Falkner and Boettcher, Appendix A

Average Error: 2.2 → 0.1

Time: 7.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{+73}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-10}{k}\right) \cdot \left(\frac{a}{k} \cdot \frac{{k}^{m}}{k}\right)\\ \end{array} \]

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+73)
   (/ (* a (pow k m)) (fma k (+ k 10.0) 1.0))
   (* (+ 1.0 (/ -10.0 k)) (* (/ a k) (/ (pow k m) 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+73) {
		tmp = (a * pow(k, m)) / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = (1.0 + (-10.0 / k)) * ((a / k) * (pow(k, m) / 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+73)
		tmp = Float64(Float64(a * (k ^ m)) / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(Float64(1.0 + Float64(-10.0 / k)) * Float64(Float64(a / k) * Float64((k ^ m) / 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+73], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-10.0 / k), $MachinePrecision]), $MachinePrecision] * N[(N[(a / k), $MachinePrecision] * N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.99999999999999983e72

   1. Initial program 0.1

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

   2. Simplified 0.0

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

   if 9.99999999999999983e72 < k

   1. Initial program 7.3

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

   2. Simplified 7.3

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

   3. Applied egg-rr 7.4

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

   4. Applied egg-rr 7.3

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

   5. Applied egg-rr 7.5

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

   6. Taylor expanded in k around inf 7.3

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{2}} + -10 \cdot \frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{{k}^{3}}} \]

   7. Simplified 0.1

      \[\leadsto \color{blue}{\left(\frac{-10}{k} + 1\right) \cdot \left(\frac{a}{k} \cdot \frac{{k}^{m}}{k}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{+73}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-10}{k}\right) \cdot \left(\frac{a}{k} \cdot \frac{{k}^{m}}{k}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022206 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))