Falkner and Boettcher, Appendix A

Average Error: 2.0 → 0.1

Time: 5.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{+149}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)} \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

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 1.05e+149)
   (* (/ a (fma k 10.0 (fma k k 1.0))) (pow k m))
   (* (/ 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 <= 1.05e+149) {
		tmp = (a / fma(k, 10.0, fma(k, k, 1.0))) * pow(k, m);
	} else {
		tmp = (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 <= 1.05e+149)
		tmp = Float64(Float64(a / fma(k, 10.0, fma(k, k, 1.0))) * (k ^ m));
	else
		tmp = 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, 1.05e+149], N[(N[(a / N[(k * 10.0 + N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] * N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

1. Split input into 2 regimes

2. if k < 1.0500000000000001e149

   1. Initial program 0.1

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

   2. Simplified 0.0

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

   3. Applied egg-rr 0.1

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

   4. Taylor expanded in a around 0 19.1

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

   5. Simplified 0.0

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

   if 1.0500000000000001e149 < k

   1. Initial program 9.8

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

   2. Simplified 9.8

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

   3. Applied egg-rr 9.9

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

   4. Taylor expanded in k around -inf 64.0

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

   5. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{+149}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)} \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))