Falkner and Boettcher, Appendix A

Average Error: 2.0 → 0.1

Time: 4.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+108}:\\ \;\;\;\;a \cdot \left({k}^{m} \cdot \frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{{k}^{\left(3 - m\right)}}, -10, \frac{a}{k} \cdot \frac{{k}^{m}}{k}\right)\\ \end{array} \]

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 5e+108)
   (* a (* (pow k m) (/ 1.0 (fma k (+ k 10.0) 1.0))))
   (fma (/ a (pow k (- 3.0 m))) -10.0 (* (/ 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 <= 5e+108) {
		tmp = a * (pow(k, m) * (1.0 / fma(k, (k + 10.0), 1.0)));
	} else {
		tmp = fma((a / pow(k, (3.0 - m))), -10.0, ((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 <= 5e+108)
		tmp = Float64(a * Float64((k ^ m) * Float64(1.0 / fma(k, Float64(k + 10.0), 1.0))));
	else
		tmp = fma(Float64(a / (k ^ Float64(3.0 - m))), -10.0, 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, 5e+108], N[(a * N[(N[Power[k, m], $MachinePrecision] * N[(1.0 / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / N[Power[k, N[(3.0 - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -10.0 + N[(N[(a / k), $MachinePrecision] * N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision]), $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 < 4.99999999999999991e108

   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 0.0

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

   if 4.99999999999999991e108 < k

   1. Initial program 7.8

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

   2. Simplified 7.8

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

   3. Taylor expanded in k around inf 7.8

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

   4. Simplified 0.1

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

   5. Applied egg-rr 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+108}:\\ \;\;\;\;a \cdot \left({k}^{m} \cdot \frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{{k}^{\left(3 - m\right)}}, -10, \frac{a}{k} \cdot \frac{{k}^{m}}{k}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))