Falkner and Boettcher, Appendix A

Average Error: 1.9 → 1.9

Time: 15.7s

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 1.9

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

2. Simplified 1.9

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

   [Start] 1.9	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   rational.json-simplify-49 [=>] 1.9	\[ \color{blue}{{k}^{m} \cdot \frac{a}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
   rational.json-simplify-1 [=>] 1.9	\[ {k}^{m} \cdot \frac{a}{\color{blue}{k \cdot k + \left(1 + 10 \cdot k\right)}} \]
   rational.json-simplify-41 [=>] 1.9	\[ {k}^{m} \cdot \frac{a}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
   rational.json-simplify-2 [=>] 1.9	\[ {k}^{m} \cdot \frac{a}{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)} \]
   rational.json-simplify-51 [=>] 1.9	\[ {k}^{m} \cdot \frac{a}{1 + \color{blue}{k \cdot \left(k + 10\right)}} \]

3. Final simplification 1.9

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

Alternatives

Alternative 1
Error	8.3
Cost	7304

\[\begin{array}{l} t_0 := a \cdot \frac{{k}^{m}}{1 + 10 \cdot k}\\ t_1 := 1 + k \cdot 10\\ \mathbf{if}\;m \leq -9.2 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 7.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{2 + k \cdot 20}{t_1 \cdot \left(t_1 \cdot \frac{2}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	8.4
Cost	6920

\[\begin{array}{l} t_0 := 1 + k \cdot 10\\ t_1 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -105:\\ \;\;\;\;t_1\\ \mathbf{elif}\;m \leq 0.052:\\ \;\;\;\;\frac{2 + k \cdot 20}{t_0 \cdot \left(t_0 \cdot \frac{2}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	29.1
Cost	2000

\[\begin{array}{l} t_0 := 2 + k \cdot 20\\ t_1 := 1 + k \cdot 10\\ t_2 := \frac{t_0}{a \cdot \left(\frac{t_1}{a} \cdot \frac{t_0}{a}\right)}\\ t_3 := a \cdot \frac{t_1}{t_1 \cdot t_1}\\ \mathbf{if}\;a \leq -5 \cdot 10^{-22}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-273}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-226}:\\ \;\;\;\;\frac{8}{\frac{t_1 \cdot 8}{a}}\\ \mathbf{elif}\;a \leq 1.38 \cdot 10^{-167}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	28.9
Cost	2000

\[\begin{array}{l} t_0 := 1 + k \cdot 10\\ t_1 := 2 + k \cdot 20\\ t_2 := \frac{t_1}{a \cdot \left(\frac{t_0}{a} \cdot \frac{t_1}{a}\right)}\\ \mathbf{if}\;a \leq -1 \cdot 10^{+114}:\\ \;\;\;\;t_1 \cdot \frac{\frac{a}{\frac{t_1}{\frac{2}{a}}}}{t_0 \cdot \frac{2}{a}}\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-273}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-226}:\\ \;\;\;\;\frac{8}{\frac{t_0 \cdot 8}{a}}\\ \mathbf{elif}\;a \leq 6.1 \cdot 10^{-167}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t_0}{t_0 \cdot t_0}\\ \end{array} \]

Alternative 5
Error	31.4
Cost	1344

\[\begin{array}{l} t_0 := 1 + k \cdot 10\\ \frac{2 + k \cdot 20}{t_0 \cdot \left(t_0 \cdot \frac{2}{a}\right)} \end{array} \]

Alternative 6
Error	31.3
Cost	1216

\[\begin{array}{l} t_0 := 1 + k \cdot 10\\ a \cdot \frac{t_0}{t_0 \cdot t_0} \end{array} \]

Alternative 7
Error	38.8
Cost	704

\[\frac{8}{\frac{\left(1 + k \cdot 10\right) \cdot 8}{a}} \]

Alternative 8
Error	39.3
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 9
Error	39.3
Cost	584

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

Alternative 10
Error	39.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;k \leq -0.1:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \end{array} \]

Alternative 11
Error	38.8
Cost	576

\[\frac{2}{\frac{2 + k \cdot 20}{a}} \]

Alternative 12
Error	39.1
Cost	448

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

Alternative 13
Error	46.8
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023068 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))