Falkner and Boettcher, Appendix A

Average Error: 2.0 → 0.1

Time: 13.0s

Precision: binary64

Cost: 7428

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k \cdot {k}^{\left(-m\right)}}\\ \end{array} \]

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 5e+52)
   (/ a (/ (+ 1.0 (+ (* k 10.0) (* k k))) (pow k m)))
   (/ (/ a k) (* k (pow k (- m))))))

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 <= 5e+52) {
		tmp = a / ((1.0 + ((k * 10.0) + (k * k))) / pow(k, m));
	} else {
		tmp = (a / k) / (k * pow(k, -m));
	}
	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 <= 5d+52) then
        tmp = a / ((1.0d0 + ((k * 10.0d0) + (k * k))) / (k ** m))
    else
        tmp = (a / k) / (k * (k ** -m))
    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 <= 5e+52) {
		tmp = a / ((1.0 + ((k * 10.0) + (k * k))) / Math.pow(k, m));
	} else {
		tmp = (a / k) / (k * Math.pow(k, -m));
	}
	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 <= 5e+52:
		tmp = a / ((1.0 + ((k * 10.0) + (k * k))) / math.pow(k, m))
	else:
		tmp = (a / k) / (k * math.pow(k, -m))
	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 <= 5e+52)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(Float64(k * 10.0) + Float64(k * k))) / (k ^ m)));
	else
		tmp = Float64(Float64(a / k) / Float64(k * (k ^ Float64(-m))));
	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 <= 5e+52)
		tmp = a / ((1.0 + ((k * 10.0) + (k * k))) / (k ^ m));
	else
		tmp = (a / k) / (k * (k ^ -m));
	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, 5e+52], 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[(a / k), $MachinePrecision] / N[(k * N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5e52

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

   if 5e52 < k

   1. Initial program 6.1

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

   2. Simplified 6.2

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

      [Start] 6.1	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-/l* [=>] 6.2	\[ \color{blue}{\frac{a}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{{k}^{m}}}} \]
      associate-+l+ [=>] 6.2	\[ \frac{a}{\frac{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}{{k}^{m}}} \]
      *-commutative [=>] 6.2	\[ \frac{a}{\frac{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)}{{k}^{m}}} \]

   3. Taylor expanded in k around inf 6.2

      \[\leadsto \frac{a}{\frac{1 + \color{blue}{{k}^{2}}}{{k}^{m}}} \]

   4. Simplified 6.2

      \[\leadsto \frac{a}{\frac{1 + \color{blue}{k \cdot k}}{{k}^{m}}} \]
      Proof

      [Start] 6.2	\[ \frac{a}{\frac{1 + {k}^{2}}{{k}^{m}}} \]
      unpow2 [=>] 6.2	\[ \frac{a}{\frac{1 + \color{blue}{k \cdot k}}{{k}^{m}}} \]

   5. Taylor expanded in k around inf 6.1

      \[\leadsto \color{blue}{\frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{{k}^{2}}} \]

   6. Simplified 0.1

      \[\leadsto \color{blue}{\frac{a}{k} \cdot \frac{\frac{1}{\frac{1}{{k}^{m}}}}{k}} \]
      Proof

      [Start] 6.1	\[ \frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{{k}^{2}} \]
      unpow2 [=>] 6.1	\[ \frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{\color{blue}{k \cdot k}} \]
      times-frac [=>] 0.1	\[ \color{blue}{\frac{a}{k} \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{k}} \]
      mul-1-neg [=>] 0.1	\[ \frac{a}{k} \cdot \frac{e^{\color{blue}{-\log \left(\frac{1}{k}\right) \cdot m}}}{k} \]
      exp-neg [=>] 0.1	\[ \frac{a}{k} \cdot \frac{\color{blue}{\frac{1}{e^{\log \left(\frac{1}{k}\right) \cdot m}}}}{k} \]
      log-rec [=>] 0.1	\[ \frac{a}{k} \cdot \frac{\frac{1}{e^{\color{blue}{\left(-\log k\right)} \cdot m}}}{k} \]
      distribute-lft-neg-in [<=] 0.1	\[ \frac{a}{k} \cdot \frac{\frac{1}{e^{\color{blue}{-\log k \cdot m}}}}{k} \]
      exp-neg [=>] 0.1	\[ \frac{a}{k} \cdot \frac{\frac{1}{\color{blue}{\frac{1}{e^{\log k \cdot m}}}}}{k} \]
      exp-to-pow [=>] 0.1	\[ \frac{a}{k} \cdot \frac{\frac{1}{\frac{1}{\color{blue}{{k}^{m}}}}}{k} \]

   7. Applied egg-rr 6.2

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

   8. Simplified 0.1

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

      [Start] 6.2	\[ \frac{a}{\left(k \cdot k\right) \cdot {k}^{\left(-m\right)}} \]
      associate-*l* [=>] 6.2	\[ \frac{a}{\color{blue}{k \cdot \left(k \cdot {k}^{\left(-m\right)}\right)}} \]
      associate-/r* [=>] 0.1	\[ \color{blue}{\frac{\frac{a}{k}}{k \cdot {k}^{\left(-m\right)}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k \cdot {k}^{\left(-m\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	39232

\[\begin{array}{l} t_0 := \sqrt{{k}^{m}}\\ t_0 \cdot \frac{\frac{a}{\mathsf{hypot}\left(1, k\right)} \cdot t_0}{\mathsf{hypot}\left(1, k\right)} \end{array} \]

Alternative 2
Error	0.6
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 10.2:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot 10}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k \cdot {k}^{\left(-m\right)}}\\ \end{array} \]

Alternative 3
Error	0.8
Cost	7108

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

Alternative 4
Error	0.8
Cost	7044

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

Alternative 5
Error	2.5
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -1.16 \cdot 10^{-12} \lor \neg \left(m \leq 0.02\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 6
Error	23.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -0.44:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 7
Error	23.3
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 8
Error	21.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq -0.65:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 9
Error	20.4
Cost	708

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

Alternative 10
Error	24.5
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 11
Error	23.5
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 12
Error	24.5
Cost	448

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

Alternative 13
Error	46.7
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023083 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))