Falkner and Boettcher, Appendix A

Average Error: 2.1 → 2.1

Time: 20.3s

Precision: binary64

Cost: 7168

Math

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

↓

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

FPCore

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

↓

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

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 pow(k, m) * (a / (1.0 + (k * (k + 10.0))));
}

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

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 math.pow(k, m) * (a / (1.0 + (k * (k + 10.0))))

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((k ^ m) * Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0)))))
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 = (k ^ m) * (a / (1.0 + (k * (k + 10.0))));
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[Power[k, m], $MachinePrecision] * N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.1

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

2. Simplified 2.1

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

   [Start] 2.1	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   rational.json-simplify-49 [=>] 2.1	\[ \color{blue}{{k}^{m} \cdot \frac{a}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
   rational.json-simplify-1 [=>] 2.1	\[ {k}^{m} \cdot \frac{a}{\color{blue}{k \cdot k + \left(1 + 10 \cdot k\right)}} \]
   rational.json-simplify-41 [=>] 2.1	\[ {k}^{m} \cdot \frac{a}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
   rational.json-simplify-2 [=>] 2.1	\[ {k}^{m} \cdot \frac{a}{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)} \]
   rational.json-simplify-51 [=>] 2.1	\[ {k}^{m} \cdot \frac{a}{1 + \color{blue}{k \cdot \left(k + 10\right)}} \]

3. Final simplification 2.1

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

Alternatives

Alternative 1
Error	2.1
Cost	7168

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

Alternative 2
Error	2.9
Cost	6920

\[\begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -230000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	11.8
Cost	2640

\[\begin{array}{l} t_0 := \left(a - -1\right) + -1\\ t_1 := k \cdot \left(k + 10\right)\\ t_2 := 1 + t_1\\ t_3 := \frac{a}{t_2}\\ \mathbf{if}\;m \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1 + \left(1 - \frac{2}{-1 - t_1}\right)}{\frac{2}{a}}\\ \mathbf{elif}\;m \leq 0.22:\\ \;\;\;\;t_3\\ \mathbf{elif}\;m \leq 1.65 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 5.6 \cdot 10^{+110}:\\ \;\;\;\;t_3 \cdot \left(\left(t_3 \cdot \frac{t_2}{a \cdot a}\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	11.2
Cost	1220

\[\begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1 + \left(1 - \frac{2}{-1 - t_0}\right)}{\frac{2}{a}}\\ \mathbf{elif}\;m \leq 0.185:\\ \;\;\;\;\frac{a}{1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(a - -1\right) + -1\\ \end{array} \]

Alternative 5
Error	11.2
Cost	1092

\[\begin{array}{l} t_0 := 1 + k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -5 \cdot 10^{+18}:\\ \;\;\;\;a \cdot \left(-1 + \left(1 - \frac{-1}{t_0}\right)\right)\\ \mathbf{elif}\;m \leq 0.22:\\ \;\;\;\;\frac{a}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(a - -1\right) + -1\\ \end{array} \]

Alternative 6
Error	16.6
Cost	964

\[\begin{array}{l} \mathbf{if}\;m \leq 0.235:\\ \;\;\;\;a \cdot \frac{a}{\left(1 + k \cdot \left(k + 10\right)\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(a - -1\right) + -1\\ \end{array} \]

Alternative 7
Error	33.0
Cost	848

\[\begin{array}{l} t_0 := \left(a - -1\right) + -1\\ \mathbf{if}\;m \leq -1.7 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq -2.7 \cdot 10^{-261}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 3.9 \cdot 10^{-173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 2 \cdot 10^{-66}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	16.6
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 0.92:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(a - -1\right) + -1\\ \end{array} \]

Alternative 9
Error	16.5
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 0.235:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a - -1\right) + -1\\ \end{array} \]

Alternative 10
Error	46.6
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023069 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))