Falkner and Boettcher, Appendix A

Average Accuracy: 67.4% → 69.6%

Time: 15.5s

Precision: binary64

Cost: 13700

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{+93}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-{\left(\frac{-1}{k}\right)}^{-1}\right)}^{m}}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

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 6e+93)
   (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k)))
   (/ (pow (- (pow (/ -1.0 k) -1.0)) 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 <= 6e+93) {
		tmp = (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = pow(-pow((-1.0 / k), -1.0), 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 <= 6d+93) then
        tmp = (a * (k ** m)) / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = (-(((-1.0d0) / k) ** (-1.0d0)) ** 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 <= 6e+93) {
		tmp = (a * Math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = Math.pow(-Math.pow((-1.0 / k), -1.0), 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 <= 6e+93:
		tmp = (a * math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = math.pow(-math.pow((-1.0 / k), -1.0), 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 <= 6e+93)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64((Float64(-(Float64(-1.0 / k) ^ -1.0)) ^ m) / 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 <= 6e+93)
		tmp = (a * (k ^ m)) / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = (-((-1.0 / k) ^ -1.0) ^ 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, 6e+93], 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[Power[(-N[Power[N[(-1.0 / k), $MachinePrecision], -1.0], $MachinePrecision]), m], $MachinePrecision] / N[(k / N[(a / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.99999999999999957e93

   1. Initial program 66.3%

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

   if 5.99999999999999957e93 < k

   1. Initial program 61.0%

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

   2. Simplified 61.0%

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

      [Start] 61.0	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-/l* [=>] 61.0	\[ \color{blue}{\frac{a}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{{k}^{m}}}} \]
      associate-+l+ [=>] 61.0	\[ \frac{a}{\frac{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}{{k}^{m}}} \]
      *-commutative [=>] 61.0	\[ \frac{a}{\frac{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)}{{k}^{m}}} \]

   3. Applied egg-rr 61.0%

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

      [Start] 61.0	\[ \frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}} \]
      distribute-lft-out [=>] 61.0	\[ \frac{a}{\frac{1 + \color{blue}{k \cdot \left(10 + k\right)}}{{k}^{m}}} \]
      add-sqr-sqrt [=>] 60.9	\[ \frac{a}{\frac{1 + \color{blue}{\left(\sqrt{k} \cdot \sqrt{k}\right)} \cdot \left(10 + k\right)}{{k}^{m}}} \]
      associate-*l* [=>] 61.0	\[ \frac{a}{\frac{1 + \color{blue}{\sqrt{k} \cdot \left(\sqrt{k} \cdot \left(10 + k\right)\right)}}{{k}^{m}}} \]
      +-commutative [=>] 61.0	\[ \frac{a}{\frac{1 + \sqrt{k} \cdot \left(\sqrt{k} \cdot \color{blue}{\left(k + 10\right)}\right)}{{k}^{m}}} \]

   4. Taylor expanded in k around -inf 0.0%

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

   5. Simplified 68.1%

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

      [Start] 0.0	\[ -1 \cdot \frac{a \cdot e^{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right) \cdot m}}{{k}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}} \]
      associate-*r/ [=>] 0.0	\[ \color{blue}{\frac{-1 \cdot \left(a \cdot e^{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right) \cdot m}\right)}{{k}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}} \]
      unpow2 [=>] 0.0	\[ \frac{-1 \cdot \left(a \cdot e^{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right) \cdot m}\right)}{\color{blue}{\left(k \cdot k\right)} \cdot {\left(\sqrt{-1}\right)}^{2}} \]
      *-commutative [=>] 0.0	\[ \frac{-1 \cdot \left(a \cdot e^{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right) \cdot m}\right)}{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \left(k \cdot k\right)}} \]
      unpow2 [=>] 0.0	\[ \frac{-1 \cdot \left(a \cdot e^{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right) \cdot m}\right)}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \left(k \cdot k\right)} \]
      rem-square-sqrt [=>] 0.0	\[ \frac{-1 \cdot \left(a \cdot e^{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right) \cdot m}\right)}{\color{blue}{-1} \cdot \left(k \cdot k\right)} \]
      times-frac [=>] 0.0	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{a \cdot e^{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right) \cdot m}}{k \cdot k}} \]
      metadata-eval [=>] 0.0	\[ \color{blue}{1} \cdot \frac{a \cdot e^{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right) \cdot m}}{k \cdot k} \]
      *-lft-identity [=>] 0.0	\[ \color{blue}{\frac{a \cdot e^{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right) \cdot m}}{k \cdot k}} \]
      *-commutative [=>] 0.0	\[ \frac{\color{blue}{e^{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right) \cdot m} \cdot a}}{k \cdot k} \]
      associate-/l* [=>] 0.0	\[ \color{blue}{\frac{e^{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{k}\right)\right) \cdot m}}{\frac{k \cdot k}{a}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{+93}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-{\left(\frac{-1}{k}\right)}^{-1}\right)}^{m}}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	69.7%
Cost	13700

\[\begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+96}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{\left(-{\left(\frac{-1}{k}\right)}^{-1}\right)}^{m}}{k}\\ \end{array} \]

Alternative 2
Accuracy	67.4%
Cost	7296

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

Alternative 3
Accuracy	67.4%
Cost	7296

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

Alternative 4
Accuracy	66.4%
Cost	7044

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

Alternative 5
Accuracy	66.3%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;m \leq -9.2 \cdot 10^{-14}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 7.1 \cdot 10^{-29}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \end{array} \]

Alternative 6
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 7
Accuracy	46.8%
Cost	716

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -9.5 \cdot 10^{-133}:\\ \;\;\;\;-1 + \left(1 + t_0\right)\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-12}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{+223}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Accuracy	44.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -1.15 \cdot 10^{+43}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-12}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 9
Accuracy	47.3%
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq -1.62 \cdot 10^{+18}:\\ \;\;\;\;-1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 10
Accuracy	27.5%
Cost	585

\[\begin{array}{l} \mathbf{if}\;k \leq -3.6 \cdot 10^{+47} \lor \neg \left(k \leq 2.1 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 11
Accuracy	43.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;k \leq -6.5 \cdot 10^{+49} \lor \neg \left(k \leq 2.1 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 12
Accuracy	44.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;k \leq -2.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-12}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 13
Accuracy	43.4%
Cost	448

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

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))))