Falkner and Boettcher, Appendix A

Average Accuracy: 96.9% → 98.2%

Time: 14.9s

Precision: binary64

Cost: 7176

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-17}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 5.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{a}{k}}{k}\\ \end{array} \]

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 2e-17)
   (* a (pow k m))
   (if (<= k 5.6e+29)
     (/ 1.0 (/ (+ 1.0 (* k (+ k 10.0))) a))
     (/ (* (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 <= 2e-17) {
		tmp = a * pow(k, m);
	} else if (k <= 5.6e+29) {
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a);
	} 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 <= 2d-17) then
        tmp = a * (k ** m)
    else if (k <= 5.6d+29) then
        tmp = 1.0d0 / ((1.0d0 + (k * (k + 10.0d0))) / a)
    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 <= 2e-17) {
		tmp = a * Math.pow(k, m);
	} else if (k <= 5.6e+29) {
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a);
	} 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 <= 2e-17:
		tmp = a * math.pow(k, m)
	elif k <= 5.6e+29:
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a)
	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 <= 2e-17)
		tmp = Float64(a * (k ^ m));
	elseif (k <= 5.6e+29)
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / a));
	else
		tmp = Float64(Float64((k ^ m) * Float64(a / k)) / 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 <= 2e-17)
		tmp = a * (k ^ m);
	elseif (k <= 5.6e+29)
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a);
	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, 2e-17], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.6e+29], N[(1.0 / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.00000000000000014e-17

   1. Initial program 99.9%

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

   2. Simplified 100.0%

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

      [Start] 99.9	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-*r/ [<=] 99.9	\[ \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      associate-+l+ [=>] 99.9	\[ a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
      +-commutative [=>] 99.9	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}} \]
      distribute-rgt-out [=>] 100.0	\[ a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]
      fma-def [=>] 100.0	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
      +-commutative [=>] 100.0	\[ a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

   3. Taylor expanded in k around 0 60.1%

      \[\leadsto \color{blue}{e^{\log k \cdot m} \cdot a} \]

   4. Simplified 100.0%

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

      [Start] 60.1	\[ e^{\log k \cdot m} \cdot a \]
      exp-to-pow [=>] 100.0	\[ \color{blue}{{k}^{m}} \cdot a \]
      *-commutative [=>] 100.0	\[ \color{blue}{a \cdot {k}^{m}} \]

   if 2.00000000000000014e-17 < k < 5.5999999999999999e29

   1. Initial program 99.6%

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

   2. Simplified 99.6%

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

      [Start] 99.6	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-*r/ [<=] 99.6	\[ \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      associate-+l+ [=>] 99.6	\[ a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
      +-commutative [=>] 99.6	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}} \]
      distribute-rgt-out [=>] 99.6	\[ a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]
      fma-def [=>] 99.6	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
      +-commutative [=>] 99.6	\[ a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

   3. Applied egg-rr 99.0%

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

      [Start] 99.6	\[ a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)} \]
      associate-*r/ [=>] 99.7	\[ \color{blue}{\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}} \]
      clear-num [=>] 99.0	\[ \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a \cdot {k}^{m}}}} \]

   4. Taylor expanded in m around 0 67.7%

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

   if 5.5999999999999999e29 < k

   1. Initial program 91.1%

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

   2. Simplified 91.0%

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

      [Start] 91.1	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-*r/ [<=] 91.0	\[ \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      associate-+l+ [=>] 91.0	\[ a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
      +-commutative [=>] 91.0	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}} \]
      distribute-rgt-out [=>] 91.0	\[ a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]
      fma-def [=>] 91.0	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
      +-commutative [=>] 91.0	\[ a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

