Falkner and Boettcher, Appendix A

Average Error: 2.2 → 2.2

Time: 27.7s

Precision: binary64

Cost: 7168

Math

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

↓

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

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

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

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

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

Derivation

1. Initial program 2.2

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

2. Simplified 2.2

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

   [Start] 2.2	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   rational.json-simplify-2 [=>] 2.2	\[ \frac{\color{blue}{{k}^{m} \cdot a}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   rational.json-simplify-49 [=>] 2.2	\[ \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
   rational.json-simplify-1 [=>] 2.2	\[ a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot k + \left(1 + 10 \cdot k\right)}} \]
   rational.json-simplify-41 [=>] 2.2	\[ a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
   rational.json-simplify-2 [=>] 2.2	\[ a \cdot \frac{{k}^{m}}{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)} \]
   rational.json-simplify-51 [=>] 2.2	\[ a \cdot \frac{{k}^{m}}{1 + \color{blue}{k \cdot \left(k + 10\right)}} \]

3. Applied egg-rr 2.2

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

4. Final simplification 2.2

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

Alternatives

Alternative 1
Error	3.0
Cost	7304

\[\begin{array}{l} \mathbf{if}\;m \leq -1.34 \cdot 10^{+17}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{elif}\;m \leq 1.85 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \left(a + a \cdot \left(k \cdot -10\right)\right)\\ \end{array} \]

Alternative 2
Error	2.9
Cost	7304

\[\begin{array}{l} \mathbf{if}\;m \leq -1.34 \cdot 10^{+17}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{elif}\;m \leq 1.85 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{1 + k \cdot 10}\\ \end{array} \]

Alternative 3
Error	2.9
Cost	7304

\[\begin{array}{l} \mathbf{if}\;m \leq -1.34 \cdot 10^{+17}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{elif}\;m \leq 3.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot 10}{{k}^{m}}}\\ \end{array} \]

Alternative 4
Error	2.2
Cost	7168

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

Alternative 5
Error	2.2
Cost	7168

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

Alternative 6
Error	2.9
Cost	7048

\[\begin{array}{l} \mathbf{if}\;m \leq -1.34 \cdot 10^{+17}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{elif}\;m \leq 2.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a}}\\ \end{array} \]

Alternative 7
Error	3.0
Cost	6920

\[\begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -1.34 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 2.3 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	13.7
Cost	1604

\[\begin{array}{l} t_0 := 1 + k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -1.34 \cdot 10^{+17}:\\ \;\;\;\;\frac{a}{t_0 \cdot \frac{t_0}{a \cdot \frac{1}{a}}}\\ \mathbf{elif}\;m \leq 0.6:\\ \;\;\;\;\frac{a}{t_0}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(-1 + \left(1 - \left(-a\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	13.3
Cost	1604

\[\begin{array}{l} t_0 := 1 + k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -1.34 \cdot 10^{+17}:\\ \;\;\;\;\frac{a}{\frac{\frac{t_0}{a}}{\frac{\frac{1}{a}}{t_0}}}\\ \mathbf{elif}\;m \leq 0.56:\\ \;\;\;\;\frac{a}{t_0}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(-1 + \left(1 - \left(-a\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	14.5
Cost	1476

\[\begin{array}{l} t_0 := 1 + k \cdot \left(k + 10\right)\\ t_1 := \frac{a}{t_0}\\ \mathbf{if}\;m \leq -1.34 \cdot 10^{+17}:\\ \;\;\;\;t_1 \cdot \frac{\frac{a}{a}}{t_0}\\ \mathbf{elif}\;m \leq 0.42:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(-1 + \left(1 - \left(-a\right)\right)\right)\\ \end{array} \]

Alternative 11
Error	14.0
Cost	1476

\[\begin{array}{l} t_0 := \frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{if}\;m \leq -1.34 \cdot 10^{+17}:\\ \;\;\;\;\frac{t_0 \cdot t_0}{a}\\ \mathbf{elif}\;m \leq 0.32:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(-1 + \left(1 - \left(-a\right)\right)\right)\\ \end{array} \]

Alternative 12
Error	14.6
Cost	1156

\[\begin{array}{l} t_0 := 1 + k \cdot \left(k + 10\right)\\ t_1 := -1 + \left(1 - \left(-a\right)\right)\\ \mathbf{if}\;m \leq -8 \cdot 10^{+22}:\\ \;\;\;\;\frac{1}{t_0} \cdot t_1\\ \mathbf{elif}\;m \leq 0.45:\\ \;\;\;\;\frac{a}{t_0}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t_1\\ \end{array} \]

Alternative 13
Error	16.4
Cost	1092

\[\begin{array}{l} \mathbf{if}\;m \leq 0.45:\\ \;\;\;\;\frac{a \cdot \frac{1}{a}}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(-1 + \left(1 - \left(-a\right)\right)\right)\\ \end{array} \]

Alternative 14
Error	16.4
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 0.31:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(-1 + \left(1 - \left(-a\right)\right)\right)\\ \end{array} \]

Alternative 15
Error	28.8
Cost	776

\[\begin{array}{l} t_0 := 1 \cdot \left(-1 + \left(1 - \left(-a\right)\right)\right)\\ \mathbf{if}\;m \leq -7 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 0.42:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	16.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 0.35:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(-1 + \left(1 - \left(-a\right)\right)\right)\\ \end{array} \]

Alternative 17
Error	38.9
Cost	448

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

Alternative 18
Error	47.0
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023073 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))