Falkner and Boettcher, Appendix A

Average Error: 2.0 → 0.2

Time: 6.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 3.9990722708710586 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{k}{\frac{a}{k}}}{{k}^{m}}\right)}^{-1}\\ \end{array} \]

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 3.9990722708710586e+132)
   (/ (/ a (fma k (+ k 10.0) 1.0)) (pow k (- m)))
   (pow (/ (/ k (/ a k)) (pow k m)) -1.0)))

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 tmp;
	if (k <= 3.9990722708710586e+132) {
		tmp = (a / fma(k, (k + 10.0), 1.0)) / pow(k, -m);
	} else {
		tmp = pow(((k / (a / k)) / pow(k, m)), -1.0);
	}
	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)
	tmp = 0.0
	if (k <= 3.9990722708710586e+132)
		tmp = Float64(Float64(a / fma(k, Float64(k + 10.0), 1.0)) / (k ^ Float64(-m)));
	else
		tmp = Float64(Float64(k / Float64(a / k)) / (k ^ m)) ^ -1.0;
	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_] := If[LessEqual[k, 3.9990722708710586e+132], N[(N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(k / N[(a / k), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

1. Split input into 2 regimes

2. if k < 3.9990722708710586e132

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   3. Applied egg-rr 0.3

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

   4. Applied egg-rr 0.1

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

   if 3.9990722708710586e132 < k

   1. Initial program 9.4

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

   2. Simplified 9.4

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

   3. Applied egg-rr 9.4

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

   4. Taylor expanded in k around -inf 64.0

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

   5. Simplified 0.6

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.9990722708710586 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{k}{\frac{a}{k}}}{{k}^{m}}\right)}^{-1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))