Falkner and Boettcher, Appendix A

Average Error: 2.1 → 0.2

Time: 10.0s

Precision: binary64

Cost: 13572

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{+91}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot \frac{\frac{k}{a}}{{k}^{m}}\right)}^{-1}\\ \end{array} \]

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 1e+91)
   (/ a (/ (fma k (+ k 10.0) 1.0) (pow k m)))
   (pow (* k (/ (/ k a) (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 <= 1e+91) {
		tmp = a / (fma(k, (k + 10.0), 1.0) / pow(k, m));
	} else {
		tmp = pow((k * ((k / a) / 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 <= 1e+91)
		tmp = Float64(a / Float64(fma(k, Float64(k + 10.0), 1.0) / (k ^ m)));
	else
		tmp = Float64(k * Float64(Float64(k / a) / (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, 1e+91], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(k * N[(N[(k / a), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000008e91

   1. Initial program 0.1

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

   2. Applied egg-rr 0.2

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

   3. Applied egg-rr 0.0

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

   if 1.00000000000000008e91 < k

   1. Initial program 7.6

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

   2. Applied egg-rr 7.7

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

   3. Taylor expanded in k around inf 7.7

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

   4. Simplified 0.5

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{+91}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot \frac{\frac{k}{a}}{{k}^{m}}\right)}^{-1}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	13508

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{+91}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot \frac{\frac{k}{a}}{{k}^{m}}\right)}^{-1}\\ \end{array} \]

Alternative 2
Error	2.1
Cost	7296

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

Alternative 3
Error	2.8
Cost	6920

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -3.4 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 8 \cdot 10^{-20}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	21.6
Cost	844

\[\begin{array}{l} t_0 := -10 \cdot \left(k \cdot a\right)\\ \mathbf{if}\;k \leq -9.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq -3.5 \cdot 10^{-300}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;a + t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 5
Error	21.6
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -9.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq -3.5 \cdot 10^{-300}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 61000000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 6
Error	16.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -19500000000:\\ \;\;\;\;-1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{elif}\;m \leq 0.0036:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 7
Error	22.8
Cost	716

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -9.8 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq -3.5 \cdot 10^{-300}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 61000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	21.9
Cost	716

\[\begin{array}{l} \mathbf{if}\;k \leq -9.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq -3.5 \cdot 10^{-300}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 61000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 9
Error	17.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq -19500000000:\\ \;\;\;\;-1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{elif}\;m \leq 0.0036:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 10
Error	20.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 0.0036:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 11
Error	42.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 0.0036:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 12
Error	46.1
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022284 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))