Falkner and Boettcher, Appendix A

Percentage Accurate: 96.4% → 99.8%

Time: 41.8s

Precision: binary64

Cost: 13572

• could not determine a ground truth

  1. a = 1.2984565629019837e+144
  2. k = -4.165148359892114e+85
  3. m = 2.698693100047449e+196

Math

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

↓

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

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 5.6e+17)
   (* a (/ (pow k m) (fma k (+ k 10.0) 1.0)))
   (/ (/ (pow k m) k) (/ k a))))

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 <= 5.6e+17) {
		tmp = a * (pow(k, m) / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = (pow(k, m) / k) / (k / a);
	}
	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 <= 5.6e+17)
		tmp = Float64(a * Float64((k ^ m) / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = Float64(Float64((k ^ m) / k) / Float64(k / a));
	end
	return 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, 5.6e+17], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision] / N[(k / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.6e17

   1. Initial program 99.4%

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

   2. Simplified 100.0%

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

      [Start] 99.4%	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-*r/ [<=] 99.4%	\[ \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      associate-+l+ [=>] 99.4%	\[ a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
      +-commutative [=>] 99.4%	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}} \]
      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%	\[ a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

   if 5.6e17 < k

   1. Initial program 85.8%

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

   2. Simplified 85.7%

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

      [Start] 85.8%	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-*r/ [<=] 85.7%	\[ \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      associate-+l+ [=>] 85.7%	\[ a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
      +-commutative [=>] 85.7%	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}} \]
      distribute-rgt-out [=>] 85.7%	\[ a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]
      fma-def [=>] 85.7%	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
      +-commutative [=>] 85.7%	\[ a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

   3. Taylor expanded in k around inf 85.7%

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

   4. Simplified 85.7%

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

      [Start] 85.7%	\[ a \cdot \frac{{k}^{m}}{{k}^{2}} \]
      unpow2 [=>] 85.7%	\[ a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot k}} \]

   5. Applied egg-rr 99.9%

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

      [Start] 85.7%	\[ a \cdot \frac{{k}^{m}}{k \cdot k} \]
      associate-*r/ [=>] 85.8%	\[ \color{blue}{\frac{a \cdot {k}^{m}}{k \cdot k}} \]
      remove-double-div [<=] 85.8%	\[ \frac{a \cdot \color{blue}{\frac{1}{\frac{1}{{k}^{m}}}}}{k \cdot k} \]
      frac-times [<=] 99.8%	\[ \color{blue}{\frac{a}{k} \cdot \frac{\frac{1}{\frac{1}{{k}^{m}}}}{k}} \]
      *-commutative [=>] 99.8%	\[ \color{blue}{\frac{\frac{1}{\frac{1}{{k}^{m}}}}{k} \cdot \frac{a}{k}} \]
      clear-num [=>] 99.8%	\[ \frac{\frac{1}{\frac{1}{{k}^{m}}}}{k} \cdot \color{blue}{\frac{1}{\frac{k}{a}}} \]
      un-div-inv [=>] 99.9%	\[ \color{blue}{\frac{\frac{\frac{1}{\frac{1}{{k}^{m}}}}{k}}{\frac{k}{a}}} \]
      remove-double-div [=>] 99.9%	\[ \frac{\frac{\color{blue}{{k}^{m}}}{k}}{\frac{k}{a}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	13572

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

Alternative 2
Accuracy	99.8%
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 1.26 \cdot 10^{+17}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{k}}{\frac{k}{a}}\\ \end{array} \]

Alternative 3
Accuracy	97.0%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{+154}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{-1}{k}}{-k}\\ \end{array} \]

Alternative 4
Accuracy	99.0%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 10.5:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{k}}{\frac{k}{a}}\\ \end{array} \]

Alternative 5
Accuracy	97.0%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{+156}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{-1}{k}}{-k}\\ \end{array} \]

Alternative 6
Accuracy	96.9%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{+155}:\\ \;\;\;\;\frac{a}{{k}^{\left(2 - m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{-1}{k}}{-k}\\ \end{array} \]

Alternative 7
Accuracy	98.7%
Cost	7044

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

Alternative 8
Accuracy	92.2%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;k \leq 6500000000000:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k}}{\frac{k}{a}}\\ \end{array} \]

Alternative 9
Accuracy	65.6%
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{-308}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 10
Accuracy	65.9%
Cost	844

\[\begin{array}{l} t_0 := -10 \cdot \left(k \cdot a\right)\\ \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{-308}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 0.098:\\ \;\;\;\;a + t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 11
Accuracy	66.0%
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{-308}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 10.5:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 12
Accuracy	63.9%
Cost	716

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{-308}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Accuracy	65.6%
Cost	716

\[\begin{array}{l} \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{-308}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 14
Accuracy	67.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 0.94:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 15
Accuracy	66.8%
Cost	580

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

Alternative 16
Accuracy	32.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 6500000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 17
Accuracy	32.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 6500000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 18
Accuracy	27.0%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023263 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))