Falkner and Boettcher, Appendix A

Average Accuracy: 68.5% → 70.6%

Time: 15.8s

Precision: binary64

Cost: 7428

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+35}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k}\\ \end{array} \]

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 5e+35)
   (/ a (/ (+ 1.0 (+ (* k 10.0) (* k k))) (pow k m)))
   (* (/ (pow k m) 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 <= 5e+35) {
		tmp = a / ((1.0 + ((k * 10.0) + (k * k))) / pow(k, m));
	} else {
		tmp = (pow(k, m) / 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 <= 5d+35) then
        tmp = a / ((1.0d0 + ((k * 10.0d0) + (k * k))) / (k ** m))
    else
        tmp = ((k ** m) / 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 <= 5e+35) {
		tmp = a / ((1.0 + ((k * 10.0) + (k * k))) / Math.pow(k, m));
	} else {
		tmp = (Math.pow(k, m) / 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 <= 5e+35:
		tmp = a / ((1.0 + ((k * 10.0) + (k * k))) / math.pow(k, m))
	else:
		tmp = (math.pow(k, m) / 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 <= 5e+35)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(Float64(k * 10.0) + Float64(k * k))) / (k ^ m)));
	else
		tmp = Float64(Float64((k ^ m) / 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 <= 5e+35)
		tmp = a / ((1.0 + ((k * 10.0) + (k * k))) / (k ^ m));
	else
		tmp = ((k ^ m) / 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, 5e+35], 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[Power[k, m], $MachinePrecision] / k), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.00000000000000021e35

   1. Initial program 68.1%

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

   2. Simplified 68.1%

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

      [Start] 68.1	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-/l* [=>] 68.1	\[ \color{blue}{\frac{a}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{{k}^{m}}}} \]
      associate-+l+ [=>] 68.1	\[ \frac{a}{\frac{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}{{k}^{m}}} \]
      *-commutative [=>] 68.1	\[ \frac{a}{\frac{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)}{{k}^{m}}} \]

   if 5.00000000000000021e35 < k

   1. Initial program 69.3%

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

   2. Simplified 69.2%

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

      [Start] 69.3	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-*r/ [<=] 69.2	\[ \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      associate-+l+ [=>] 69.2	\[ a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
      +-commutative [=>] 69.2	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}} \]
      distribute-rgt-out [=>] 69.2	\[ a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]
      fma-def [=>] 69.2	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
      +-commutative [=>] 69.2	\[ a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

   3. Taylor expanded in k around -inf 0.0%

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

   4. Simplified 75.9%

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

      [Start] 0.0	\[ \frac{a \cdot e^{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right) \cdot m}}{{k}^{2}} \]
      unpow2 [=>] 0.0	\[ \frac{a \cdot e^{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right) \cdot m}}{\color{blue}{k \cdot k}} \]
      times-frac [=>] 0.0	\[ \color{blue}{\frac{a}{k} \cdot \frac{e^{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right) \cdot m}}{k}} \]
      exp-prod [=>] 0.0	\[ \frac{a}{k} \cdot \frac{\color{blue}{{\left(e^{\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)}\right)}^{m}}}{k} \]
      exp-sum [=>] 0.0	\[ \frac{a}{k} \cdot \frac{{\color{blue}{\left(e^{\log -1} \cdot e^{-1 \cdot \log \left(\frac{-1}{k}\right)}\right)}}^{m}}{k} \]
      rem-exp-log [=>] 0.0	\[ \frac{a}{k} \cdot \frac{{\left(\color{blue}{-1} \cdot e^{-1 \cdot \log \left(\frac{-1}{k}\right)}\right)}^{m}}{k} \]
      *-commutative [=>] 0.0	\[ \frac{a}{k} \cdot \frac{{\left(-1 \cdot e^{\color{blue}{\log \left(\frac{-1}{k}\right) \cdot -1}}\right)}^{m}}{k} \]
      exp-to-pow [=>] 75.9	\[ \frac{a}{k} \cdot \frac{{\left(-1 \cdot \color{blue}{{\left(\frac{-1}{k}\right)}^{-1}}\right)}^{m}}{k} \]

