Falkner and Boettcher, Appendix A

Average Error: 3.36% → 0.14%

Time: 13.4s

Precision: binary64

Cost: 7428

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \left(-\frac{a}{k}\right)}{-k}\\ \end{array} \]

FPCore

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

↓

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.5e+17)
   (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k)))
   (/ (* (pow k m) (- (/ a k))) (- k))))

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) {
	double tmp;
	if (k <= 1.5e+17) {
		tmp = (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = (pow(k, m) * -(a / k)) / -k;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (k <= 1.5d+17) then
        tmp = (a * (k ** m)) / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = ((k ** m) * -(a / k)) / -k
    end if
    code = tmp
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) {
	double tmp;
	if (k <= 1.5e+17) {
		tmp = (a * Math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = (Math.pow(k, m) * -(a / k)) / -k;
	}
	return tmp;
}

Python

def code(a, k, m):
	return (a * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))

↓

def code(a, k, m):
	tmp = 0
	if k <= 1.5e+17:
		tmp = (a * math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = (math.pow(k, m) * -(a / k)) / -k
	return tmp

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)
	tmp = 0.0
	if (k <= 1.5e+17)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64(Float64((k ^ m) * Float64(-Float64(a / k))) / Float64(-k));
	end
	return tmp
end

MATLAB

function tmp = code(a, k, m)
	tmp = (a * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
end

↓

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.5e+17)
		tmp = (a * (k ^ m)) / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = ((k ^ m) * -(a / k)) / -k;
	end
	tmp_2 = tmp;
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_] := If[LessEqual[k, 1.5e+17], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] * (-N[(a / k), $MachinePrecision])), $MachinePrecision] / (-k)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.5e17

   1. Initial program 0.11

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

   if 1.5e17 < k

   1. Initial program 9.08

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

   2. Taylor expanded in k around inf 9.08

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{{k}^{2}}} \]

   3. Simplified 9.08

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]
      Proof

      [Start] 9.08	\[ \frac{a \cdot {k}^{m}}{{k}^{2}} \]
      unpow2 [=>] 9.08	\[ \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]

   4. Applied egg-rr 0.2

      \[\leadsto \color{blue}{-\frac{{k}^{m} \cdot \frac{a}{k}}{-k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.14

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \left(-\frac{a}{k}\right)}{-k}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.15%
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{+17}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \left(-\frac{a}{k}\right)}{-k}\\ \end{array} \]

Alternative 2
Error	0.19%
Cost	7300

\[\begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{+17}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \left(-\frac{a}{k}\right)}{-k}\\ \end{array} \]

Alternative 3
Error	1.34%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 0.00015:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \left(-\frac{a}{k}\right)}{-k}\\ \end{array} \]

Alternative 4
Error	1.1%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 0.00015:\\ \;\;\;\;\left(a \cdot {k}^{m}\right) \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \left(-\frac{a}{k}\right)}{-k}\\ \end{array} \]

Alternative 5
Error	1.04%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 0.00015:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \left(-\frac{a}{k}\right)}{-k}\\ \end{array} \]

Alternative 6
Error	3.05%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;k \leq 0.00015:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+174}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 7
Error	3.21%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;k \leq 0.00015:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 3.9 \cdot 10^{+173}:\\ \;\;\;\;\frac{a}{{k}^{\left(2 - m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 8
Error	1.36%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;k \leq 0.00015:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

Alternative 9
Error	4.3%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -35 \lor \neg \left(m \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 10
Error	23.06%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;m \leq -35:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 0.205:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{\left(k + 10\right) + \left(\frac{100}{k} + \frac{1000}{k \cdot k}\right)}\\ \end{array} \]

Alternative 11
Error	27.22%
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -100:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 0.1:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{a}{k} \cdot 0.1\right) + -1\\ \end{array} \]

Alternative 12
Error	37.17%
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -0.44:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.098:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 13
Error	37.09%
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -10:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 10.2:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 14
Error	36.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -10:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k + 10}\\ \end{array} \]

Alternative 15
Error	33.16%
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -3 \cdot 10^{-210}:\\ \;\;\;\;\left(1 + \frac{a}{k} \cdot 0.1\right) + -1\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k + 10}\\ \end{array} \]

Alternative 16
Error	61.98%
Cost	585

\[\begin{array}{l} \mathbf{if}\;k \leq -0.1 \lor \neg \left(k \leq 0.145\right):\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 17
Error	39.05%
Cost	585

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

Alternative 18
Error	37.44%
Cost	584

\[\begin{array}{l} \mathbf{if}\;k \leq -1:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 19
Error	74.01%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023125 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))