Falkner and Boettcher, Appendix A

Average Error: 1.9 → 0.1

Time: 5.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{+108}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k}\\ \end{array} \]

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 1e+108)
   (/ a (/ (fma k k (fma k 10.0 1.0)) (pow k m)))
   (* (/ (pow k 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 <= 1e+108) {
		tmp = a / (fma(k, k, fma(k, 10.0, 1.0)) / pow(k, m));
	} else {
		tmp = (pow(k, 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 <= 1e+108)
		tmp = Float64(a / Float64(fma(k, k, fma(k, 10.0, 1.0)) / (k ^ m)));
	else
		tmp = Float64(Float64((k ^ m) / k) * Float64(a / 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+108], N[(a / N[(N[(k * k + N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1e108

   1. Initial program 0.1

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

   2. Applied egg-rr 10.9

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{\sqrt[3]{a \cdot {k}^{m}}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}} \]

   3. Applied egg-rr 0.1

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

   if 1e108 < k

   1. Initial program 7.4

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

   2. Taylor expanded in k around inf 7.4

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

   3. Simplified 0.1

      \[\leadsto \color{blue}{\frac{{k}^{m}}{k} \cdot \frac{a}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{+108}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k}\\ \end{array} \]

Reproduce

herbie shell --seed 2022211 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))