Falkner and Boettcher, Appendix A

Average Error: 2.0 → 0.1

Time: 1.9min

Precision: binary64

Cost: 13764

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{+24}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k, 10 + k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot \frac{\frac{a}{k}}{k}\\ \end{array} \]

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 1e+24)
   (/ (* (pow k m) a) (fma k (+ 10.0 k) 1.0))
   (* (exp (* -1.0 (* (log (/ 1.0 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 <= 1e+24) {
		tmp = (pow(k, m) * a) / fma(k, (10.0 + k), 1.0);
	} else {
		tmp = exp((-1.0 * (log((1.0 / 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 <= 1e+24)
		tmp = Float64(Float64((k ^ m) * a) / fma(k, Float64(10.0 + k), 1.0));
	else
		tmp = Float64(exp(Float64(-1.0 * Float64(log(Float64(1.0 / k)) * m))) * Float64(Float64(a / k) / k));
	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, 1e+24], N[(N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision] / N[(k * N[(10.0 + k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(-1.0 * N[(N[Log[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.9999999999999998e23

   1. Initial program 0.0

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

   2. Simplified 0.0

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

   if 9.9999999999999998e23 < k

   1. Initial program 5.6

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

   2. Taylor expanded in k around inf 5.4

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

   3. Taylor expanded in a around 0 5.6

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

   4. Simplified 0.1

      \[\leadsto e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot \color{blue}{\frac{\frac{a}{k}}{k}} \]
      Proof
3. Recombined 2 regimes into one program.

Alternatives

Alternative 1
Error	1.2
Cost	13572

\[\begin{array}{l} \mathbf{if}\;k \leq 2.55 \cdot 10^{+163}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k, 10 + k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 2
Error	5.5
Cost	7436

\[\begin{array}{l} t_0 := \frac{{k}^{m} \cdot a}{1 + 10 \cdot k}\\ t_1 := 1 + \begin{array}{l} \mathbf{if}\;10 + k \ne 0:\\ \;\;\;\;\frac{k}{\frac{-1}{-10 - k}}\\ \mathbf{else}:\\ \;\;\;\;\left(10 + k\right) \cdot k\\ \end{array}\\ \mathbf{if}\;k \leq 0.000115:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{+49}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;a \ne 0:\\ \;\;\;\;\frac{1}{\frac{t_1}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{t_1}\\ \end{array}\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{+98}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 3
Error	1.2
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{+163}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 4
Error	2.5
Cost	7048

\[\begin{array}{l} t_0 := \frac{{k}^{m} \cdot a}{1}\\ \mathbf{if}\;m \leq -7 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 8.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	21.3
Cost	840

\[\begin{array}{l} \mathbf{if}\;k \leq -4.5 \cdot 10^{-64}:\\ \;\;\;\;3 - \left(3 + \frac{a \cdot -0.1}{k}\right)\\ \mathbf{elif}\;k \leq 2.55 \cdot 10^{+163}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 6
Error	16.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -380000000000:\\ \;\;\;\;3 - \left(3 + \frac{\frac{a}{k}}{-k}\right)\\ \mathbf{elif}\;m \leq 7000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;3 - \left(3 + \frac{a \cdot -0.1}{k}\right)\\ \end{array} \]

Alternative 7
Error	23.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -0.43:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + \left(-10 \cdot k\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 8
Error	23.0
Cost	712

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

Alternative 9
Error	21.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -2.85 \cdot 10^{-67}:\\ \;\;\;\;3 - \left(3 + \frac{a \cdot -0.1}{k}\right)\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + 10 \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 10
Error	38.8
Cost	584

\[\begin{array}{l} t_0 := \frac{a}{10 \cdot k}\\ \mathbf{if}\;k \leq -0.1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	24.4
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{a}{k}}{k}\\ \mathbf{if}\;k \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	23.2
Cost	584

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

Alternative 13
Error	46.4
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023033 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))