   3. Taylor expanded in k around inf 91.0%

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

   4. Simplified 91.1%

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

      [Start] 91.0	\[ \frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{{k}^{2}} \]
      associate-/l* [=>] 91.1	\[ \color{blue}{\frac{a}{\frac{{k}^{2}}{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}}} \]
      associate-/r/ [=>] 91.1	\[ \color{blue}{\frac{a}{{k}^{2}} \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}} \]
      unpow2 [=>] 91.1	\[ \frac{a}{\color{blue}{k \cdot k}} \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \]
      associate-*r* [=>] 91.1	\[ \frac{a}{k \cdot k} \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{1}{k}\right)\right) \cdot m}} \]
      exp-prod [=>] 91.1	\[ \frac{a}{k \cdot k} \cdot \color{blue}{{\left(e^{-1 \cdot \log \left(\frac{1}{k}\right)}\right)}^{m}} \]
      mul-1-neg [=>] 91.1	\[ \frac{a}{k \cdot k} \cdot {\left(e^{\color{blue}{-\log \left(\frac{1}{k}\right)}}\right)}^{m} \]
      log-rec [=>] 91.1	\[ \frac{a}{k \cdot k} \cdot {\left(e^{-\color{blue}{\left(-\log k\right)}}\right)}^{m} \]
      remove-double-neg [=>] 91.1	\[ \frac{a}{k \cdot k} \cdot {\left(e^{\color{blue}{\log k}}\right)}^{m} \]
      rem-exp-log [=>] 91.1	\[ \frac{a}{k \cdot k} \cdot {\color{blue}{k}}^{m} \]
      remove-double-div [<=] 91.1	\[ \frac{a}{k \cdot k} \cdot \color{blue}{\frac{1}{\frac{1}{{k}^{m}}}} \]

   5. Applied egg-rr 99.9%

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

      [Start] 91.1	\[ \frac{a}{k \cdot k} \cdot \frac{1}{\frac{1}{{k}^{m}}} \]
      remove-double-div [=>] 91.1	\[ \frac{a}{k \cdot k} \cdot \color{blue}{{k}^{m}} \]
      *-commutative [=>] 91.1	\[ \color{blue}{{k}^{m} \cdot \frac{a}{k \cdot k}} \]
      associate-/r* [=>] 99.9	\[ {k}^{m} \cdot \color{blue}{\frac{\frac{a}{k}}{k}} \]
      associate-*r/ [=>] 99.9	\[ \color{blue}{\frac{{k}^{m} \cdot \frac{a}{k}}{k}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-17}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 5.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{a}{k}}{k}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	13572

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{+30}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{a}{k}}{k}\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{+31}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{a}{k}}{k}\\ \end{array} \]

Alternative 3
Accuracy	97.2%
Cost	7308

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 2 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 3500000000000:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{elif}\;k \leq 7 \cdot 10^{+153}:\\ \;\;\;\;\frac{t_0}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 4
Accuracy	97.1%
Cost	7180

\[\begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-17}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 2100000000000:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{elif}\;k \leq 7 \cdot 10^{+154}:\\ \;\;\;\;\frac{a}{{k}^{\left(2 - m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 5
Accuracy	92.5%
Cost	6788

\[\begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ \mathbf{if}\;k \leq 2 \cdot 10^{-17}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{1}{\frac{1 + t_0}{a}}\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{a}{\frac{1 - k \cdot \left(\left(k + 10\right) \cdot t_0\right)}{1 - t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 6
Accuracy	69.5%
Cost	2132

\[\begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ t_1 := \frac{a}{\frac{1 - k \cdot \left(\left(k + 10\right) \cdot t_0\right)}{1 - t_0}}\\ \mathbf{if}\;k \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq -1.4 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -8.6 \cdot 10^{-228}:\\ \;\;\;\;\left(1 + \frac{a}{k} \cdot 0.1\right) + -1\\ \mathbf{elif}\;k \leq 3.4 \cdot 10^{+131}:\\ \;\;\;\;\frac{a}{1 + t_0}\\ \mathbf{elif}\;k \leq 3.7 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 7
Accuracy	70.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -13500 \lor \neg \left(m \leq 1750\right):\\ \;\;\;\;\left(1 + \frac{a}{k} \cdot 0.1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 8
Accuracy	63.8%
Cost	712

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

Alternative 9
Accuracy	63.9%
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 10
Accuracy	64.2%
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 11
Accuracy	67.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -8.5 \cdot 10^{-228}:\\ \;\;\;\;\left(1 + \frac{a}{k} \cdot 0.1\right) + -1\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k + 10}\\ \end{array} \]

Alternative 12
Accuracy	39.2%
Cost	585

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

Alternative 13
Accuracy	39.6%
Cost	585

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

Alternative 14
Accuracy	62.2%
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 15
Accuracy	63.6%
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 16
Accuracy	28.0%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023147 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))