Falkner and Boettcher, Appendix A

Average Error: 2.2 → 2.2

Time: 14.5s

Precision: binary64

Cost: 7296

Math

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

↓

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

FPCore

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

↓

(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* 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) {
	return (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}

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
    code = (a * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
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) {
	return (a * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}

Python

def code(a, k, m):
	return (a * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))

↓

def code(a, k, m):
	return (a * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))

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)
	return Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
end

MATLAB

function tmp = code(a, k, m)
	tmp = (a * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
end

↓

function tmp = code(a, k, m)
	tmp = (a * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
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_] := 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]

Derivation

1. Initial program 2.2

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

2. Final simplification 2.2

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

Alternatives

Alternative 1
Error	2.2
Cost	7168

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

Alternative 2
Error	2.2
Cost	7168

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

Alternative 3
Error	2.8
Cost	6920

\[\begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -65000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 2.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	12.7
Cost	1220

\[\begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -8 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot \left(-1 + \left(1 - a \cdot \frac{-2}{t_0 + 1}\right)\right)\\ \mathbf{elif}\;m \leq 1.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-1 + \left(1 - \frac{a}{-0.5}\right)\right)\\ \end{array} \]

Alternative 5
Error	10.9
Cost	1220

\[\begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -65000:\\ \;\;\;\;\frac{-1 + \left(1 - \frac{2}{-1 - t_0}\right)}{\frac{2}{a}}\\ \mathbf{elif}\;m \leq 1.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-1 + \left(1 - \frac{a}{-0.5}\right)\right)\\ \end{array} \]

Alternative 6
Error	14.5
Cost	1092

\[\begin{array}{l} t_0 := 1 + k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -5 \cdot 10^{+34}:\\ \;\;\;\;a \cdot \left(a \cdot \frac{1}{a \cdot t_0}\right)\\ \mathbf{elif}\;m \leq 3.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{t_0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-1 + \left(1 - \frac{a}{-0.5}\right)\right)\\ \end{array} \]

Alternative 7
Error	15.3
Cost	964

\[\begin{array}{l} t_0 := \frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{if}\;m \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\frac{a \cdot t_0}{a}\\ \mathbf{elif}\;m \leq 1.1 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-1 + \left(1 - \frac{a}{-0.5}\right)\right)\\ \end{array} \]

Alternative 8
Error	28.2
Cost	840

\[\begin{array}{l} t_0 := 0.5 \cdot \left(-1 + \left(1 - \frac{a}{-0.5}\right)\right)\\ \mathbf{if}\;m \leq -2 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 1.75 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	35.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 150000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{2}{a} \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 10
Error	16.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 1.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-1 + \left(1 - \frac{a}{-0.5}\right)\right)\\ \end{array} \]

Alternative 11
Error	38.7
Cost	448

\[\frac{a}{1 + k \cdot 10} \]

Alternative 12
Error	46.5
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023075 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))