Falkner and Boettcher, Appendix A

Average Accuracy: 96.5% → 99.9%

Time: 14.2s

Precision: binary64

Cost: 7364

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{a}{-1 + k \cdot \left(-10 - k\right)} \cdot \left(-{k}^{m}\right)\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{a}{k}}{k}\\ \end{array} \]

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 2.2e+17)
   (* (/ a (+ -1.0 (* k (- -10.0 k)))) (- (pow k m)))
   (* (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 <= 2.2e+17) {
		tmp = (a / (-1.0 + (k * (-10.0 - k)))) * -pow(k, m);
	} 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 <= 2.2d+17) then
        tmp = (a / ((-1.0d0) + (k * ((-10.0d0) - k)))) * -(k ** m)
    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 <= 2.2e+17) {
		tmp = (a / (-1.0 + (k * (-10.0 - k)))) * -Math.pow(k, m);
	} 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 <= 2.2e+17:
		tmp = (a / (-1.0 + (k * (-10.0 - k)))) * -math.pow(k, m)
	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 <= 2.2e+17)
		tmp = Float64(Float64(a / Float64(-1.0 + Float64(k * Float64(-10.0 - k)))) * Float64(-(k ^ m)));
	else
		tmp = Float64((k ^ m) * Float64(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 <= 2.2e+17)
		tmp = (a / (-1.0 + (k * (-10.0 - k)))) * -(k ^ m);
	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, 2.2e+17], N[(N[(a / N[(-1.0 + N[(k * N[(-10.0 - k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Power[k, m], $MachinePrecision])), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.2e17

   1. Initial program 99.9%

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

   2. Simplified 99.9%

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

      [Start] 99.9	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-/l* [=>] 99.9	\[ \color{blue}{\frac{a}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{{k}^{m}}}} \]
      associate-+l+ [=>] 99.9	\[ \frac{a}{\frac{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}{{k}^{m}}} \]
      *-commutative [=>] 99.9	\[ \frac{a}{\frac{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)}{{k}^{m}}} \]

   3. Applied egg-rr 100.0%

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

      [Start] 99.9	\[ \frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}} \]
      frac-2neg [=>] 99.9	\[ \frac{a}{\color{blue}{\frac{-\left(1 + \left(k \cdot 10 + k \cdot k\right)\right)}{-{k}^{m}}}} \]
      associate-/r/ [=>] 99.9	\[ \color{blue}{\frac{a}{-\left(1 + \left(k \cdot 10 + k \cdot k\right)\right)} \cdot \left(-{k}^{m}\right)} \]
      neg-sub0 [=>] 99.9	\[ \frac{a}{\color{blue}{0 - \left(1 + \left(k \cdot 10 + k \cdot k\right)\right)}} \cdot \left(-{k}^{m}\right) \]
      metadata-eval [<=] 99.9	\[ \frac{a}{\color{blue}{\log 1} - \left(1 + \left(k \cdot 10 + k \cdot k\right)\right)} \cdot \left(-{k}^{m}\right) \]
      associate--r+ [=>] 99.9	\[ \frac{a}{\color{blue}{\left(\log 1 - 1\right) - \left(k \cdot 10 + k \cdot k\right)}} \cdot \left(-{k}^{m}\right) \]
      metadata-eval [=>] 99.9	\[ \frac{a}{\left(\color{blue}{0} - 1\right) - \left(k \cdot 10 + k \cdot k\right)} \cdot \left(-{k}^{m}\right) \]
      metadata-eval [=>] 99.9	\[ \frac{a}{\color{blue}{-1} - \left(k \cdot 10 + k \cdot k\right)} \cdot \left(-{k}^{m}\right) \]
      distribute-lft-out [=>] 100.0	\[ \frac{a}{-1 - \color{blue}{k \cdot \left(10 + k\right)}} \cdot \left(-{k}^{m}\right) \]
      +-commutative [=>] 100.0	\[ \frac{a}{-1 - k \cdot \color{blue}{\left(k + 10\right)}} \cdot \left(-{k}^{m}\right) \]

   if 2.2e17 < k

   1. Initial program 90.3%

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

   2. Simplified 90.2%

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

      [Start] 90.3	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-/l* [=>] 90.2	\[ \color{blue}{\frac{a}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{{k}^{m}}}} \]
      associate-+l+ [=>] 90.2	\[ \frac{a}{\frac{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}{{k}^{m}}} \]
      *-commutative [=>] 90.2	\[ \frac{a}{\frac{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)}{{k}^{m}}} \]