   5. Taylor expanded in a around 0 69.3%

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

   6. Simplified 75.9%

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

      [Start] 69.3	\[ \frac{e^{\log k \cdot m} \cdot a}{{k}^{2}} \]
      associate-/l* [=>] 69.1	\[ \color{blue}{\frac{e^{\log k \cdot m}}{\frac{{k}^{2}}{a}}} \]
      *-commutative [=>] 69.1	\[ \frac{e^{\color{blue}{m \cdot \log k}}}{\frac{{k}^{2}}{a}} \]
      exp-prod [=>] 69.0	\[ \frac{\color{blue}{{\left(e^{m}\right)}^{\log k}}}{\frac{{k}^{2}}{a}} \]
      remove-double-neg [<=] 69.0	\[ \frac{{\left(e^{m}\right)}^{\log \color{blue}{\left(-\left(-k\right)\right)}}}{\frac{{k}^{2}}{a}} \]
      neg-mul-1 [=>] 69.0	\[ \frac{{\left(e^{m}\right)}^{\log \color{blue}{\left(-1 \cdot \left(-k\right)\right)}}}{\frac{{k}^{2}}{a}} \]
      log-prod [=>] 41.2	\[ \frac{{\left(e^{m}\right)}^{\color{blue}{\left(\log -1 + \log \left(-k\right)\right)}}}{\frac{{k}^{2}}{a}} \]
      neg-mul-1 [=>] 41.2	\[ \frac{{\left(e^{m}\right)}^{\left(\log -1 + \log \color{blue}{\left(-1 \cdot k\right)}\right)}}{\frac{{k}^{2}}{a}} \]
      metadata-eval [<=] 41.2	\[ \frac{{\left(e^{m}\right)}^{\left(\log -1 + \log \left(\color{blue}{\frac{1}{-1}} \cdot k\right)\right)}}{\frac{{k}^{2}}{a}} \]
      associate-/r/ [<=] 41.2	\[ \frac{{\left(e^{m}\right)}^{\left(\log -1 + \log \color{blue}{\left(\frac{1}{\frac{-1}{k}}\right)}\right)}}{\frac{{k}^{2}}{a}} \]
      log-rec [=>] 41.2	\[ \frac{{\left(e^{m}\right)}^{\left(\log -1 + \color{blue}{\left(-\log \left(\frac{-1}{k}\right)\right)}\right)}}{\frac{{k}^{2}}{a}} \]
      mul-1-neg [<=] 41.2	\[ \frac{{\left(e^{m}\right)}^{\left(\log -1 + \color{blue}{-1 \cdot \log \left(\frac{-1}{k}\right)}\right)}}{\frac{{k}^{2}}{a}} \]
      exp-prod [<=] 0.0	\[ \frac{\color{blue}{e^{m \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right)}}}{\frac{{k}^{2}}{a}} \]
      *-commutative [<=] 0.0	\[ \frac{e^{\color{blue}{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right) \cdot m}}}{\frac{{k}^{2}}{a}} \]
      associate-/l* [<=] 0.0	\[ \color{blue}{\frac{e^{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right) \cdot m} \cdot a}{{k}^{2}}} \]
      unpow2 [=>] 0.0	\[ \frac{e^{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right) \cdot m} \cdot a}{\color{blue}{k \cdot k}} \]
      times-frac [=>] 0.0	\[ \color{blue}{\frac{e^{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right) \cdot m}}{k} \cdot \frac{a}{k}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+35}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	70.0%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot 10}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 2
Accuracy	69.8%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 3
Accuracy	67.9%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -1.05 \cdot 10^{-7} \lor \neg \left(m \leq 6.5 \cdot 10^{-10}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 4
Accuracy	67.8%
Cost	6920

\[\begin{array}{l} \mathbf{if}\;m \leq -8.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{elif}\;m \leq 2.05 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 5
Accuracy	49.6%
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -6.5 \cdot 10^{+49} \lor \neg \left(m \leq 180000\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 6
Accuracy	50.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -6.2 \cdot 10^{+49} \lor \neg \left(m \leq 400000\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 7
Accuracy	45.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -1.25 \cdot 10^{+64}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 8
Accuracy	28.3%
Cost	585

\[\begin{array}{l} \mathbf{if}\;k \leq -5.6 \cdot 10^{+67} \lor \neg \left(k \leq 1.4 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 9
Accuracy	44.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;k \leq -8.6 \cdot 10^{-79} \lor \neg \left(k \leq 1.4 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 10
Accuracy	45.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;k \leq -1.06 \cdot 10^{-78}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 11
Accuracy	45.7%
Cost	580

\[\begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 12
Accuracy	19.9%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023153 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))