Falkner and Boettcher, Appendix A

Average Accuracy: 67.4% → 69.7%

Time: 15.1s

Precision: binary64

Cost: 7428

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{+96}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

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 3e+96)
   (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k)))
   (* (/ a k) (/ (pow k m) 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 <= 3e+96) {
		tmp = (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = (a / k) * (pow(k, m) / 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 <= 3d+96) then
        tmp = (a * (k ** m)) / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = (a / k) * ((k ** m) / 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 <= 3e+96) {
		tmp = (a * Math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = (a / k) * (Math.pow(k, m) / 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 <= 3e+96:
		tmp = (a * math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = (a / k) * (math.pow(k, m) / 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 <= 3e+96)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64(Float64(a / k) * Float64((k ^ m) / 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 <= 3e+96)
		tmp = (a * (k ^ m)) / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = (a / k) * ((k ^ m) / 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, 3e+96], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] * N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3e96

   1. Initial program 66.6%

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

   if 3e96 < k

   1. Initial program 59.7%

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

   2. Simplified 59.7%

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

      [Start] 59.7	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-/l* [=>] 59.7	\[ \color{blue}{\frac{a}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{{k}^{m}}}} \]
      associate-+l+ [=>] 59.7	\[ \frac{a}{\frac{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}{{k}^{m}}} \]
      *-commutative [=>] 59.7	\[ \frac{a}{\frac{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)}{{k}^{m}}} \]

   3. Taylor expanded in k around inf 59.7%

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

   4. Simplified 67.1%

      \[\leadsto \color{blue}{\frac{a}{k} \cdot \frac{{k}^{m}}{k}} \]
      Step-by-step derivation

      [Start] 59.7	\[ \frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{{k}^{2}} \]
      unpow2 [=>] 59.7	\[ \frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{\color{blue}{k \cdot k}} \]
      times-frac [=>] 67.1	\[ \color{blue}{\frac{a}{k} \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{k}} \]
      rem-exp-log [<=] 65.9	\[ \frac{a}{k} \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{\color{blue}{e^{\log k}}} \]
      div-exp [=>] 65.9	\[ \frac{a}{k} \cdot \color{blue}{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right) - \log k}} \]
      mul-1-neg [=>] 65.9	\[ \frac{a}{k} \cdot e^{\color{blue}{\left(-\log \left(\frac{1}{k}\right) \cdot m\right)} - \log k} \]
      log-rec [=>] 65.9	\[ \frac{a}{k} \cdot e^{\left(-\color{blue}{\left(-\log k\right)} \cdot m\right) - \log k} \]
      distribute-lft-neg-in [<=] 65.9	\[ \frac{a}{k} \cdot e^{\left(-\color{blue}{\left(-\log k \cdot m\right)}\right) - \log k} \]
      remove-double-neg [=>] 65.9	\[ \frac{a}{k} \cdot e^{\color{blue}{\log k \cdot m} - \log k} \]
      div-exp [<=] 65.9	\[ \frac{a}{k} \cdot \color{blue}{\frac{e^{\log k \cdot m}}{e^{\log k}}} \]
      exp-to-pow [=>] 65.9	\[ \frac{a}{k} \cdot \frac{\color{blue}{{k}^{m}}}{e^{\log k}} \]
      rem-exp-log [=>] 67.1	\[ \frac{a}{k} \cdot \frac{{k}^{m}}{\color{blue}{k}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{+96}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	69.7%
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

Alternative 2
Accuracy	68.7%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

Alternative 3
Accuracy	66.3%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;m \leq -1.02 \cdot 10^{-13}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 7.1 \cdot 10^{-29}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \end{array} \]

Alternative 4
Accuracy	66.3%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -1.02 \cdot 10^{-13} \lor \neg \left(m \leq 7.1 \cdot 10^{-29}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 5
Accuracy	66.4%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(2 - m\right)}}\\ \end{array} \]

Alternative 6
Accuracy	49.4%
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -3.7 \cdot 10^{+42}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 4.2 \cdot 10^{-307}:\\ \;\;\;\;99 \cdot \left(a \cdot \left(k \cdot k\right)\right)\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-12}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 7
Accuracy	57.6%
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -3.15 \cdot 10^{+18}:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 1.1:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;99 \cdot \left(a \cdot \left(k \cdot k\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	49.3%
Cost	716

\[\begin{array}{l} \mathbf{if}\;k \leq -3.7 \cdot 10^{+42}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{-306}:\\ \;\;\;\;99 \cdot \left(a \cdot \left(k \cdot k\right)\right)\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-12}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 9
Accuracy	56.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq -2.1 \cdot 10^{+18}:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 0.62:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;99 \cdot \left(a \cdot \left(k \cdot k\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	43.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;k \leq -3.7 \cdot 10^{+42} \lor \neg \left(k \leq 2.1 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 11
Accuracy	44.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;k \leq -3.7 \cdot 10^{+42}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-12}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 12
Accuracy	53.7%
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 1.25:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;99 \cdot \left(a \cdot \left(k \cdot k\right)\right)\\ \end{array} \]

Alternative 13
Accuracy	23.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 1050000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 14
Accuracy	19.7%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023157 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))