   3. Applied egg-rr 90.2%

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

      [Start] 90.2	\[ \frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}} \]
      frac-2neg [=>] 90.2	\[ \frac{a}{\color{blue}{\frac{-\left(1 + \left(k \cdot 10 + k \cdot k\right)\right)}{-{k}^{m}}}} \]
      associate-/r/ [=>] 90.2	\[ \color{blue}{\frac{a}{-\left(1 + \left(k \cdot 10 + k \cdot k\right)\right)} \cdot \left(-{k}^{m}\right)} \]
      neg-sub0 [=>] 90.2	\[ \frac{a}{\color{blue}{0 - \left(1 + \left(k \cdot 10 + k \cdot k\right)\right)}} \cdot \left(-{k}^{m}\right) \]
      metadata-eval [<=] 90.2	\[ \frac{a}{\color{blue}{\log 1} - \left(1 + \left(k \cdot 10 + k \cdot k\right)\right)} \cdot \left(-{k}^{m}\right) \]
      associate--r+ [=>] 90.2	\[ \frac{a}{\color{blue}{\left(\log 1 - 1\right) - \left(k \cdot 10 + k \cdot k\right)}} \cdot \left(-{k}^{m}\right) \]
      metadata-eval [=>] 90.2	\[ \frac{a}{\left(\color{blue}{0} - 1\right) - \left(k \cdot 10 + k \cdot k\right)} \cdot \left(-{k}^{m}\right) \]
      metadata-eval [=>] 90.2	\[ \frac{a}{\color{blue}{-1} - \left(k \cdot 10 + k \cdot k\right)} \cdot \left(-{k}^{m}\right) \]
      distribute-lft-out [=>] 90.2	\[ \frac{a}{-1 - \color{blue}{k \cdot \left(10 + k\right)}} \cdot \left(-{k}^{m}\right) \]
      +-commutative [=>] 90.2	\[ \frac{a}{-1 - k \cdot \color{blue}{\left(k + 10\right)}} \cdot \left(-{k}^{m}\right) \]

   4. Taylor expanded in k around inf 90.2%

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

   5. Simplified 99.7%

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

      [Start] 90.2	\[ \left(-1 \cdot \frac{a}{{k}^{2}}\right) \cdot \left(-{k}^{m}\right) \]
      associate-*r/ [=>] 90.2	\[ \color{blue}{\frac{-1 \cdot a}{{k}^{2}}} \cdot \left(-{k}^{m}\right) \]
      mul-1-neg [=>] 90.2	\[ \frac{\color{blue}{-a}}{{k}^{2}} \cdot \left(-{k}^{m}\right) \]
      unpow2 [=>] 90.2	\[ \frac{-a}{\color{blue}{k \cdot k}} \cdot \left(-{k}^{m}\right) \]
      associate-/r* [=>] 99.7	\[ \color{blue}{\frac{\frac{-a}{k}}{k}} \cdot \left(-{k}^{m}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{a}{-1 + k \cdot \left(-10 - k\right)} \cdot \left(-{k}^{m}\right)\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	7300

\[\begin{array}{l} \mathbf{if}\;k \leq 10000000000000:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 2
Accuracy	95.7%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;m \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 3.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{\frac{{k}^{m}}{k}}{k}\\ \end{array} \]

Alternative 3
Accuracy	99.0%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 2.7:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot 10}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 4
Accuracy	98.6%
Cost	7044

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

Alternative 5
Accuracy	95.7%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -2.4 \cdot 10^{-7} \lor \neg \left(m \leq 0.00023\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \end{array} \]

Alternative 6
Accuracy	67.9%
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -1.1 \cdot 10^{+53} \lor \neg \left(m \leq 2.55 \cdot 10^{+25}\right):\\ \;\;\;\;-1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 7
Accuracy	69.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -1.12 \cdot 10^{+35} \lor \neg \left(m \leq 4.2 \cdot 10^{+23}\right):\\ \;\;\;\;-1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \end{array} \]

Alternative 8
Accuracy	61.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;k \leq -9.6 \lor \neg \left(k \leq 10\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \end{array} \]

Alternative 9
Accuracy	60.7%
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 10
Accuracy	60.7%
Cost	448

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

Alternative 11
Accuracy	26.7%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023133 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))