Falkner and Boettcher, Appendix A

Average Error: 2.2 → 2.2

Time: 23.7s

Precision: binary64

Cost: 7168

Math

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

↓

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

FPCore

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

↓

(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ 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 (a * pow(k, m)) / (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 = (a * (k ** m)) / (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 (a * Math.pow(k, m)) / (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 (a * math.pow(k, m)) / (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(Float64(a * (k ^ m)) / 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 = (a * (k ^ m)) / (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[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $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

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

   [Start] 2.2	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   rational_best.json-simplify-1 [=>] 2.2	\[ \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k + \left(1 + 10 \cdot k\right)}} \]
   rational_best.json-simplify-1 [=>] 2.2	\[ \frac{a \cdot {k}^{m}}{k \cdot k + \color{blue}{\left(10 \cdot k + 1\right)}} \]
   rational_best.json-simplify-43 [=>] 2.2	\[ \frac{a \cdot {k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
   rational_best.json-simplify-2 [=>] 2.2	\[ \frac{a \cdot {k}^{m}}{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)} \]
   rational_best.json-simplify-47 [=>] 2.2	\[ \frac{a \cdot {k}^{m}}{1 + \color{blue}{k \cdot \left(k + 10\right)}} \]

3. Final simplification 2.2

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

Alternatives

Alternative 1
Error	3.0
Cost	7304

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -7.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{t_0}{1}\\ \mathbf{elif}\;m \leq 1.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 + k \cdot 10}\\ \end{array} \]

Alternative 2
Error	2.7
Cost	7048

\[\begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{1}\\ \mathbf{if}\;m \leq -7.8 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	16.7
Cost	1988

\[\begin{array}{l} t_0 := \frac{a}{1 + k \cdot \left(k + 10\right)}\\ t_1 := -10 \cdot \left(k \cdot a\right)\\ \mathbf{if}\;m \leq -1.78:\\ \;\;\;\;\frac{1}{a + t_1} \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{elif}\;m \leq 0.9:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	16.7
Cost	1732

\[\begin{array}{l} t_0 := \frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{if}\;m \leq -0.68:\\ \;\;\;\;\frac{-0.1}{k \cdot a} \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{elif}\;m \leq 0.64:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 5
Error	17.8
Cost	1604

\[\begin{array}{l} t_0 := \frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{if}\;m \leq -1.05:\\ \;\;\;\;\frac{1}{a} \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{elif}\;m \leq 0.98:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 6
Error	20.2
Cost	708

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

Alternative 7
Error	35.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 1.7:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 8
Error	43.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 4.5:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 9
Error	47.2
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023090 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))