Falkner and Boettcher, Appendix A

Average Error: 3.05% → 0.15%

Time: 15.0s

Precision: binary64

Cost: 7428

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 1.12 \cdot 10^{+17}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k} \cdot \left(-{k}^{m}\right)}{-k}\\ \end{array} \]

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 1.12e+17)
   (/ a (/ (+ 1.0 (+ (* k 10.0) (* k k))) (pow k m)))
   (/ (* (/ a k) (- (pow k m))) (- 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 <= 1.12e+17) {
		tmp = a / ((1.0 + ((k * 10.0) + (k * k))) / pow(k, m));
	} else {
		tmp = ((a / k) * -pow(k, m)) / -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 <= 1.12d+17) then
        tmp = a / ((1.0d0 + ((k * 10.0d0) + (k * k))) / (k ** m))
    else
        tmp = ((a / k) * -(k ** m)) / -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 <= 1.12e+17) {
		tmp = a / ((1.0 + ((k * 10.0) + (k * k))) / Math.pow(k, m));
	} else {
		tmp = ((a / k) * -Math.pow(k, m)) / -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 <= 1.12e+17:
		tmp = a / ((1.0 + ((k * 10.0) + (k * k))) / math.pow(k, m))
	else:
		tmp = ((a / k) * -math.pow(k, m)) / -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 <= 1.12e+17)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(Float64(k * 10.0) + Float64(k * k))) / (k ^ m)));
	else
		tmp = Float64(Float64(Float64(a / k) * Float64(-(k ^ m))) / 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 <= 1.12e+17)
		tmp = a / ((1.0 + ((k * 10.0) + (k * k))) / (k ^ m));
	else
		tmp = ((a / k) * -(k ^ m)) / -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, 1.12e+17], N[(a / N[(N[(1.0 + N[(N[(k * 10.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a / k), $MachinePrecision] * (-N[Power[k, m], $MachinePrecision])), $MachinePrecision] / (-k)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.12e17

   1. Initial program 0.07

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

   2. Simplified 0.09

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

      [Start] 0.07	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-/l* [=>] 0.09	\[ \color{blue}{\frac{a}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{{k}^{m}}}} \]
      associate-+l+ [=>] 0.09	\[ \frac{a}{\frac{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}{{k}^{m}}} \]
      *-commutative [=>] 0.09	\[ \frac{a}{\frac{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)}{{k}^{m}}} \]

   if 1.12e17 < k

   1. Initial program 8.24

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

   2. Simplified 8.24

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

      [Start] 8.24	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-+l+ [=>] 8.24	\[ \frac{a \cdot {k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
      +-commutative [=>] 8.24	\[ \frac{a \cdot {k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}} \]
      distribute-rgt-out [=>] 8.24	\[ \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]
      fma-def [=>] 8.24	\[ \frac{a \cdot {k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
      +-commutative [=>] 8.24	\[ \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

   3. Taylor expanded in k around inf 8.24

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

   4. Simplified 8.24

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

      [Start] 8.24	\[ \frac{a \cdot {k}^{m}}{{k}^{2}} \]
      unpow2 [=>] 8.24	\[ \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]

   5. Applied egg-rr 0.24

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

4. Final simplification 0.15

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.12 \cdot 10^{+17}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k} \cdot \left(-{k}^{m}\right)}{-k}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.11%
Cost	7300

\[\begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{+16}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k} \cdot \left(-{k}^{m}\right)}{-k}\\ \end{array} \]

Alternative 2
Error	1.25%
Cost	7172

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

Alternative 3
Error	0.9%
Cost	7172

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

Alternative 4
Error	3.06%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{+162}:\\ \;\;\;\;\frac{a}{{k}^{\left(2 - m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 5
Error	1.26%
Cost	7044

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

Alternative 6
Error	3.85%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -0.0008 \lor \neg \left(m \leq 0.82\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + \left(k \cdot 10 + k \cdot k\right)}\\ \end{array} \]

Alternative 7
Error	28.73%
Cost	969

\[\begin{array}{l} \mathbf{if}\;m \leq -0.6 \lor \neg \left(m \leq 3.5 \cdot 10^{+23}\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + \left(k \cdot 10 + k \cdot k\right)}\\ \end{array} \]

Alternative 8
Error	29.82%
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -0.8 \lor \neg \left(m \leq 1.55 \cdot 10^{+31}\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 9
Error	28.73%
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -0.46 \lor \neg \left(m \leq 4 \cdot 10^{+23}\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 10
Error	36%
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -0.43:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + a \cdot \left(k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 11
Error	35.93%
Cost	712

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

Alternative 12
Error	60.44%
Cost	585

\[\begin{array}{l} \mathbf{if}\;k \leq -0.1 \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 13
Error	37.58%
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 14
Error	36.25%
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 15
Error	36.47%
Cost	580

\[\begin{array}{l} \mathbf{if}\;k \leq 2.55 \cdot 10^{+162}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 16
Error	72.77%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023088 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))