Falkner and Boettcher, Appendix A

Average Error: 2.0 → 0.1

Time: 16.2s

Precision: binary64

Cost: 7300

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{k}}{\frac{k}{a}}\\ \end{array} \]

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 2e+97)
   (/ a (/ (+ 1.0 (* k (+ k 10.0))) (pow k m)))
   (/ (/ (pow k m) k) (/ k a))))

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 <= 2e+97) {
		tmp = a / ((1.0 + (k * (k + 10.0))) / pow(k, m));
	} else {
		tmp = (pow(k, m) / k) / (k / a);
	}
	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 <= 2d+97) then
        tmp = a / ((1.0d0 + (k * (k + 10.0d0))) / (k ** m))
    else
        tmp = ((k ** m) / k) / (k / a)
    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 <= 2e+97) {
		tmp = a / ((1.0 + (k * (k + 10.0))) / Math.pow(k, m));
	} else {
		tmp = (Math.pow(k, m) / k) / (k / a);
	}
	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 <= 2e+97:
		tmp = a / ((1.0 + (k * (k + 10.0))) / math.pow(k, m))
	else:
		tmp = (math.pow(k, m) / k) / (k / a)
	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 <= 2e+97)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / (k ^ m)));
	else
		tmp = Float64(Float64((k ^ m) / k) / Float64(k / a));
	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 <= 2e+97)
		tmp = a / ((1.0 + (k * (k + 10.0))) / (k ^ m));
	else
		tmp = ((k ^ m) / k) / (k / a);
	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, 2e+97], N[(a / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision] / N[(k / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.0000000000000001e97

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-/l* [=>] 0.1	\[ \color{blue}{\frac{a}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{{k}^{m}}}} \]
      associate-+l+ [=>] 0.1	\[ \frac{a}{\frac{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}{{k}^{m}}} \]
      *-commutative [=>] 0.1	\[ \frac{a}{\frac{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)}{{k}^{m}}} \]

   3. Applied egg-rr 0.0

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

   if 2.0000000000000001e97 < k

   1. Initial program 7.4

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

   2. Simplified 7.4

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

      [Start] 7.4	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-*r/ [<=] 7.4	\[ \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      associate-+l+ [=>] 7.4	\[ a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
      +-commutative [=>] 7.4	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}} \]
      distribute-rgt-out [=>] 7.4	\[ a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]
      fma-def [=>] 7.4	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
      +-commutative [=>] 7.4	\[ a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

   3. Taylor expanded in k around inf 7.4

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

   4. Simplified 7.4

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

      [Start] 7.4	\[ a \cdot \frac{{k}^{m}}{{k}^{2}} \]
      unpow2 [=>] 7.4	\[ a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot k}} \]

   5. Applied egg-rr 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{k}}{\frac{k}{a}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 0.105:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot 10}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{k}}{\frac{k}{a}}\\ \end{array} \]

Alternative 2
Error	2.4
Cost	7049

\[\begin{array}{l} \mathbf{if}\;m \leq -3.2 \cdot 10^{-7} \lor \neg \left(m \leq 1.8 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{a}{{k}^{\left(2 - m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 3
Error	2.4
Cost	7048

\[\begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{-7}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(2 - m\right)}}\\ \end{array} \]

Alternative 4
Error	0.8
Cost	7044

\[\begin{array}{l} \mathbf{if}\;k \leq 0.105:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{k}}{\frac{k}{a}}\\ \end{array} \]

Alternative 5
Error	2.8
Cost	6984

\[\begin{array}{l} \mathbf{if}\;m \leq -2.85 \cdot 10^{+18}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 1.15 \cdot 10^{-16}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \end{array} \]

Alternative 6
Error	2.8
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -2.85 \cdot 10^{+18} \lor \neg \left(m \leq 1.15 \cdot 10^{-16}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 7
Error	19.5
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -2.85 \cdot 10^{+18} \lor \neg \left(m \leq 1.15\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 8
Error	18.7
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -2.85 \cdot 10^{+18} \lor \neg \left(m \leq 0.95\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 9
Error	23.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;k \leq -0.435 \lor \neg \left(k \leq 0.11\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 - k \cdot 10\right)\\ \end{array} \]

Alternative 10
Error	23.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;k \leq -0.435 \lor \neg \left(k \leq 0.11\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 11
Error	23.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -0.435:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;k \leq 0.11:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 12
Error	23.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -0.435:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;k \leq 0.11:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 13
Error	23.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -9.8:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 14
Error	38.9
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 15
Error	24.1
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 16
Error	24.1
Cost	448

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

Alternative 17
Error	46.3
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022364 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))