Falkner and Boettcher, Appendix A

Average Error: 2.1 → 2.0

Time: 24.0s

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 2.0

   \[\leadsto \color{blue}{{k}^{m} \cdot \frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   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 [=>] 2.0	\[ {k}^{m} \cdot \frac{a}{1 + \color{blue}{k \cdot \left(k + 10\right)}} \]

3. Final simplification 2.0

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

Alternatives

Alternative 1
Error	2.5
Cost	7304

\[\begin{array}{l} \mathbf{if}\;m \leq -2000:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{elif}\;m \leq 5.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{\frac{2 + k \cdot \left(k + \left(k + 20\right)\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot 10}\\ \end{array} \]

Alternative 2
Error	2.0
Cost	7168

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

Alternative 3
Error	2.6
Cost	6920

\[\begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -2000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 6.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{\frac{2 + k \cdot \left(k + \left(k + 20\right)\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	12.2
Cost	1744

\[\begin{array}{l} t_0 := 1 + k \cdot \left(k + 10\right)\\ t_1 := 2 + k \cdot \left(k + \left(k + 20\right)\right)\\ \mathbf{if}\;m \leq -2000:\\ \;\;\;\;\frac{a}{\frac{1}{-1 + \left(1 - \left(-\frac{1}{t_0}\right)\right)}}\\ \mathbf{elif}\;m \leq 0.42:\\ \;\;\;\;\frac{a}{\frac{t_1}{2}}\\ \mathbf{elif}\;m \leq 1.02 \cdot 10^{+41}:\\ \;\;\;\;-1 + \left(1 - \frac{a}{t_1 \cdot -0.5}\right)\\ \mathbf{elif}\;m \leq 1.45 \cdot 10^{+132}:\\ \;\;\;\;\frac{a}{\frac{1}{\frac{a}{t_0} \cdot \frac{a}{a \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - \frac{a}{-1 - k \cdot 10}\right)\\ \end{array} \]

Alternative 5
Error	13.5
Cost	1352

\[\begin{array}{l} t_0 := 2 + k \cdot \left(k + \left(k + 20\right)\right)\\ t_1 := -1 + \left(1 - \frac{a}{t_0 \cdot -0.5}\right)\\ \mathbf{if}\;m \leq -41000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;m \leq 0.17:\\ \;\;\;\;\frac{a}{\frac{t_0}{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	11.3
Cost	1352

\[\begin{array}{l} t_0 := 2 + k \cdot \left(k + \left(k + 20\right)\right)\\ \mathbf{if}\;m \leq -2000:\\ \;\;\;\;\frac{a}{\frac{1}{-1 + \left(1 - \left(-\frac{1}{1 + k \cdot \left(k + 10\right)}\right)\right)}}\\ \mathbf{elif}\;m \leq 0.42:\\ \;\;\;\;\frac{a}{\frac{t_0}{2}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - \frac{a}{t_0 \cdot -0.5}\right)\\ \end{array} \]

Alternative 7
Error	15.9
Cost	1096

\[\begin{array}{l} t_0 := -1 + \left(1 - \frac{a}{-1 - k \cdot 10}\right)\\ \mathbf{if}\;m \leq -5.8 \cdot 10^{+57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 0.31:\\ \;\;\;\;\frac{a}{\frac{2 + k \cdot \left(k + \left(k + 20\right)\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	15.8
Cost	968

\[\begin{array}{l} t_0 := -1 + \left(1 - \frac{a}{-1 - k \cdot 10}\right)\\ \mathbf{if}\;m \leq -6.8 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 0.43:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	39.3
Cost	712

\[\begin{array}{l} t_0 := 0.1 \cdot \frac{a}{k}\\ \mathbf{if}\;k \leq -0.062:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + a \cdot \left(k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	39.5
Cost	584

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

Alternative 11
Error	23.8
Cost	576

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

Alternative 12
Error	39.2
Cost	448

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

Alternative 13
Error	46.5
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023067 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))