Falkner and Boettcher, Appendix A

Average Error: 1.8 → 0.2

Time: 12.8s

Precision: binary64

Cost: 13636

Math

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

↓

\[\begin{array}{l} t_0 := {k}^{\left(-m\right)}\\ \mathbf{if}\;k \leq 50000000:\\ \;\;\;\;\frac{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{a}{k}}{k}}{t_0}\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))

↓

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (pow k (- m))))
   (if (<= k 50000000.0)
     (/ (/ a (fma k (+ k 10.0) 1.0)) t_0)
     (/ (/ (/ a k) k) t_0))))

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 t_0 = pow(k, -m);
	double tmp;
	if (k <= 50000000.0) {
		tmp = (a / fma(k, (k + 10.0), 1.0)) / t_0;
	} else {
		tmp = ((a / k) / k) / t_0;
	}
	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)
	t_0 = k ^ Float64(-m)
	tmp = 0.0
	if (k <= 50000000.0)
		tmp = Float64(Float64(a / fma(k, Float64(k + 10.0), 1.0)) / t_0);
	else
		tmp = Float64(Float64(Float64(a / k) / k) / t_0);
	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_] := Block[{t$95$0 = N[Power[k, (-m)], $MachinePrecision]}, If[LessEqual[k, 50000000.0], N[(N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5e7

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-/l* [=>] 0.1	\[ \color{blue}{\frac{a}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{{k}^{m}}}} \]
      associate-+l+ [=>] 0.1	\[ \frac{a}{\frac{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}{{k}^{m}}} \]
      *-commutative [=>] 0.1	\[ \frac{a}{\frac{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)}{{k}^{m}}} \]

   3. Applied egg-rr 0.0

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

   4. Simplified 0.0

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

      [Start] 0.0	\[ \frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot \frac{a}{{k}^{\left(-m\right)}} \]
      associate-*r/ [=>] 0.0	\[ \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot a}{{k}^{\left(-m\right)}}} \]
      associate-*l/ [=>] 0.0	\[ \frac{\color{blue}{\frac{1 \cdot a}{\mathsf{fma}\left(k, k + 10, 1\right)}}}{{k}^{\left(-m\right)}} \]
      *-commutative [<=] 0.0	\[ \frac{\frac{\color{blue}{a \cdot 1}}{\mathsf{fma}\left(k, k + 10, 1\right)}}{{k}^{\left(-m\right)}} \]
      associate-*l/ [<=] 0.0	\[ \frac{\color{blue}{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot 1}}{{k}^{\left(-m\right)}} \]
      *-rgt-identity [=>] 0.0	\[ \frac{\color{blue}{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}}}{{k}^{\left(-m\right)}} \]

   if 5e7 < k

   1. Initial program 4.7

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

   2. Simplified 4.7

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

      [Start] 4.7	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-/l* [=>] 4.7	\[ \color{blue}{\frac{a}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{{k}^{m}}}} \]
      associate-+l+ [=>] 4.7	\[ \frac{a}{\frac{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}{{k}^{m}}} \]
      *-commutative [=>] 4.7	\[ \frac{a}{\frac{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)}{{k}^{m}}} \]

   3. Applied egg-rr 4.7

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

   4. Simplified 4.7

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

      [Start] 4.7	\[ \frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot \frac{a}{{k}^{\left(-m\right)}} \]
      associate-*r/ [=>] 4.7	\[ \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot a}{{k}^{\left(-m\right)}}} \]
      associate-*l/ [=>] 4.7	\[ \frac{\color{blue}{\frac{1 \cdot a}{\mathsf{fma}\left(k, k + 10, 1\right)}}}{{k}^{\left(-m\right)}} \]
      *-commutative [<=] 4.7	\[ \frac{\frac{\color{blue}{a \cdot 1}}{\mathsf{fma}\left(k, k + 10, 1\right)}}{{k}^{\left(-m\right)}} \]
      associate-*l/ [<=] 4.7	\[ \frac{\color{blue}{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot 1}}{{k}^{\left(-m\right)}} \]
      *-rgt-identity [=>] 4.7	\[ \frac{\color{blue}{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}}}{{k}^{\left(-m\right)}} \]

   5. Taylor expanded in k around inf 4.9

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

   6. Simplified 0.4

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

      [Start] 4.9	\[ \frac{\frac{a}{{k}^{2}}}{{k}^{\left(-m\right)}} \]
      unpow2 [=>] 4.9	\[ \frac{\frac{a}{\color{blue}{k \cdot k}}}{{k}^{\left(-m\right)}} \]
      associate-/r* [=>] 0.4	\[ \frac{\color{blue}{\frac{\frac{a}{k}}{k}}}{{k}^{\left(-m\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 50000000:\\ \;\;\;\;\frac{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{a}{k}}{k}}{{k}^{\left(-m\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 50000000:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{a}{k}}{k}}{{k}^{\left(-m\right)}}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	7172

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

Alternative 3
Error	0.8
Cost	7108

\[\begin{array}{l} t_0 := {k}^{\left(-m\right)}\\ \mathbf{if}\;k \leq 1:\\ \;\;\;\;\frac{a}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{a}{k}}{k}}{t_0}\\ \end{array} \]

Alternative 4
Error	2.4
Cost	6984

\[\begin{array}{l} \mathbf{if}\;m \leq -8.7 \cdot 10^{-10}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 7.1 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \end{array} \]

Alternative 5
Error	2.4
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -1.85 \cdot 10^{-5} \lor \neg \left(m \leq 7.8 \cdot 10^{-10}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 6
Error	19.9
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -2.1 \cdot 10^{+28} \lor \neg \left(m \leq 13.2\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 7
Error	19.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -2.1 \cdot 10^{+28} \lor \neg \left(m \leq 6\right):\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 8
Error	24.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -1.26:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a - a \cdot \left(k \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 9
Error	24.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -1.26:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a - a \cdot \left(k \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 10
Error	24.0
Cost	712

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

Alternative 11
Error	24.7
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 12
Error	24.1
Cost	584

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

Alternative 13
Error	24.7
Cost	448

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

Alternative 14
Error	46.7
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023060 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))