

   3. Simplified 6.7

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

   4. Applied egg-rr 1.0

      \[\leadsto \color{blue}{\frac{\frac{{k}^{m} \cdot a}{k}}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.5:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m} \cdot a}{k}}{k}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.8
Cost	6920

\[\begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -0.0012:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 1.3:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	20.8
Cost	976

\[\begin{array}{l} \mathbf{if}\;k \leq -3.9 \cdot 10^{+149}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq -4.6 \cdot 10^{-76}:\\ \;\;\;\;\left(1 + 0.1 \cdot \frac{a}{k}\right) + -1\\ \mathbf{elif}\;k \leq -6.5 \cdot 10^{-299}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 110000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 3
Error	22.8
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.061:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq -6.5 \cdot 10^{-299}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 0.035:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 4
Error	22.9
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.061:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq -6.5 \cdot 10^{-299}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 110000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 5
Error	22.8
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.061:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;k \leq -6.5 \cdot 10^{-299}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 110000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 6
Error	16.3
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -25000000:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 3.7:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 7
Error	23.1
Cost	716

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -0.061:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq -6.5 \cdot 10^{-299}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 110000000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	20.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 3.7:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 9
Error	38.9
Cost	584

\[\begin{array}{l} t_0 := \frac{a \cdot 0.1}{k}\\ \mathbf{if}\;k \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 0.035:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	24.7
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{a}{k}}{k}\\ \mathbf{if}\;k \leq -0.061:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 110000000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	24.9
Cost	584

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -0.061:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 110000000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	46.6
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022228 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))