Falkner and Boettcher, Appendix A

Average Error: 2.1 → 1.2

Time: 18.4s

Precision: binary64

Cost: 7428

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{+155}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

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 1.05e+155)
   (/ a (/ (+ 1.0 (+ (* k 10.0) (* k k))) (pow k m)))
   (/ 1.0 (/ k (/ a 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 <= 1.05e+155) {
		tmp = a / ((1.0 + ((k * 10.0) + (k * k))) / pow(k, m));
	} else {
		tmp = 1.0 / (k / (a / 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 <= 1.05d+155) then
        tmp = a / ((1.0d0 + ((k * 10.0d0) + (k * k))) / (k ** m))
    else
        tmp = 1.0d0 / (k / (a / 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 <= 1.05e+155) {
		tmp = a / ((1.0 + ((k * 10.0) + (k * k))) / Math.pow(k, m));
	} else {
		tmp = 1.0 / (k / (a / 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 <= 1.05e+155:
		tmp = a / ((1.0 + ((k * 10.0) + (k * k))) / math.pow(k, m))
	else:
		tmp = 1.0 / (k / (a / 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 <= 1.05e+155)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(Float64(k * 10.0) + Float64(k * k))) / (k ^ m)));
	else
		tmp = Float64(1.0 / Float64(k / Float64(a / 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 <= 1.05e+155)
		tmp = a / ((1.0 + ((k * 10.0) + (k * k))) / (k ^ m));
	else
		tmp = 1.0 / (k / (a / 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, 1.05e+155], 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[(1.0 / N[(k / N[(a / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.05e155

   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 1.05e155 < k

   1. Initial program 10.4

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

   2. Simplified 10.4

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

      [Start] 10.4	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-*r/ [<=] 10.4	\[ \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      associate-+l+ [=>] 10.4	\[ a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
      +-commutative [=>] 10.4	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}} \]
      distribute-rgt-out [=>] 10.4	\[ a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]
      fma-def [=>] 10.4	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
      +-commutative [=>] 10.4	\[ a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

   3. Taylor expanded in m around 0 10.4

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

   4. Taylor expanded in k around inf 10.4

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

   5. Simplified 10.4

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

      [Start] 10.4	\[ \frac{a}{{k}^{2}} \]
      unpow2 [=>] 10.4	\[ \frac{a}{\color{blue}{k \cdot k}} \]

   6. Applied egg-rr 5.3

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

   7. Applied egg-rr 5.7

      \[\leadsto \color{blue}{\frac{1}{\frac{k}{\frac{a}{k}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{+155}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
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	3.0
Cost	7305

\[\begin{array}{l} \mathbf{if}\;m \leq -3 \cdot 10^{-19} \lor \neg \left(m \leq 5.4 \cdot 10^{-8}\right):\\ \;\;\;\;a \cdot \left({k}^{m} \cdot \left(1 + k \cdot -10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 3
Error	2.9
Cost	7304

\[\begin{array}{l} \mathbf{if}\;m \leq -3 \cdot 10^{-19}:\\ \;\;\;\;a \cdot \left({k}^{m} \cdot \left(1 + k \cdot -10\right)\right)\\ \mathbf{elif}\;m \leq 6.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot 10}{{k}^{m}}}\\ \end{array} \]

Alternative 4
Error	0.6
Cost	7300

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

Alternative 5
Error	2.0
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{+155}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot k}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

Alternative 6
Error	2.8
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -8.8 \cdot 10^{-17} \lor \neg \left(m \leq 2.4 \cdot 10^{-8}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 7
Error	11.9
Cost	1873

\[\begin{array}{l} \mathbf{if}\;m \leq -9500:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 0.172:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 2.2 \cdot 10^{+156} \lor \neg \left(m \leq 4.4 \cdot 10^{+201}\right):\\ \;\;\;\;\frac{a}{1 + k \cdot \left(\left(100 - k \cdot k\right) \cdot \left(\frac{-1}{k} + \frac{-10}{k \cdot k}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \left(1 + a \cdot 0.1\right)}{k}\\ \end{array} \]

Alternative 8
Error	21.9
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{-295}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 9
Error	22.0
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{-295}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

Alternative 10
Error	21.9
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{-295}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

Alternative 11
Error	22.1
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{-295}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{+155}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

Alternative 12
Error	16.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -9500:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 13
Error	13.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -9500:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 0.24:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \left(1 + a \cdot 0.1\right)}{k}\\ \end{array} \]

Alternative 14
Error	23.1
Cost	716

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{-295}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Error	22.1
Cost	716

\[\begin{array}{l} \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{-295}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 16
Error	17.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq -9500:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 1.75:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 17
Error	43.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 0.65:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 18
Error	43.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 0.62:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 19
Error	47.0
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023039 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))